Patent Document

BACKGROUND OF THE INVENTION 
       [0001]    The background of the present invention is laid out in detail in the book BICs 4 Derivatives Volume I: Theory and in PCT international application publication no. WO03107137 hereby incorporated by reference. Unless otherwise indicated, definitions, notations, chapters, sections references or other parts not internally described are by default those of the book  BICs  4  Derivatives Volume I: Theory    
     
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
         [0002]      FIG. 1 . illustrates the correspondence between implied density function and implied volatility function as described in Part A of the Detailed Description. 
           [0003]      FIG. 2 . illustrates the correspondence between implied bivariate density function and implied correlation function as described in Part A of the Detailed Description. 
           [0004]      FIG. 3 . illustrates the correspondence between implied multivariate joint density and implied correlation matrix function, implied volatility vector function as described in Part A of the Detailed Description. 
       
    
    
     DETAILED DESCRIPTION  
       [0005]    This detailed description is subdivided in two parts. Part A is titled The Options BICs Format and Distributional Linkage—An Alternative to Copulas and is essentially chapter IX of the book BICs 4 Derivatives Volume I: Theory; Part B is titled Functional Representation For BICs Pricing and is essentially chapter XI of the book BICs 4 Derivatives Volume I: Theory. The material here is intended to be read independently. 
       A. The Options BICS Format And Distributional Linkage—An Alternative To Copulas 
     1. Introduction 
       [0006]    In this chapter, we investigate a surprising theoretical application of one of the BICs formulas reviewed in chapter VI of the book BICs 4 Derivatives, Volume I, ISBN 0-9764253-0-0, the Options BIC format. Traditionally, when one has the distribution of two underlyings given independently and one needs to relate them to one another, the only available mathematical tool at one&#39;s disposal are the mathematical objects called copulas. 
         [0007]    After briefly reviewing what copulas stand for, we outline their main properties. Then we show how the Options BICs formula can be used to achieve the same more naturally. We move even further and show how we can link any given bivariate distributions to form a multivariate distribution whose bivariate marginal distributions coincide with the given bivariate distributions. Further, we are able to propose a definition of implied correlation that naturally and intuitively extends the concept of implied volatility well known by derivatives markets practitioners. 
       2. Copulas: Definition, Main Results &amp; Examples 
     2.1. Definition &amp; Main Results 
       [0008]    Definition 1. Let (X 1 , . . . , X n ) be a random vector with cumulative distribution function F(x 1 , . . . , x n )=Pr ob(X 1 ≦x 1 , . . . , X n ≦x n )and marginal functions F i (x i )=Pr ob(X i ≦x i ) 
         [0009]    A copula function of F is defined as the cumulative distribution function of a probability measure taking the value zero everywhere but on [0,1]n and verifying: 
         [0010]    (1) For every 1≦i≦n,0≦u i ≦1, C i (u i )=C(1, . . . , 1, u i , 1, . . . , 1)=u i    
         [0011]    (2) For every (x 1 , . . . , x n ) such that xi is a continuity point of Fi(xi), (1≦i≦n), F(x 1 , . . . , x n )=C(F 1 (x 1 ), . . . , F n (x n )), 
         [0012]    Copulas satisfy the following properties: 
       Theorem 1 (Sklar). 
       [0013]    Every cumulative distribution function F has at least one copula function. When all the marginal cumulative distribution functions Fi (1≦i≦n) are continuous, the copula function C of F is unique. 
       2.2. Examples of Copulas 
     Definition 2. Gaussian Copula 
       [0014]    The Gaussian Copula C ρ   Gauss  is defined as: C ρ   Gauss (u,v)=M(N −1 (u), N −1 (v), ρ) Definition.3. 
       Definition 3. Gumbel-Hougaard Copula 
       [0015]    The Gumbel-Hougaard Copula C θ   GumHou  is defined as: C θ   GumHou  (u,v)=e−(((−Log(u)) θ +(−Log(v)) θ ) 1−θ)    
         [0016]    As can be seen in the definition above, copulas are tools for consistently modeling the relationship of the distribution of individual underlyings with one another. In other words, they link together individual distributions in a way that allows them to fit together 
         [0017]    Copulas emerged in the last decade as a useful tool in financial engineering spurred by the growth of multi-underlyings derivatives contracts, and in particular credit derivatives. Credit derivatives typically, depend at least on two underlyings representing credit market risks as well as interest rate risks. 
         [0018]    In most problems of interest, information is available about the cumulative distribution function of individual underlyings, but information about the cumulative joint distribution may not be available. Hence, there are infinitely many ways to fit together any given individual distributions, creating in practical modeling applications, the following problem. 
         [0019]    Given any sequence of underlyings distribution, which copula function should one choose to give a realistic picture of the joint underlyings distribution? 
         [0020]    In traditional models for derivatives pricing, geometric Brownian motion assumptions are made to model the distribution of underlyings, but under such assumptions volatility is the key parameter necessary to infer the price of vanilla call/puts. 
         [0021]    Despite the fact that empirical evidence does not support that assumption, allowing implied volatility to be a function of the strike level allows one to recover the price yielded by any alternative model. As a result implied volatility is a market standard for quoting options prices. 
         [0022]    A natural approach—which further illustrate our philosophy of approaching normative issues from a strong positive standpoint—more inclusive of market prices structure to infer joint distributions would desirably extend rather than change the structure of these accepted features. 
         [0023]    In such a way, a concept of imply correlation would provide a better standard to rely on than the current alternatives developed via a copula analysis, the “compound correlation” or even the improvement provided by “base correlations.” 
         [0024]    The BICs decomposition in the options format provides a natural method for reaching such a goal. As an alternative to copulas its salient benefits are as follows: 
         [0025]    (1) Underlyings marginal distribution information is provided directly by options prices or implied volatilities rather than after processing to obtain joint distribution functions. 
         [0026]    (2) Joint distribution information relationships can be provided in the form of implied correlation functions that naturally extend the concept of implied volatility functions. In such a manner joint distribution information is provided immediately in a format that makes it suitable for pricing of derivatives depending on multiple underlyings, without the need for further processing. 
         [0027]    (3) Derivatives prices may display a more meaningful sensitivity to implied correlation than the alternative Gaussian Copula. Such sensitivity may be more comparable to the sensitivity of options prices to volatility. 
         [0028]    This definition of implied correlation is indeed the prime candidate to solve the correlation conundrum created by the use of alternative copula functions. 
         [0029]    How all this works is the subject of the next section. 
       3. Options BICs And Distributional Linkage 
     3.1. Option BICs Payout Payment Amount Format And Properties 
       [0030]    Let us recall that the generic payout payment amount of such contracts at a given payout payment time T is: (δ 1 (S T   1 −K 1 )) +  . . . (δ n (S T   n −K n ) +   
         [0031]    where δ 1 , . . . , δ n ∈{−1,1}; K 1 , . . . , K n  ∈□ +   
       Assumption 
       [0032]    We assume the underlyings S T   1 , . . . , S T   n  are driven by a geometric brownian motion, i.e. 
         [0000]    
       
         
           
             
               
                 
                   dS 
                   t 
                   1 
                 
                 
                   S 
                   t 
                   1 
                 
               
               = 
               
                 
                   
                     ( 
                     
                       
                         r 
                         t 
                         d 
                       
                       - 
                       
                         r 
                         t 
                         
                           f 
                           1 
                         
                       
                     
                     ) 
                   
                    
                   dt 
                 
                 + 
                 
                   
                     σ 
                     t 
                     1 
                   
                    
                   
                     dW 
                     t 
                     1 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 we 
                  
                 
                     
                 
                  
                 note 
                  
                 
                     
                 
                  
                 
                   F 
                   T 
                   1 
                 
               
               = 
               
                 
                   S 
                   0 
                   1 
                 
                  
                 
                    
                   
                     
                       ∫ 
                       0 
                       T 
                     
                      
                     
                       
                         ( 
                         
                           
                             r 
                             t 
                             d 
                           
                           - 
                           
                             r 
                             t 
                             
                               f 
                               1 
                             
                           
                         
                         ) 
                       
                        
                       
                           
                       
                        
                       
                          
                         t 
                       
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               σ 
               1 
             
             = 
             
               
                 
                   1 
                   T 
                 
                  
                 
                   
                     ∫ 
                     0 
                     T 
                   
                    
                   
                     
                       
                         ( 
                         
                           σ 
                           t 
                           1 
                         
                         ) 
                       
                       2 
                     
                      
                     
                         
                     
                      
                     
                        
                       t 
                     
                   
                 
               
             
           
         
       
       
         
           ⋮ 
         
       
       
         
           
             
               
                 
                   dS 
                   t 
                   n 
                 
                 
                   S 
                   t 
                   n 
                 
               
               = 
               
                 
                   
                     ( 
                     
                       
                         r 
                         t 
                         d 
                       
                       - 
                       
                         r 
                         t 
                         
                           f 
                           n 
                         
                       
                     
                     ) 
                   
                    
                   dt 
                 
                 + 
                 
                   
                     σ 
                     t 
                     n 
                   
                    
                   
                     dW 
                     t 
                     n 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 we 
                  
                 
                     
                 
                  
                 note 
                  
                 
                     
                 
                  
                 
                   F 
                   T 
                   n 
                 
               
               = 
               
                 
                   S 
                   0 
                   1 
                 
                  
                 
                    
                   
                     
                       ∫ 
                       0 
                       T 
                     
                      
                     
                       
                         ( 
                         
                           
                             r 
                             t 
                             d 
                           
                           - 
                           
                             r 
                             t 
                             
                               f 
                               n 
                             
                           
                         
                         ) 
                       
                        
                       
                           
                       
                        
                       
                          
                         t 
                       
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               σ 
               n 
             
             = 
             
               
                 
                   1 
                   T 
                 
                  
                 
                   
                     ∫ 
                     0 
                     T 
                   
                    
                   
                     
                       
                         ( 
                         
                           σ 
                           t 
                           n 
                         
                         ) 
                       
                       2 
                     
                      
                     
                         
                     
                      
                     
                        
                       t 
                     
                   
                 
               
             
           
         
       
     
         [0000]    where r t   d  is the instantaneous domestic risk free deposit rate and r t   f     i    is the instantaneous repo/dividend/foreign rate of underlying i. 
         [0000]    
       
         
           
             
               
                 For 
                  
                 
                     
                 
                  
                 1 
               
               ≤ 
               i 
               ≤ 
               n 
             
             , 
             
               
 
             
              
             
               
                 W 
                 t 
                 i 
               
               = 
               
                 
                   ∑ 
                   
                     j 
                     = 
                     1 
                   
                   i 
                 
                  
                 
                   
                     ρ 
                     t 
                     
                       i 
                       , 
                       j 
                     
                   
                    
                   
                     B 
                     t 
                     j 
                   
                 
               
             
           
         
       
       
         
           with 
         
       
       
         
           
             
               ρ 
               t 
               
                 1 
                 , 
                 1 
               
             
             = 
             1 
           
         
       
     
         [0000]    and for 
         [0000]    
       
         
           
             1 
             &lt; 
             i 
             ≤ 
             m 
           
         
       
       
         
           
             
               ρ 
               t 
               
                 i 
                 , 
                 i 
               
             
             = 
             
               
                 1 
                 - 
                 
                   
                     ∑ 
                     
                       j 
                       = 
                       1 
                     
                     
                       i 
                       - 
                       1 
                     
                   
                    
                   
                     
                       ( 
                       
                         ρ 
                         t 
                         
                           i 
                           , 
                           j 
                         
                       
                       ) 
                     
                     2 
                   
                 
               
             
           
         
       
     
         [0000]    where the (B t   j ) 1≦j≦m  are i.i.d normal variables with mean 0 and variance t. 
         [0033]    For 
         [0000]    
       
         
           
             
               1 
               ≤ 
               j 
               ≤ 
               i 
               ≤ 
               n 
             
             , 
             
               
                 
                    
                   
                     ρ 
                     ij 
                   
                    
                 
                 ≤ 
                 
                   1 
                    
                   
                       
                   
                    
                   
                     S 
                     t 
                     i 
                   
                 
               
               = 
               
                 
                   F 
                   t 
                   i 
                 
                  
                 
                    
                   
                     
                       - 
                       
                         1 
                         2 
                       
                     
                      
                     
                       
                         t 
                          
                         
                           ( 
                           
                             σ 
                             t 
                             i 
                           
                           ) 
                         
                       
                       2 
                     
                   
                 
                  
                 
                   
                      
                     
                       
                         σ 
                         t 
                         i 
                       
                        
                       
                         t 
                       
                        
                       
                         ( 
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               1 
                             
                             i 
                           
                            
                           
                             
                               ρ 
                               t 
                               
                                 i 
                                 , 
                                 j 
                               
                             
                              
                             
                               x 
                               j 
                             
                           
                         
                         ) 
                       
                     
                   
                   . 
                 
               
             
           
         
       
     
       Hence if 
       [0034]    
       
         
           
             
               ρ 
               _ 
             
             = 
             
               
                 ( 
                 
                   
                     
                       ρ 
                       _ 
                     
                     ij 
                   
                   = 
                   
                     
                       E 
                        
                       
                         ( 
                         
                           
                             W 
                             t 
                             i 
                           
                           , 
                           
                             W 
                             t 
                             i 
                           
                         
                         ) 
                       
                     
                     
                       
                         
                           Var 
                            
                           
                             ( 
                             
                               W 
                               t 
                               i 
                             
                             ) 
                           
                         
                          
                         
                           Var 
                            
                           
                             ( 
                             
                               W 
                               t 
                               j 
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 ) 
               
               
                 
                   1 
                   ≤ 
                   j 
                 
                 , 
                 
                   i 
                   ≤ 
                   n 
                 
               
             
           
         
       
     
         [0000]    is the matrix of correlations betwen the (W t   i , W t   i ) 1≦j, i≦n ,  ρ = t ρρ with ρ=(ρ ij   1   {j≦i} ) 1≦j, i≦n . 
         [0035]    So it is important to bear in mind that  ρ  and ρ are two related but distinct objects. 
         [0036]    We provide here a “close form” solution that extends the well-known Black-Scholes formula in the case of a single risky asset following a geometric Brownian motion. In order to do so we first provide the definition of extended cumulative multivariate normal distributions that we will use here. Indeed, beyond n=2, there is no definition of the cumulative multivariate normal distribution that explicitly describes the functional relationship with the correlation coefficients. Hence, we propose here a clarifying definition that will not be simply based on the inverse of the correlation matrix and that is more suited for our purposes. 
       Definition.4 
       [0037]    For δ 1 , . . . , δ n  ∈{−1, 1}; x 1 , . . . , x n  ∈□; L is a lower triangular and invertible matrix, we define: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       N 
                       n 
                     
                     ( 
                     
                       
                         x 
                         1 
                       
                       , 
                       … 
                        
                       
                           
                       
                       , 
                       
                         
                           x 
                           n 
                         
                         ; 
                         
                           δ 
                           1 
                         
                       
                       , 
                       … 
                        
                       
                           
                       
                       , 
                       
                         
                           δ 
                           n 
                         
                         ; 
                         L 
                       
                     
                     ) 
                   
                   = 
                   
                     
                       
                         
                           
                             ∏ 
                             
                               k 
                               = 
                               1 
                             
                             n 
                           
                            
                           
                               
                           
                            
                           
                             L 
                             kk 
                           
                         
                         
                           
                             ( 
                             
                               2 
                                
                               
                                   
                               
                                
                               π 
                             
                             ) 
                           
                           
                             n 
                             / 
                             2 
                           
                         
                       
                        
                       
                         
                           ∫ 
                           
                             x 
                             1 
                           
                           
                             
                               δ 
                               1 
                             
                              
                             ∞ 
                           
                         
                          
                         
                           … 
                            
                           
                               
                           
                            
                           
                             
                               ∫ 
                               
                                 x 
                                 n 
                               
                               
                                 
                                   δ 
                                   n 
                                 
                                  
                                 ∞ 
                               
                             
                              
                             
                               
                                 Exp 
                                  
                                 
                                   ( 
                                   
                                     
                                       - 
                                       
                                         
                                           1 
                                           t 
                                         
                                         2 
                                       
                                     
                                      
                                     
                                       ( 
                                       LT 
                                       ) 
                                     
                                      
                                     
                                       ( 
                                       LT 
                                       ) 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                  
                                 
                                   t 
                                   n 
                                 
                               
                                
                               
                                   
                               
                                
                               … 
                                
                               
                                   
                               
                                
                               
                                  
                                 
                                   t 
                                   1 
                                 
                               
                             
                           
                         
                       
                     
                     = 
                     
                       
                         
                           
                             ∏ 
                             
                               i 
                               = 
                               1 
                             
                             n 
                           
                            
                           
                               
                           
                            
                           
                             L 
                             ii 
                           
                         
                         
                           
                             ( 
                             
                               2 
                                
                               
                                   
                               
                                
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                             ) 
                           
                           
                             n 
                             / 
                             2 
                           
                         
                       
                        
                       
                         
                           ∫ 
                           
                             x 
                             1 
                           
                           
                             
                               δ 
                               1 
                             
                              
                             ∞ 
                           
                         
                          
                         
                           … 
                            
                           
                               
                           
                            
                           
                             
                               ∫ 
                               
                                 x 
                                 n 
                               
                               
                                 
                                   δ 
                                   n 
                                 
                                  
                                 ∞ 
                               
                             
                              
                             
                               
                                 Exp 
                                  
                                 
                                   ( 
                                   
                                     
                                       - 
                                       
                                         1 
                                         2 
                                       
                                     
                                      
                                     
                                       
                                         ∑ 
                                         
                                           i 
                                           = 
                                           1 
                                         
                                         n 
                                       
                                        
                                       
                                         
                                           ( 
                                           
                                             
                                               ∑ 
                                               
                                                 j 
                                                 = 
                                                 1 
                                               
                                               i 
                                             
                                              
                                             
                                               
                                                 L 
                                                 ij 
                                               
                                                
                                               
                                                 t 
                                                 j 
                                               
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                  
                                 
                                   t 
                                   n 
                                 
                               
                                
                               
                                   
                               
                                
                               … 
                                
                               
                                   
                               
                                
                               
                                  
                                 
                                   t 
                                   1 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   i 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           N 
                           n 
                         
                          
                         
                           ( 
                           
                             
                               x 
                               1 
                             
                             , 
                             … 
                              
                             
                                 
                             
                             , 
                             
                               
                                 x 
                                 n 
                               
                               ; 
                               L 
                             
                           
                           ) 
                         
                       
                       = 
                       
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                         n 
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           
                             N 
                             n 
                           
                            
                           
                             ( 
                             
                               
                                 x 
                                 1 
                               
                               , 
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                                
                               
                                   
                               
                               , 
                               
                                 
                                   x 
                                   n 
                                 
                                 ; 
                                 
                                   - 
                                   1 
                                 
                               
                               , 
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                                
                               
                                   
                               
                               , 
                               
                                 
                                   - 
                                   1 
                                 
                                 ; 
                                 L 
                               
                             
                             ) 
                           
                         
                         . 
                         
                             
                         
                          
                         
                             
                         
                          
                         In 
                       
                        
                       
                           
                       
                        
                       particular 
                        
                       
                           
                       
                        
                       we 
                        
                       
                           
                       
                        
                       define 
                        
                       
                         : 
                       
                     
                   
                 
               
               
                 
                   ( 
                   ii 
                   ) 
                 
               
             
             
               
                 
                   
                     M 
                      
                     
                       ( 
                       
                         
                           x 
                           1 
                         
                         , 
                         
                           x 
                           2 
                         
                         , 
                         ρ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           N 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               x 
                               1 
                             
                             , 
                             
                               
                                 x 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       1 
                                     
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       
                                         - 
                                         
                                           ρ 
                                           
                                             
                                               1 
                                               - 
                                               
                                                 ρ 
                                                 2 
                                               
                                             
                                           
                                         
                                       
                                     
                                     
                                       
                                         1 
                                         
                                           
                                             1 
                                             - 
                                             
                                               ρ 
                                               2 
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                        
                       
                           
                       
                        
                       and 
                        
                       
                           
                       
                        
                       
                         N 
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                     
                     = 
                     
                       
                         1 
                         
                           
                             2 
                              
                             
                                 
                             
                              
                             π 
                           
                         
                       
                        
                       
                         
                           ∫ 
                           
                             - 
                             ∞ 
                           
                           x 
                         
                          
                         
                           
                              
                             
                               - 
                               
                                 
                                   t 
                                   2 
                                 
                                 2 
                               
                             
                           
                            
                           
                               
                           
                            
                           
                              
                             t 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   iii 
                   ) 
                 
               
             
           
         
       
     
       Lemma.1 
       [0038]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       For 
                        
                       
                         
                             
                         
                          
                         
                             
                         
                       
                        
                       α 
                     
                     , 
                     a 
                     , 
                     
                       b 
                       ∈ 
                       • 
                     
                     , 
                     
                       δ 
                       ∈ 
                       
                         { 
                         
                           
                             - 
                             1 
                           
                           , 
                           1 
                         
                         } 
                       
                     
                     , 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       M 
                        
                       
                         ( 
                         
                           a 
                           , 
                           b 
                           , 
                           ρ 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         N 
                          
                         
                           ( 
                           b 
                           ) 
                         
                       
                       - 
                       
                         M 
                          
                         
                           ( 
                           
                             
                               - 
                               a 
                             
                             , 
                             b 
                             , 
                             
                               - 
                               ρ 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   i 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         1 
                         
                           
                             2 
                              
                             
                                 
                             
                              
                             π 
                           
                         
                       
                        
                       
                         
                           ∫ 
                           α 
                           
                             δ 
                              
                             
                                 
                             
                              
                             ∞ 
                           
                         
                          
                         
                           
                              
                             
                               - 
                               
                                 
                                   x 
                                   2 
                                 
                                 2 
                               
                             
                           
                            
                           
                               
                           
                            
                           
                              
                             x 
                           
                         
                       
                     
                     = 
                     
                       δ 
                        
                       
                           
                       
                        
                       
                         N 
                          
                         
                           ( 
                           
                             
                               - 
                               δ 
                             
                              
                             
                                 
                             
                              
                             α 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   ii 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         1 
                         
                           
                             2 
                              
                             
                                 
                             
                              
                             π 
                           
                         
                       
                        
                       
                         
                           ∫ 
                           α 
                           
                             δ 
                              
                             
                                 
                             
                              
                             ∞ 
                           
                         
                          
                         
                           
                              
                             
                               - 
                               
                                 
                                   x 
                                   2 
                                 
                                 2 
                               
                             
                           
                            
                           
                             
                               N 
                               1 
                             
                              
                             
                               ( 
                               
                                 ax 
                                 + 
                                 b 
                               
                               ) 
                             
                           
                            
                           
                               
                           
                            
                           
                              
                             x 
                           
                         
                       
                     
                     = 
                     
                       δ 
                        
                       
                           
                       
                        
                       
                         M 
                          
                         
                           ( 
                           
                             
                               
                                 - 
                                 δ 
                               
                                
                               
                                   
                               
                                
                               α 
                             
                             , 
                             
                               b 
                               
                                 
                                   1 
                                   + 
                                   
                                     a 
                                     2 
                                   
                                 
                               
                             
                             , 
                             
                               
                                 δ 
                                  
                                 
                                     
                                 
                                  
                                 a 
                               
                               
                                 
                                   1 
                                   + 
                                   
                                     a 
                                     2 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   iii 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         N 
                         2 
                       
                        
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                           , 
                           
                             δ 
                             1 
                           
                           , 
                           
                             δ 
                             2 
                           
                           , 
                           
                             ( 
                             
                               
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   
                                     ρ 
                                     
                                       
                                         1 
                                         - 
                                         
                                           ρ 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         δ 
                         1 
                       
                        
                       
                         δ 
                         2 
                       
                        
                       
                         M 
                          
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     
                                       - 
                                       
                                         δ 
                                         1 
                                       
                                     
                                      
                                     
                                       x 
                                       1 
                                     
                                   
                                   , 
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       - 
                                       
                                         δ 
                                         2 
                                       
                                     
                                      
                                     
                                       
                                         1 
                                         - 
                                         
                                           ρ 
                                           2 
                                         
                                       
                                     
                                      
                                     
                                       x 
                                       2 
                                     
                                   
                                   , 
                                 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     
                                       δ 
                                       1 
                                     
                                   
                                    
                                   
                                     δ 
                                     2 
                                   
                                    
                                   ρ 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   iv 
                   ) 
                 
               
             
           
         
       
     
       Theorem 2 
       [0039]    
       
         
           
             
               
                 ∏ 
                 0 
                 BS 
               
                
               
                   
               
                
               
                 [ 
                 
                   
                     
                       ( 
                       
                         
                           δ 
                           1 
                         
                          
                         
                           ( 
                           
                             
                               S 
                               T 
                               1 
                             
                             - 
                             
                               K 
                               1 
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     + 
                   
                    
                   
                       
                   
                    
                   … 
                    
                   
                       
                   
                    
                   
                     
                       ( 
                       
                         
                           δ 
                           n 
                         
                          
                         
                           ( 
                           
                             
                               S 
                               T 
                               n 
                             
                             - 
                             
                               K 
                               n 
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     + 
                   
                 
                 ] 
               
             
             = 
             
               
                 
                   C 
                   0 
                   
                     BS 
                     , 
                     n 
                   
                 
                  
                 
                   ( 
                   
                     
                       ( 
                       
                         
                           δ 
                           1 
                         
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           δ 
                           n 
                         
                       
                       ) 
                     
                     , 
                     
                       ( 
                       
                         
                           K 
                           1 
                         
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           K 
                           n 
                         
                       
                       ) 
                     
                     , 
                     
                       ( 
                       
                         
                           F 
                           T 
                           1 
                         
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           F 
                           T 
                           n 
                         
                       
                       ) 
                     
                     , 
                     
                       ( 
                       
                         
                           σ 
                           1 
                         
                         , 
                         
                           … 
                            
                           
                               
                           
                            
                           
                             σ 
                             n 
                           
                         
                       
                       ) 
                     
                     , 
                     
                       
                         ( 
                         
                           ρ 
                           T 
                           
                             i 
                             , 
                             j 
                           
                         
                         ) 
                       
                       
                         j 
                         &lt; 
                         i 
                       
                     
                     , 
                     T 
                   
                   ) 
                 
               
               = 
               
                 
                   B 
                   T 
                   0 
                 
                  
                 
                   δ 
                   1 
                 
                  
                 
                     
                 
                  
                 … 
                  
                 
                     
                 
                  
                 
                   δ 
                   n 
                 
                 × 
                 
                   
                     ∑ 
                     
                       k 
                       = 
                       0 
                     
                     n 
                   
                    
                   
                     
                       
                         ( 
                         
                           - 
                           1 
                         
                         ) 
                       
                       k 
                     
                      
                     
                       
                         ∑ 
                         
                           
                             
                               1 
                               ≤ 
                               
                                 i 
                                 1 
                               
                               &lt; 
                               … 
                               &lt; 
                               
                                 i 
                                 k 
                               
                               ≤ 
                               
                                 
                                   n 
                                   / 
                                   k 
                                 
                                 + 
                                 p 
                               
                             
                             = 
                             n 
                           
                           
                             
                               { 
                               
                                 
                                   j 
                                   1 
                                 
                                 , 
                                 … 
                                  
                                 
                                     
                                 
                                 , 
                                 
                                   j 
                                   p 
                                 
                               
                               } 
                             
                             = 
                             
                               
                                 { 
                                 
                                   1 
                                   , 
                                   … 
                                    
                                   
                                       
                                   
                                   , 
                                   n 
                                 
                                 } 
                               
                                
                               \ 
                                
                               
                                 { 
                                 
                                   
                                     i 
                                     1 
                                   
                                   , 
                                   … 
                                    
                                   
                                       
                                   
                                   , 
                                   
                                     i 
                                     k 
                                   
                                 
                                 } 
                               
                             
                           
                         
                         
                             
                         
                       
                        
                       
                         
                           K 
                           
                             i 
                             1 
                           
                         
                          
                         
                             
                         
                          
                         … 
                          
                         
                             
                         
                          
                         
                           K 
                           
                             i 
                             k 
                           
                         
                          
                         
                           F 
                           T 
                           
                             j 
                             1 
                           
                         
                          
                         
                             
                         
                          
                         … 
                          
                         
                             
                         
                          
                         
                           F 
                           T 
                           
                             j 
                             p 
                           
                         
                          
                         
                            
                           
                             
                               - 
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         q 
                                         = 
                                         1 
                                       
                                       p 
                                     
                                      
                                     
                                       
                                         ( 
                                         
                                           σ 
                                           
                                             j 
                                             q 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                   - 
                                   
                                     
                                       ∑ 
                                       
                                         k 
                                         = 
                                         1 
                                       
                                       n 
                                     
                                      
                                     
                                       
                                         ( 
                                         
                                           
                                             ∑ 
                                             
                                               q 
                                               = 
                                               1 
                                             
                                             p 
                                           
                                            
                                           
                                             
                                               σ 
                                               
                                                 j 
                                                 q 
                                               
                                             
                                              
                                             
                                               ρ 
                                               T 
                                               
                                                 
                                                   j 
                                                   q 
                                                 
                                                 , 
                                                 k 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                                 2 
                               
                             
                              
                             T 
                           
                         
                         × 
                         
                           
                             N 
                             n 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   κ 
                                   1 
                                 
                                 - 
                                 
                                   
                                     ∑ 
                                     
                                       q 
                                       = 
                                       1 
                                     
                                     p 
                                   
                                    
                                   
                                     
                                       σ 
                                       
                                         j 
                                         q 
                                       
                                     
                                      
                                     
                                       ρ 
                                       t 
                                       
                                         
                                           j 
                                           q 
                                         
                                         , 
                                         1 
                                       
                                     
                                      
                                     
                                       T 
                                     
                                   
                                 
                               
                               , 
                               … 
                                
                               
                                   
                               
                               , 
                               
                                 
                                   
                                     κ 
                                     n 
                                   
                                   - 
                                   
                                     
                                       ∑ 
                                       
                                         q 
                                         = 
                                         1 
                                       
                                       p 
                                     
                                      
                                     
                                       
                                         σ 
                                         
                                           j 
                                           q 
                                         
                                       
                                        
                                       
                                         ρ 
                                         t 
                                         
                                           
                                             j 
                                             q 
                                           
                                           , 
                                           n 
                                         
                                       
                                        
                                       
                                         T 
                                       
                                     
                                   
                                 
                                 ; 
                                 
                                   δ 
                                   1 
                                 
                               
                               , 
                               … 
                                
                               
                                   
                               
                               , 
                               
                                 
                                   δ 
                                   n 
                                 
                                 ; 
                                 L 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   with 
                    
                   
                       
                   
                    
                   
                     L 
                     
                       i 
                       , 
                       i 
                     
                   
                 
                 = 
                 1 
               
               , 
               
                 
 
               
                
               
                   
               
                
               
                 
                   and 
                    
                   
                       
                   
                    
                   for 
                    
                   
                       
                   
                    
                   j 
                 
                 &lt; 
                 i 
               
               , 
               
                 
 
               
                
               
                   
               
                
               
                 
                   
                     L 
                     
                       i 
                       , 
                       j 
                     
                   
                   = 
                   
                     
                       ρ 
                       T 
                       
                         i 
                         , 
                         j 
                       
                     
                     
                       
                         1 
                         - 
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               1 
                             
                             
                               i 
                               - 
                               1 
                             
                           
                            
                           
                             
                               ( 
                               
                                 ρ 
                                 T 
                                 
                                   i 
                                   , 
                                   j 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
                 ; 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   κ 
                   i 
                 
                 = 
                 
                   
                     1 
                     
                       
                         σ 
                         T 
                         i 
                       
                        
                       
                         T 
                       
                     
                   
                    
                   
                     ( 
                     
                       
                         Log 
                          
                         
                           ( 
                           
                             
                               K 
                               i 
                             
                             
                               F 
                               T 
                               i 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           1 
                           2 
                         
                          
                         
                           
                             T 
                              
                             
                               ( 
                               
                                 σ 
                                 T 
                                 i 
                               
                               ) 
                             
                           
                           2 
                         
                       
                     
                     ) 
                   
                 
               
               ; 
             
           
         
       
     
       3.1.1. Computational Considerations 
       [0040]    It is important to note that for any given n, the number of terms in the sum yielding the price with cumulative multivariate normal terms is 2 n , an exponential growth that renders the practical use of the formula above impossible beyond a few values of n. Further compounding the complexity is the non trivial task of computing the values of the cumulative multivariate normal functions. 
         [0041]    The description of the cumulative multivariate normal function as a function of L may facilitate the numerical approximation of these functions via limited development as a function of the values of diagonal matrices and in particular, the identity matrix as is more useful in the case at hand where the diagonal terms of L are all equal to one. 
         [0042]    However, the formula lends itself to various qualitative analysis. A particular critical one is the generation of multivariate distributions from their univariate marginal and bivariate marginal counterparts. We will see here that this represents a very effective and less cumbersome alternative to the copula approach. 
         [0043]    Corollary 1: If n=2, 
         [0000]    
       
         
           
             
               
                 C 
                 0 
                 BS 
               
                
               
                 ( 
                 
                   
                     
                       
                         
                           δ 
                           1 
                         
                         , 
                         
                           δ 
                           2 
                         
                         , 
                         
                           K 
                           1 
                         
                         , 
                         
                           K 
                           2 
                         
                         , 
                         
                           σ 
                           1 
                         
                         , 
                         
                           σ 
                           2 
                         
                         , 
                       
                     
                   
                   
                     
                       
                         ρ 
                         , 
                         T 
                         , 
                         
                           B 
                           T 
                           0 
                         
                         , 
                         
                           F 
                           T 
                           1 
                         
                         , 
                         
                           F 
                           T 
                           2 
                         
                       
                     
                   
                 
                 ) 
               
             
             = 
             
               
                 
                   ∏ 
                   0 
                   BS 
                 
                  
                 
                     
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 δ 
                                 1 
                               
                                
                               
                                 ( 
                                 
                                   
                                     S 
                                     T 
                                     1 
                                   
                                   - 
                                   
                                     K 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                             ) 
                           
                           + 
                         
                       
                     
                     
                       
                         
                           
                             ( 
                             
                               
                                 δ 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   
                                     S 
                                     T 
                                     2 
                                   
                                   - 
                                   
                                     K 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             ) 
                           
                           + 
                         
                       
                     
                   
                   ] 
                 
               
               = 
               
                 
                   δ 
                   1 
                 
                  
                 
                   δ 
                   2 
                 
                  
                 
                   
                     B 
                     T 
                     0 
                   
                    
                   
                     
 
                   
                   ( 
                   
                     
                       
                         
                           
                             F 
                             T 
                             1 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       F 
                                       T 
                                       2 
                                     
                                      
                                     
                                        
                                       
                                         ρ 
                                          
                                         
                                             
                                         
                                          
                                         
                                           σ 
                                           1 
                                         
                                          
                                         
                                           σ 
                                           2 
                                         
                                          
                                         T 
                                       
                                     
                                      
                                     
                                       M 
                                       ( 
                                       
                                         
                                           - 
                                           
                                             
                                               δ 
                                               1 
                                             
                                              
                                             
                                               ( 
                                               
                                                 
                                                   κ 
                                                   1 
                                                 
                                                 - 
                                                 
                                                   
                                                     σ 
                                                     2 
                                                   
                                                    
                                                   ρ 
                                                    
                                                   
                                                     T 
                                                   
                                                 
                                                 - 
                                                 
                                                   
                                                     σ 
                                                     1 
                                                   
                                                    
                                                   
                                                     T 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                         , 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         
                                           - 
                                           
                                             δ 
                                             2 
                                           
                                         
                                          
                                         
                                           
                                             1 
                                             - 
                                             
                                               ρ 
                                               2 
                                             
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               κ 
                                               2 
                                             
                                             - 
                                             
                                               
                                                 σ 
                                                 2 
                                               
                                                
                                               
                                                 
                                                   T 
                                                    
                                                   
                                                     ( 
                                                     
                                                       1 
                                                       - 
                                                       
                                                         ρ 
                                                         2 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       , 
                                       
                                         
                                           - 
                                           
                                             δ 
                                             1 
                                           
                                         
                                          
                                         
                                           δ 
                                           2 
                                         
                                          
                                         ρ 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       
                                         K 
                                         2 
                                       
                                     
                                      
                                     
                                       M 
                                        
                                       
                                         ( 
                                         
                                           
                                             - 
                                             
                                               
                                                 δ 
                                                 1 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     κ 
                                                     1 
                                                   
                                                   - 
                                                   
                                                     
                                                       σ 
                                                       1 
                                                     
                                                      
                                                     
                                                       T 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           , 
                                           
                                             
                                               - 
                                               
                                                 δ 
                                                 2 
                                               
                                             
                                              
                                             
                                               κ 
                                               2 
                                             
                                              
                                             
                                               
                                                 1 
                                                 - 
                                                 
                                                   ρ 
                                                   2 
                                                 
                                               
                                             
                                           
                                           , 
                                           
                                             
                                               - 
                                               
                                                 δ 
                                                 1 
                                               
                                             
                                              
                                             
                                               δ 
                                               2 
                                             
                                              
                                             ρ 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           - 
                           
                             
                               K 
                               1 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         F 
                                         T 
                                         2 
                                       
                                        
                                       
                                         M 
                                         ( 
                                         
                                           
                                             - 
                                             
                                               
                                                 δ 
                                                 1 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     κ 
                                                     1 
                                                   
                                                   - 
                                                   
                                                     
                                                       σ 
                                                       2 
                                                     
                                                      
                                                     ρ 
                                                      
                                                     
                                                       T 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           , 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         
                                           - 
                                           
                                             δ 
                                             2 
                                           
                                         
                                          
                                         
                                           
                                             1 
                                             - 
                                             
                                               ρ 
                                               2 
                                             
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               κ 
                                               2 
                                             
                                             - 
                                             
                                               
                                                 σ 
                                                 2 
                                               
                                                
                                               
                                                 
                                                   T 
                                                    
                                                   
                                                     ( 
                                                     
                                                       1 
                                                       - 
                                                       
                                                         ρ 
                                                         2 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       , 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         
                                           - 
                                           
                                             δ 
                                             1 
                                           
                                         
                                          
                                         
                                           δ 
                                           2 
                                         
                                          
                                         ρ 
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         - 
                                         
                                           K 
                                           2 
                                         
                                       
                                        
                                       
                                         M 
                                          
                                         
                                           ( 
                                           
                                             
                                               
                                                 - 
                                                 
                                                   δ 
                                                   1 
                                                 
                                               
                                                
                                               
                                                 κ 
                                                 1 
                                               
                                             
                                             , 
                                             
                                               
                                                 - 
                                                 
                                                   δ 
                                                   2 
                                                 
                                               
                                                
                                               
                                                 κ 
                                                 2 
                                               
                                                
                                               
                                                 
                                                   1 
                                                   - 
                                                   
                                                     ρ 
                                                     2 
                                                   
                                                 
                                               
                                             
                                             , 
                                             
                                               
                                                 - 
                                                 
                                                   δ 
                                                   1 
                                                 
                                               
                                                
                                               
                                                 δ 
                                                 2 
                                               
                                                
                                               ρ 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             with 
              
             
               : 
             
           
         
       
       
         
           
             
               
                 κ 
                 1 
               
               = 
               
                 
                   1 
                   
                     
                       σ 
                       1 
                     
                      
                     
                       T 
                     
                   
                 
                  
                 
                   ( 
                   
                     
                       Log 
                        
                       
                         ( 
                         
                           
                             K 
                             1 
                           
                           
                             F 
                             T 
                             1 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         1 
                         2 
                       
                        
                       
                         σ 
                         1 
                         2 
                       
                        
                       T 
                     
                   
                   ) 
                 
               
             
             ; 
           
         
       
       
         
           
             
               κ 
               2 
             
             = 
             
               
                 1 
                 
                   
                     σ 
                     2 
                   
                    
                   
                     T 
                   
                 
               
                
               
                 ( 
                 
                   
                     Log 
                      
                     
                       ( 
                       
                         
                           K 
                           2 
                         
                         
                           F 
                           T 
                           2 
                         
                       
                       ) 
                     
                   
                   + 
                   
                     
                       1 
                       2 
                     
                      
                     
                       σ 
                       2 
                       2 
                     
                      
                     T 
                   
                 
                 ) 
               
             
           
         
       
     
         [0000]    with: K 1 =K 1  and K 2 =K 2  and are used interchangeably. 
       3.2. The Implied Correlation Function 
     3.2.1. Sensitivity of Second Order BICs With Respect To ρ 
     Definition &amp; Proposition. (i) 
       [0044]    
       
         
           
             
               
                 
                   
                     
                       
                         M 
                         1 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           y 
                           , 
                           z 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           ∂ 
                           M 
                         
                         
                           ∂ 
                           x 
                         
                       
                       = 
                       
                         
                           
                              
                             
                               - 
                               
                                 
                                   x 
                                   2 
                                 
                                 2 
                               
                             
                           
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               π 
                             
                           
                         
                          
                         
                           N 
                            
                           
                             ( 
                             
                               
                                 y 
                                 - 
                                 zx 
                               
                               
                                 
                                   1 
                                   - 
                                   
                                     z 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   i 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         M 
                         2 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           y 
                           , 
                           z 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           ∂ 
                           M 
                         
                         
                           ∂ 
                           y 
                         
                       
                       = 
                       
                         
                           
                             
                                
                               
                                 - 
                                 
                                   
                                     y 
                                     2 
                                   
                                   2 
                                 
                               
                             
                             
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 π 
                               
                             
                           
                            
                           
                             N 
                              
                             
                               ( 
                               
                                 
                                   x 
                                   - 
                                   zy 
                                 
                                 
                                   
                                     1 
                                     - 
                                     
                                       z 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                         = 
                         
                           
                             M 
                             1 
                           
                            
                           
                             ( 
                             
                               y 
                               , 
                               x 
                               , 
                               z 
                             
                             ) 
                           
                         
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   ii 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             M 
                             3 
                           
                            
                           
                             ( 
                             
                               x 
                               , 
                               y 
                               , 
                               z 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             ∂ 
                             
                               ∂ 
                               z 
                             
                           
                            
                           
                             M 
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 y 
                                 , 
                                 z 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             1 
                             
                               2 
                                
                               
                                   
                               
                                
                               π 
                                
                               
                                 
                                   1 
                                   - 
                                   
                                     z 
                                     2 
                                   
                                 
                               
                             
                           
                            
                           
                              
                             
                               - 
                               
                                 
                                   
                                     x 
                                     2 
                                   
                                   - 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     xyz 
                                   
                                   + 
                                   
                                     y 
                                     2 
                                   
                                 
                                 
                                   2 
                                    
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       
                                         z 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             ∂ 
                             
                               ∂ 
                               x 
                             
                           
                            
                           
                             
                               ∂ 
                               
                                   
                               
                             
                             
                               ∂ 
                               y 
                             
                           
                            
                           
                             M 
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 y 
                                 , 
                                 z 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   iii 
                   ) 
                 
               
             
           
         
       
     
       Proposition 1 
       [0045]    
       
         
           
             
               
                 
                   ∂ 
                   
                       
                   
                 
                 
                   ∂ 
                   ρ 
                 
               
                
               
                 
                   C 
                   0 
                   BS 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           
                             δ 
                             1 
                           
                           , 
                           
                             δ 
                             2 
                           
                           , 
                           
                             K 
                             1 
                           
                           , 
                           
                             K 
                             2 
                           
                           , 
                           
                             σ 
                             1 
                           
                           , 
                           
                             σ 
                             2 
                           
                           , 
                         
                       
                     
                     
                       
                         
                           ρ 
                           , 
                           T 
                           , 
                           
                             B 
                             T 
                             0 
                           
                           , 
                           
                             F 
                             T 
                             1 
                           
                           , 
                           
                             F 
                             T 
                             2 
                           
                         
                       
                     
                   
                   ) 
                 
               
             
             = 
             
               
                 δ 
                 1 
               
                
               
                 δ 
                 2 
               
                
               
                 B 
                 T 
                 0 
               
               × 
               
                 
 
               
                
               
                 ( 
                 
                   
                     
                       
                         
                           F 
                           T 
                           1 
                         
                          
                         
                           F 
                           T 
                           2 
                         
                          
                         
                           σ 
                           1 
                         
                          
                         
                           σ 
                           2 
                         
                          
                         T 
                          
                         
                             
                         
                          
                         
                            
                           
                             ρ 
                              
                             
                                 
                             
                              
                             
                               σ 
                               1 
                             
                              
                             
                               σ 
                               2 
                             
                              
                             T 
                           
                         
                          
                         
                           M 
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       - 
                                       
                                         
                                           δ 
                                           1 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               κ 
                                               1 
                                             
                                             - 
                                             
                                               
                                                 σ 
                                                 2 
                                               
                                                
                                               ρ 
                                                
                                               
                                                 T 
                                               
                                             
                                             - 
                                             
                                               
                                                 σ 
                                                 1 
                                               
                                                
                                               
                                                 T 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     , 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         - 
                                         
                                           δ 
                                           2 
                                         
                                       
                                        
                                       
                                         
                                           1 
                                           - 
                                           
                                             ρ 
                                             2 
                                           
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             κ 
                                             2 
                                           
                                           - 
                                           
                                             
                                               σ 
                                               2 
                                             
                                              
                                             
                                               
                                                 T 
                                                  
                                                 
                                                   ( 
                                                   
                                                     1 
                                                     - 
                                                     
                                                       ρ 
                                                       2 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     , 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       
                                         δ 
                                         1 
                                       
                                     
                                      
                                     
                                       δ 
                                       2 
                                     
                                      
                                     ρ 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           δ 
                           1 
                         
                          
                         
                           σ 
                           2 
                         
                          
                         
                           T 
                         
                          
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     F 
                                     T 
                                     1 
                                   
                                    
                                   
                                     F 
                                     T 
                                     2 
                                   
                                    
                                   
                                      
                                     
                                       ρ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         σ 
                                         1 
                                       
                                        
                                       
                                         σ 
                                         2 
                                       
                                        
                                       T 
                                     
                                   
                                    
                                   
                                     
                                       M 
                                       1 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               
                                                 - 
                                                 
                                                   
                                                     δ 
                                                     1 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         κ 
                                                         1 
                                                       
                                                       - 
                                                       
                                                         
                                                           σ 
                                                           2 
                                                         
                                                          
                                                         ρ 
                                                          
                                                         
                                                           T 
                                                         
                                                       
                                                       - 
                                                       
                                                         
                                                           σ 
                                                           1 
                                                         
                                                          
                                                         
                                                           T 
                                                         
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               , 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 
                                                   - 
                                                   
                                                     δ 
                                                     2 
                                                   
                                                 
                                                  
                                                 
                                                   
                                                     1 
                                                     - 
                                                     
                                                       ρ 
                                                       2 
                                                     
                                                   
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       κ 
                                                       2 
                                                     
                                                     - 
                                                     
                                                       
                                                         σ 
                                                         2 
                                                       
                                                        
                                                       
                                                         
                                                           T 
                                                            
                                                           
                                                             ( 
                                                             
                                                             1 
                                                             - 
                                                             
                                                             ρ 
                                                             2 
                                                             
                                                             
                                                             ) 
                                                           
                                                         
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               , 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 - 
                                                 
                                                   δ 
                                                   1 
                                                 
                                               
                                                
                                               
                                                 δ 
                                                 2 
                                               
                                                
                                               ρ 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     
                                       F 
                                       T 
                                       2 
                                     
                                   
                                    
                                   
                                     K 
                                     1 
                                   
                                    
                                   
                                     
                                       M 
                                       1 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               
                                                 - 
                                                 
                                                   
                                                     δ 
                                                     1 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         κ 
                                                         1 
                                                       
                                                       - 
                                                       
                                                         
                                                           σ 
                                                           2 
                                                         
                                                          
                                                         ρ 
                                                          
                                                         
                                                           T 
                                                         
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               , 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 
                                                   - 
                                                   
                                                     δ 
                                                     2 
                                                   
                                                 
                                                  
                                                 
                                                   
                                                     1 
                                                     - 
                                                     
                                                       ρ 
                                                       2 
                                                     
                                                   
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       κ 
                                                       2 
                                                     
                                                     - 
                                                     
                                                       
                                                         σ 
                                                         2 
                                                       
                                                        
                                                       
                                                         
                                                           T 
                                                            
                                                           
                                                             ( 
                                                             
                                                             1 
                                                             - 
                                                             
                                                             ρ 
                                                             2 
                                                             
                                                             
                                                             ) 
                                                           
                                                         
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               , 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 - 
                                                 
                                                   δ 
                                                   1 
                                                 
                                               
                                                
                                               
                                                 δ 
                                                 2 
                                               
                                                
                                               ρ 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         ρ 
                          
                         
                             
                         
                          
                         
                           
                             δ 
                             2 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       F 
                                       T 
                                       1 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     κ 
                                                     2 
                                                   
                                                   
                                                     
                                                       1 
                                                       - 
                                                       
                                                         ρ 
                                                         2 
                                                       
                                                     
                                                   
                                                 
                                                 - 
                                                 
                                                   2 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     σ 
                                                     2 
                                                   
                                                    
                                                   
                                                     T 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 F 
                                                 T 
                                                 2 
                                               
                                                
                                               
                                                  
                                                 
                                                   ρ 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     σ 
                                                     1 
                                                   
                                                    
                                                   
                                                     σ 
                                                     2 
                                                   
                                                    
                                                   T 
                                                 
                                               
                                                
                                               
                                                 M 
                                                 2 
                                               
                                             
                                           
                                         
                                         
                                           
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     
                                                       
                                                         - 
                                                         
                                                           
                                                             δ 
                                                             1 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             
                                                             κ 
                                                             1 
                                                             
                                                             - 
                                                             
                                                             
                                                             σ 
                                                             2 
                                                             
                                                              
                                                             ρ 
                                                              
                                                             
                                                             T 
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             σ 
                                                             1 
                                                             
                                                              
                                                             
                                                             T 
                                                             
                                                             
                                                             
                                                             ) 
                                                           
                                                         
                                                       
                                                       , 
                                                     
                                                   
                                                 
                                                 
                                                   
                                                     
                                                       
                                                         
                                                           - 
                                                           
                                                             δ 
                                                             2 
                                                           
                                                         
                                                          
                                                         
                                                           
                                                             1 
                                                             - 
                                                             
                                                             ρ 
                                                             2 
                                                             
                                                           
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             
                                                             κ 
                                                             2 
                                                             
                                                             - 
                                                             
                                                             
                                                             σ 
                                                             2 
                                                             
                                                              
                                                             
                                                             
                                                             T 
                                                              
                                                             
                                                             ( 
                                                             
                                                             1 
                                                             - 
                                                             
                                                             ρ 
                                                             2 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                       , 
                                                     
                                                   
                                                 
                                                 
                                                   
                                                     
                                                       
                                                         - 
                                                         
                                                           δ 
                                                           1 
                                                         
                                                       
                                                        
                                                       
                                                         δ 
                                                         2 
                                                       
                                                        
                                                       ρ 
                                                     
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 - 
                                                 
                                                   
                                                     κ 
                                                     2 
                                                   
                                                   
                                                     
                                                       1 
                                                       - 
                                                       
                                                         ρ 
                                                         2 
                                                       
                                                     
                                                   
                                                 
                                               
                                                
                                               
                                                 K 
                                                 2 
                                               
                                                
                                               
                                                 M 
                                                 2 
                                               
                                             
                                           
                                         
                                         
                                           
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     
                                                       
                                                         - 
                                                         
                                                           
                                                             δ 
                                                             1 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             
                                                             κ 
                                                             1 
                                                             
                                                             - 
                                                             
                                                             
                                                             σ 
                                                             1 
                                                             
                                                              
                                                             
                                                             T 
                                                             
                                                             
                                                             
                                                             ) 
                                                           
                                                         
                                                       
                                                       , 
                                                     
                                                   
                                                 
                                                 
                                                   
                                                     
                                                       
                                                         
                                                           - 
                                                           
                                                             δ 
                                                             2 
                                                           
                                                         
                                                          
                                                         
                                                           κ 
                                                           2 
                                                         
                                                          
                                                         
                                                           
                                                             1 
                                                             - 
                                                             
                                                             ρ 
                                                             2 
                                                             
                                                           
                                                         
                                                       
                                                       , 
                                                     
                                                   
                                                 
                                                 
                                                   
                                                     
                                                       
                                                         - 
                                                         
                                                           δ 
                                                           1 
                                                         
                                                       
                                                        
                                                       
                                                         δ 
                                                         2 
                                                       
                                                        
                                                       ρ 
                                                     
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     - 
                                     
                                       
                                         K 
                                         1 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     
                                                       κ 
                                                       2 
                                                     
                                                     
                                                       
                                                         1 
                                                         - 
                                                         
                                                           ρ 
                                                           2 
                                                         
                                                       
                                                     
                                                   
                                                   - 
                                                   
                                                     2 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     
                                                       σ 
                                                       2 
                                                     
                                                      
                                                     
                                                       T 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   F 
                                                   T 
                                                   2 
                                                 
                                                  
                                                 
                                                   M 
                                                   2 
                                                 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     
                                                       
                                                         
                                                           - 
                                                           
                                                             
                                                             δ 
                                                             1 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             κ 
                                                             1 
                                                             
                                                             - 
                                                             
                                                             
                                                             σ 
                                                             2 
                                                             
                                                              
                                                             ρ 
                                                              
                                                             
                                                             T 
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                           
                                                         
                                                         , 
                                                       
                                                     
                                                   
                                                   
                                                     
                                                       
                                                         
                                                           
                                                             - 
                                                             
                                                             δ 
                                                             2 
                                                             
                                                           
                                                            
                                                           
                                                             
                                                             1 
                                                             - 
                                                             
                                                             ρ 
                                                             2 
                                                             
                                                             
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             
                                                             κ 
                                                             2 
                                                             
                                                             - 
                                                             
                                                             
                                                             σ 
                                                             2 
                                                             
                                                              
                                                             
                                                             
                                                             T 
                                                              
                                                             
                                                             ( 
                                                             
                                                             1 
                                                             - 
                                                             
                                                             ρ 
                                                             2 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                           
                                                         
                                                         , 
                                                       
                                                     
                                                   
                                                   
                                                     
                                                       
                                                         
                                                           - 
                                                           
                                                             δ 
                                                             1 
                                                           
                                                         
                                                          
                                                         
                                                           δ 
                                                           2 
                                                         
                                                          
                                                         ρ 
                                                       
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   - 
                                                   
                                                     
                                                       κ 
                                                       2 
                                                     
                                                     
                                                       
                                                         1 
                                                         - 
                                                         
                                                           ρ 
                                                           2 
                                                         
                                                       
                                                     
                                                   
                                                 
                                                  
                                                 
                                                   K 
                                                   2 
                                                 
                                                  
                                                 
                                                   M 
                                                   2 
                                                 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     
                                                       - 
                                                       
                                                         δ 
                                                         1 
                                                       
                                                     
                                                      
                                                     
                                                       κ 
                                                       1 
                                                     
                                                   
                                                   , 
                                                   
                                                     
                                                       - 
                                                       
                                                         δ 
                                                         2 
                                                       
                                                     
                                                      
                                                     
                                                       κ 
                                                       2 
                                                     
                                                      
                                                     
                                                       
                                                         1 
                                                         - 
                                                         
                                                           ρ 
                                                           2 
                                                         
                                                       
                                                     
                                                   
                                                   , 
                                                 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   
                                                     - 
                                                     
                                                       δ 
                                                       1 
                                                     
                                                   
                                                    
                                                   
                                                     δ 
                                                     2 
                                                   
                                                    
                                                   ρ 
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           
                             δ 
                             1 
                           
                         
                          
                         
                           
                             δ 
                             2 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       F 
                                       T 
                                       1 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               
                                                 F 
                                                 T 
                                                 2 
                                               
                                                
                                               
                                                  
                                                 
                                                   ρ 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   
                                                     σ 
                                                     1 
                                                   
                                                    
                                                   
                                                     σ 
                                                     2 
                                                   
                                                    
                                                   T 
                                                 
                                               
                                                
                                               
                                                 
                                                   M 
                                                   3 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       
                                                         
                                                           
                                                             - 
                                                             
                                                             
                                                             δ 
                                                             1 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             κ 
                                                             1 
                                                             
                                                             - 
                                                             
                                                             
                                                             σ 
                                                             2 
                                                             
                                                              
                                                             ρ 
                                                              
                                                             
                                                             T 
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             σ 
                                                             1 
                                                             
                                                              
                                                             
                                                             T 
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                           
                                                           , 
                                                         
                                                       
                                                     
                                                     
                                                       
                                                         
                                                           
                                                             
                                                             - 
                                                             
                                                             δ 
                                                             2 
                                                             
                                                             
                                                              
                                                             
                                                             
                                                             1 
                                                             - 
                                                             
                                                             ρ 
                                                             2 
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             κ 
                                                             2 
                                                             
                                                             - 
                                                             
                                                             
                                                             σ 
                                                             2 
                                                             
                                                              
                                                             
                                                             
                                                             T 
                                                              
                                                             
                                                             ( 
                                                             
                                                             1 
                                                             - 
                                                             
                                                             ρ 
                                                             2 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                           
                                                           , 
                                                         
                                                       
                                                     
                                                     
                                                       
                                                         
                                                           
                                                             - 
                                                             
                                                             δ 
                                                             1 
                                                             
                                                           
                                                            
                                                           
                                                             δ 
                                                             2 
                                                           
                                                            
                                                           ρ 
                                                         
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                 - 
                                                 
                                                   K 
                                                   2 
                                                 
                                               
                                                
                                               
                                                 
                                                   M 
                                                   3 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       
                                                         
                                                           
                                                             - 
                                                             
                                                             
                                                             δ 
                                                             1 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             κ 
                                                             1 
                                                             
                                                             - 
                                                             
                                                             
                                                             σ 
                                                             1 
                                                             
                                                              
                                                             
                                                             T 
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                           
                                                           , 
                                                         
                                                       
                                                     
                                                     
                                                       
                                                         
                                                           
                                                             
                                                             - 
                                                             
                                                             δ 
                                                             2 
                                                             
                                                             
                                                              
                                                             
                                                             κ 
                                                             2 
                                                             
                                                              
                                                             
                                                             
                                                             1 
                                                             - 
                                                             
                                                             ρ 
                                                             2 
                                                             
                                                             
                                                             
                                                           
                                                           , 
                                                           
                                                             
                                                             - 
                                                             
                                                             δ 
                                                             1 
                                                             
                                                             
                                                              
                                                             
                                                             δ 
                                                             2 
                                                             
                                                              
                                                             ρ 
                                                           
                                                         
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     - 
                                     
                                       
                                         K 
                                         1 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               
                                                 
                                                   F 
                                                   T 
                                                   2 
                                                 
                                                  
                                                 
                                                   
                                                     M 
                                                     3 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         
                                                           
                                                             
                                                             - 
                                                             
                                                             
                                                             δ 
                                                             1 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             κ 
                                                             1 
                                                             
                                                             - 
                                                             
                                                             
                                                             σ 
                                                             2 
                                                             
                                                              
                                                             ρ 
                                                              
                                                             
                                                             T 
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             , 
                                                           
                                                         
                                                       
                                                       
                                                         
                                                           
                                                             
                                                             
                                                             - 
                                                             
                                                             δ 
                                                             2 
                                                             
                                                             
                                                              
                                                             
                                                             
                                                             1 
                                                             - 
                                                             
                                                             ρ 
                                                             2 
                                                             
                                                             
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             κ 
                                                             2 
                                                             
                                                             - 
                                                             
                                                             
                                                             σ 
                                                             2 
                                                             
                                                              
                                                             
                                                             
                                                             T 
                                                              
                                                             
                                                             ( 
                                                             
                                                             1 
                                                             - 
                                                             
                                                             ρ 
                                                             2 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             , 
                                                           
                                                         
                                                       
                                                       
                                                         
                                                           
                                                             
                                                             - 
                                                             
                                                             δ 
                                                             1 
                                                             
                                                             
                                                              
                                                             
                                                             δ 
                                                             2 
                                                             
                                                              
                                                             ρ 
                                                           
                                                         
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   - 
                                                   
                                                     K 
                                                     2 
                                                   
                                                 
                                                  
                                                 
                                                   
                                                     M 
                                                     3 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         
                                                           - 
                                                           
                                                             δ 
                                                             1 
                                                           
                                                         
                                                          
                                                         
                                                           κ 
                                                           1 
                                                         
                                                       
                                                       , 
                                                       
                                                         
                                                           - 
                                                           
                                                             δ 
                                                             2 
                                                           
                                                         
                                                          
                                                         
                                                           κ 
                                                           2 
                                                         
                                                          
                                                         
                                                           
                                                             1 
                                                             - 
                                                             
                                                             ρ 
                                                             2 
                                                             
                                                           
                                                         
                                                       
                                                       , 
                                                       
                                                         
                                                           - 
                                                           
                                                             δ 
                                                             1 
                                                           
                                                         
                                                          
                                                         
                                                           δ 
                                                           2 
                                                         
                                                          
                                                         ρ 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
         [0046]    We assume C 0   BS (δ 1 , δ 2 , K 1 , K 2 , σ 1 , σ 2 , ρ, T, B T   0 , F T   1 , F T   2 )as a function of ρ is monotonous. This is intuitively sensible and we can infer that for 
         [0000]    
       
         
           
             
               
                 
                   ∂ 
                   
                       
                   
                 
                 
                   ∂ 
                   ρ 
                 
               
                
               
                 
                   C 
                   0 
                   BS 
                 
                  
                 
                   ( 
                   
                     
                       δ 
                       1 
                     
                     , 
                     
                       δ 
                       2 
                     
                     , 
                     
                       K 
                       1 
                     
                     , 
                     
                       K 
                       2 
                     
                     , 
                     
                       σ 
                       1 
                     
                     , 
                     
                       σ 
                       2 
                     
                     , 
                     ρ 
                     , 
                     T 
                     , 
                     
                       B 
                       T 
                       0 
                     
                     , 
                     
                       F 
                       T 
                       1 
                     
                     , 
                     
                       F 
                       T 
                       2 
                     
                   
                   ) 
                 
               
               × 
               
                 δ 
                 1 
               
                
               
                 δ 
                 2 
               
             
             &gt; 
             0 
           
         
       
     
       3.2.2. Implied Correlation Function Definition 
       [0047]    Definition 5. For any given K 1 K 2 , B T   0 , F T   1 , F T   2 , the implied correlation function ρ(K 1 , K 2 , T) is defined by the equation 
         [0000]        C   0   BS (1,1,  K   1   , K   2 , σ 1 ( K   1   , T ), σ 2 ( K   2   , T ), ρ( K   1   , K   2   , T ),  T, B   T   0   , F   T   1   , F   T   2 )= C   0 (1,1, K   1   , K   2 , T, B T   0   , F   T   1   , F   T   2 ) 
         [0000]    and the inequalities −1≦ρ(K 1 , K 2 , T)≦1. If 
         [0000]    
       
         
           
             
               
                 
                   ∂ 
                   
                       
                   
                 
                 
                   ∂ 
                   ρ 
                 
               
                
               
                 
                   C 
                   0 
                   BS 
                 
                  
                 
                   ( 
                   
                     1 
                     , 
                     1 
                     , 
                     
                       K 
                       1 
                     
                     , 
                     
                       K 
                       2 
                     
                     , 
                     
                       σ 
                       1 
                     
                     , 
                     
                       σ 
                       2 
                     
                     , 
                     ρ 
                     , 
                     T 
                     , 
                     
                       B 
                       T 
                       0 
                     
                     , 
                     
                       F 
                       T 
                       1 
                     
                     , 
                     
                       F 
                       T 
                       2 
                     
                   
                   ) 
                 
               
             
             ≠ 
             0 
           
         
       
     
         [0000]    then the implied correlation function is uniquely determined. 
       3.2.3. Computing the Implied Correlation Function 
       [0048]    When the implied correlation function is uniquely determined, 
         [0000]    
       
         
           
             
               
                 ∂ 
                 
                   ∂ 
                   ρ 
                 
               
                
               
                 
                   C 
                   0 
                   BS 
                 
                  
                 
                   ( 
                   
                     1 
                     , 
                     1 
                     , 
                     
                       K 
                       1 
                     
                     , 
                     
                       K 
                       2 
                     
                     , 
                     
                       σ 
                       1 
                     
                     , 
                     
                       σ 
                       2 
                     
                     , 
                     ρ 
                     , 
                     T 
                     , 
                     
                       B 
                       T 
                       0 
                     
                     , 
                     
                       F 
                       T 
                       1 
                     
                     , 
                     
                       F 
                       T 
                       2 
                     
                   
                   ) 
                 
               
             
             ≠ 
             0 
           
         
       
     
         [0000]    it may thus be computed by any of the following numerical methods: 
       Newton Raphson Algorithm 
       [0049]      ρ   0 =0; (or any other good seed value) 
         [0050]    Do 
         [0000]    
       
         
           
             
               
                 ρ 
                 _ 
               
               
                 i 
                 + 
                 1 
               
             
             = 
             
               
                 
                   ρ 
                   _ 
                 
                 i 
               
               + 
               
                 
                   
                     
                       
                         
                           
                             C 
                             0 
                             Market 
                           
                            
                           
                             ( 
                             
                               
                                 δ 
                                 1 
                               
                               , 
                               
                                 δ 
                                 2 
                               
                               , 
                               
                                 K 
                                 1 
                               
                               , 
                               
                                 K 
                                 2 
                               
                               , 
                               T 
                             
                             ) 
                           
                         
                         - 
                       
                     
                   
                   
                     
                       
                         
                           C 
                           0 
                           
                             BS 
                             , 
                             2 
                           
                         
                          
                         
                           ( 
                           
                             
                               δ 
                               1 
                             
                             , 
                             
                               δ 
                               2 
                             
                             , 
                             
                               K 
                               1 
                             
                             , 
                             
                               K 
                               2 
                             
                             , 
                             
                               σ 
                               1 
                             
                             , 
                             
                               σ 
                               2 
                             
                             , 
                             
                               
                                 ρ 
                                 _ 
                               
                               i 
                             
                             , 
                             T 
                             , 
                             
                               B 
                               T 
                               0 
                             
                             , 
                             
                               F 
                               T 
                               1 
                             
                             , 
                             
                               F 
                               T 
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     ∂ 
                     
                       ∂ 
                       
                         ρ 
                         _ 
                       
                     
                   
                    
                   
                     
                       C 
                       0 
                       
                         BS 
                         , 
                         2 
                       
                     
                      
                     
                       ( 
                       
                         
                           δ 
                           1 
                         
                         , 
                         
                           δ 
                           2 
                         
                         , 
                         
                           K 
                           1 
                         
                         , 
                         
                           K 
                           2 
                         
                         , 
                         
                           σ 
                           1 
                         
                         , 
                         
                           σ 
                           2 
                         
                         , 
                         
                           
                             ρ 
                             _ 
                           
                           
                             i 
                             + 
                             1 
                           
                         
                         , 
                         T 
                         , 
                         
                           B 
                           T 
                           0 
                         
                         , 
                         
                           F 
                           T 
                           1 
                         
                         , 
                         
                           F 
                           T 
                           2 
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           until 
         
       
       
         
           
             
               
                  
                 
                   ( 
                   
                     
                       
                         
                           
                             
                               C 
                               0 
                               Market 
                             
                              
                             
                               ( 
                               
                                 
                                   δ 
                                   1 
                                 
                                 , 
                                 
                                   δ 
                                   2 
                                 
                                 , 
                                 
                                   K 
                                   1 
                                 
                                 , 
                                 
                                   K 
                                   2 
                                 
                                 , 
                                 T 
                               
                               ) 
                             
                           
                           - 
                         
                       
                     
                     
                       
                         
                           
                             C 
                             0 
                             
                               BS 
                               , 
                               2 
                             
                           
                            
                           
                             ( 
                             
                               
                                 δ 
                                 1 
                               
                               , 
                               
                                 δ 
                                 2 
                               
                               , 
                               
                                 K 
                                 1 
                               
                               , 
                               
                                 K 
                                 2 
                               
                               , 
                               
                                 σ 
                                 1 
                               
                               , 
                               
                                 σ 
                                 2 
                               
                               , 
                               
                                 
                                   ρ 
                                   _ 
                                 
                                 i 
                               
                               , 
                               T 
                               , 
                               
                                 B 
                                 T 
                                 0 
                               
                               , 
                               
                                 F 
                                 T 
                                 1 
                               
                               , 
                               
                                 F 
                                 T 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   ) 
                 
                  
               
               ≤ 
               ɛ 
             
             ; 
           
         
       
       
         
           
             
               
                 
                   ρ 
                   _ 
                 
                 impl 
               
                
               
                 ( 
                 
                   
                     K 
                     1 
                   
                   , 
                   
                     K 
                     2 
                   
                 
                 ) 
               
             
             = 
             
               
                 ρ 
                 _ 
               
               
                 i 
                 + 
                 1 
               
             
           
         
       
     
       Bisection Algorithm 
       [0051]      ρ   L =−1;  ρ   H =1; (or any other good seed value) 
         [0052]    Do 
         [0000]    
       
         
           
             
               
                 ρ 
                 _ 
               
               
                 i 
                 + 
                 1 
               
             
             = 
             
               
                 
                   ρ 
                   _ 
                 
                 L 
               
               + 
               
                 
                   
                     ( 
                     
                       
                         
                           ρ 
                           _ 
                         
                         H 
                       
                       - 
                       
                         
                           ρ 
                           _ 
                         
                         L 
                       
                     
                     ) 
                   
                    
                   
                     ( 
                     
                       
                         
                           
                             
                               
                                 C 
                                 0 
                                 Market 
                               
                                
                               
                                 ( 
                                 
                                   
                                     δ 
                                     1 
                                   
                                   , 
                                   
                                     δ 
                                     2 
                                   
                                   , 
                                   
                                     K 
                                     1 
                                   
                                   , 
                                   
                                     K 
                                     2 
                                   
                                   , 
                                   T 
                                 
                                 ) 
                               
                             
                             - 
                           
                         
                       
                       
                         
                           
                             
                               C 
                               0 
                               
                                 BS 
                                 , 
                                 2 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   δ 
                                   1 
                                 
                                 , 
                                 
                                   δ 
                                   2 
                                 
                                 , 
                                 
                                   K 
                                   1 
                                 
                                 , 
                                 
                                   K 
                                   2 
                                 
                                 , 
                                 
                                   σ 
                                   1 
                                 
                                 , 
                                 
                                   σ 
                                   2 
                                 
                                 , 
                                 
                                   
                                     ρ 
                                     _ 
                                   
                                   L 
                                 
                                 , 
                                 T 
                                 , 
                                 
                                   B 
                                   T 
                                   0 
                                 
                                 , 
                                 
                                   F 
                                   T 
                                   1 
                                 
                                 , 
                                 
                                   F 
                                   T 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     ) 
                   
                 
                 
                   
                     
                       
                         
                           
                             C 
                             0 
                             
                               BS 
                               , 
                               2 
                             
                           
                            
                           
                             ( 
                             
                               
                                 δ 
                                 1 
                               
                               , 
                               
                                 δ 
                                 2 
                               
                               , 
                               
                                 K 
                                 1 
                               
                               , 
                               
                                 K 
                                 2 
                               
                               , 
                               
                                 σ 
                                 1 
                               
                               , 
                               
                                 σ 
                                 2 
                               
                               , 
                               
                                 
                                   ρ 
                                   _ 
                                 
                                 H 
                               
                               , 
                               T 
                               , 
                               
                                 B 
                                 T 
                                 0 
                               
                               , 
                               
                                 F 
                                 T 
                                 1 
                               
                               , 
                               
                                 F 
                                 T 
                                 2 
                               
                             
                             ) 
                           
                         
                         - 
                       
                     
                   
                   
                     
                       
                         
                           C 
                           0 
                           
                             BS 
                             , 
                             2 
                           
                         
                          
                         
                           ( 
                           
                             
                               δ 
                               1 
                             
                             , 
                             
                               δ 
                               2 
                             
                             , 
                             
                               K 
                               1 
                             
                             , 
                             
                               K 
                               2 
                             
                             , 
                             
                               σ 
                               1 
                             
                             , 
                             
                               σ 
                               2 
                             
                             , 
                             
                               
                                 ρ 
                                 _ 
                               
                               L 
                             
                             , 
                             T 
                             , 
                             
                               B 
                               T 
                               0 
                             
                             , 
                             
                               F 
                               T 
                               1 
                             
                             , 
                             
                               F 
                               T 
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 If 
                  
                 
                     
                 
                  
                 
                   ( 
                   
                     
                       
                         C 
                         0 
                         Market 
                       
                        
                       
                         ( 
                         
                           
                             δ 
                             1 
                           
                           , 
                           
                             δ 
                             2 
                           
                           , 
                           
                             K 
                             1 
                           
                           , 
                           
                             K 
                             2 
                           
                           , 
                           T 
                         
                         ) 
                       
                     
                     - 
                     
                       
                         C 
                         0 
                         
                           BS 
                           , 
                           2 
                         
                       
                        
                       
                         ( 
                         
                           
                             δ 
                             1 
                           
                           , 
                           
                             δ 
                             2 
                           
                           , 
                           
                             K 
                             1 
                           
                           , 
                           
                             K 
                             2 
                           
                           , 
                           
                             σ 
                             1 
                           
                           , 
                           
                             σ 
                             2 
                           
                           , 
                           
                             
                               ρ 
                               _ 
                             
                             
                               i 
                               + 
                               1 
                             
                           
                           , 
                           T 
                           , 
                           
                             B 
                             T 
                             0 
                           
                           , 
                           
                             F 
                             T 
                             1 
                           
                           , 
                           
                             F 
                             T 
                             2 
                           
                         
                         ) 
                       
                     
                   
                   ) 
                 
               
               &gt; 
               0 
             
             , 
             
               
 
             
              
             
                 
             
              
             
               
                 
                   ρ 
                   _ 
                 
                 L 
               
               = 
               
                 
                   ρ 
                   _ 
                 
                 
                   i 
                   + 
                   1 
                 
               
             
             , 
             
               
                 else 
                  
                 
                     
                 
                  
                 
                   
                     ρ 
                     _ 
                   
                   H 
                 
               
               = 
               
                 
                   ρ 
                   _ 
                 
                 
                   i 
                   + 
                   1 
                 
               
             
           
         
       
       
         
           
             
               
                 Until 
                  
                 
                     
                 
                  
                 
                    
                   
                     ( 
                     
                       
                         
                           
                             
                               
                                 C 
                                 0 
                                 Market 
                               
                                
                               
                                 ( 
                                 
                                   
                                     δ 
                                     1 
                                   
                                   , 
                                   
                                     δ 
                                     2 
                                   
                                   , 
                                   
                                     K 
                                     1 
                                   
                                   , 
                                   
                                     K 
                                     2 
                                   
                                   , 
                                   T 
                                 
                                 ) 
                               
                             
                             - 
                           
                         
                       
                       
                         
                           
                             
                               C 
                               0 
                               
                                 BS 
                                 , 
                                 2 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   δ 
                                   1 
                                 
                                 , 
                                 
                                   δ 
                                   2 
                                 
                                 , 
                                 
                                   K 
                                   1 
                                 
                                 , 
                                 
                                   K 
                                   2 
                                 
                                 , 
                                 
                                   σ 
                                   1 
                                 
                                 , 
                                 
                                   σ 
                                   2 
                                 
                                 , 
                                 
                                   
                                     ρ 
                                     _ 
                                   
                                   
                                     i 
                                     + 
                                     1 
                                   
                                 
                                 , 
                                 T 
                                 , 
                                 
                                   B 
                                   T 
                                   0 
                                 
                                 , 
                                 
                                   F 
                                   T 
                                   1 
                                 
                                 , 
                                 
                                   F 
                                   T 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     ) 
                   
                    
                 
               
               ≤ 
               ɛ 
             
             ; 
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   
                     
                       ρ 
                       _ 
                     
                     impl 
                   
                    
                   
                     ( 
                     
                       
                         K 
                         1 
                       
                       , 
                       
                         K 
                         2 
                       
                     
                     ) 
                   
                 
                 = 
                 
                   
                     ρ 
                     _ 
                   
                   
                     i 
                     + 
                     1 
                   
                 
               
               ; 
             
           
         
       
     
         [0053]    Remark 1. There are several other implied correlation computation methods, and various approximations. Many of these approximation methods may simply adapt to the dimension 2 the approximation methods of the one dimension case. 
         [0054]    In particular, adaptation of methods such as the at the money forward approximation of Brenner and Subramanyam (1988), the Corrado and Miller extended moneyness approximation (1996) maybe particularly appropriate. See p. 172 [113]. 
         [0055]    In addition, we can also adapt the seeding value derivation method of Manaster and Koehler (1982) to speed the convergence of the Newton Raphson or bisection iterative numerical algorithms. 
       3.3. Application of the Formula For the Generation of Non Trivial Multidimensional Densities 
       [0056]    We know that in the case of a single asset, the density can be expressed in the form of implied volatility functions by inversion of the Black Scholes formula. 
         [0057]    Recall that if the price of the option is, 
         [0058]    Via the Newton raphson algorithm, if S&gt;Call(S,K,T,)&gt;B0(F−K) the algorithm: 
         [0059]    Given ε arbitrarily small 
         [0000]    
       
         
           
             
               σ 
               1 
               0 
             
             = 
             
               
                  
                 
                   
                     2 
                     T 
                   
                    
                   
                       
                   
                    
                   Log 
                    
                   
                       
                   
                    
                   
                     ( 
                     
                       
                         F 
                         T 
                         1 
                       
                       
                         K 
                         1 
                       
                     
                     ) 
                   
                 
                  
               
             
           
         
       
     
         [0000]    (or any other given good seed value) 
         [0060]    Do 
         [0000]    
       
         
           
             
               σ 
               1 
               
                 i 
                 + 
                 1 
               
             
             = 
             
               
                 σ 
                 1 
                 i 
               
               + 
               
                 
                   
                     
                       C 
                       0 
                       Market 
                     
                      
                     
                       ( 
                       
                         
                           K 
                           1 
                         
                          
                         T 
                       
                       ) 
                     
                   
                   - 
                   
                     
                       C 
                       0 
                       BS 
                     
                      
                     
                       ( 
                       
                         
                           K 
                           1 
                         
                         , 
                         
                           σ 
                           1 
                           i 
                         
                         , 
                         T 
                         , 
                         
                           B 
                           T 
                           0 
                         
                         , 
                         
                           F 
                           T 
                           1 
                         
                       
                       ) 
                     
                   
                 
                 
                   
                     VegaC 
                     0 
                     BS 
                   
                    
                   
                     ( 
                     
                       
                         K 
                         1 
                       
                       , 
                       
                         σ 
                         1 
                         i 
                       
                       , 
                       T 
                       , 
                       
                         B 
                         T 
                         0 
                       
                       , 
                       
                         F 
                         T 
                         1 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    until |C 0   Market  (K 1 ,T)−C 0   BS  (K 1 , σ 1   i+1 , T, B T   0 , F T   1 )≦ε; σ 1   implied  (K 1 )=σ 1   i+1  yields in very few steps the implied volatility as a function of K 
         [0061]    Let&#39;s suppose we have n assets and via the prices of calls and puts on the we have their implied volatility function σ 1   impl  (K 1 ) and σ 2   impl  (K 2 ). 
         [0062]    We plug those implied volatility function in the second order BIC price formula C 0   BS  (δ 1 , δ 2 , K 1 , K 2 , σ 1   impl  (K 1 ), σ 2   impl  (K 2 ), ρ, T, B T   0 , F T   1 , F T   2 ). Given a continuum of second order BICs prices, we then compute the implied correlation function as indicated above. 
         [0000]        C   0   M, n ((δ 1 , . . . , δ n ), ( K   1   , . . . , K   n   ), T)=C   0   BS, n ((δ 1 , . . . , δ n ), ( K   1   , . . . , K   n   ), (F   T   1   , . . . , F   T   n ), (σ 1 ( K   1 ), . . . , σ n ( K   n )),(  ρ   T   i,j ( K   i   , K   j )) j&lt;i≦n   , T)    
         [0063]    See also  FIG. 2 . 
         [0064]    For n&gt;3, it appears that we no longer have parametric freedom from which to calibrate to observed prices. This tempts us to conjecture the following: 
       Theorem 3 (Conjecture) 
       [0065]    If fn(x) n&gt;2 is a density function such that the marginal density of any two variables is known. Then fn(x) is uniquely determined. 
       3.3.1. Implied Distribution Derivation 
       [0066]    The multivariate distribution function can be inferred from the Options BICs prices provided in a continuum via the formula 
         [0067]    When Option BICs prices are given by the implied volatility and correlation functions, the multivariate density is uniquely determined. 
         [0068]    However, while performing this derivation in an analytical manner, the number of terms quickly explodes. It is however clear that in practice, for pricing purposes, there will be very little need to compute the density when the BICs prices are available 
       B. Functional Representations For BICs Pricing 
     1. Problem Statement 
       [0069]    The main purpose of this chapter is to answer the question: How do we deal computationally with premium payment functions used and obtained in the backward recursive procedure for the pricing of derivatives contracts? 
         [0070]    Hence, we rephrase the issue as: How do we most efficiently represent functions we are likely to face here in computing terms? How do we perform operations on them, how do we add or multiply them? 
         [0071]    The problem is not trivial. Naive choices may result in computational problems that grow exponentially with the number of time steps. 
         [0072]    When the derivatives contract is so-called path-dependent, there is a problem of price storage as one must compute this through time for all the possible values the underlying upon which the derivatives contract depend at each time step. 
       2. Solution Philosophy: The Law of the Few 
       [0073]    Prior to any quantitative analysis, it is perhaps useful to synthesize our solution argument in philosophical principles that one can intuitively relate to in many other circumstances. 
         [0074]    The value of such an approach is to enable the inquisitive reader identify in a number of similar yet seemingly unrelated situations, the predictable methodological pathway to solutions of apparently intractable problems. 
         [0075]    The terminology The Law of the Few is borrowed from the Chapter II title of Science Journalist Malclom Gladwell&#39;s book The Tipping Point How Little things can make a big difference. In this brilliant essay, Mr. Gladwell explains in a very accessible pedagogical style how a small number of properly targeted connectors, mavens and salesmen can spread news or information that trigger unprecedented chains of events. He illustrates this analysis with such examples as the comparative narrative of Paul Revere&#39;s ride and Stanley Milgrom&#39;s famous six degrees of separation experiment. 
         [0076]    The American political landscape provides numerous other examples where the Law of the Few proposition is repeatedly verified. The incredibly critical role that two relatively minor states such as Iowa and New Hampshire play in determining who eventually gets the nomination of the two leading political parties is illustrative. The outcome of Fall presidential elections often reduces to a very narrow fight in a limited number of key states and this may confound the novice blindly looking at their relatively limited number of electoral votes. A substantial part of political pundits&#39; research aims at identifying very quickly, which few factors can be acted upon to decisively tip the balance in favor of their candidate. 
         [0077]    In business or mass media marketing, the almost sacred law of effective communication is to reduce all messages one would want to transmit to a few carefully selected single bullet points repeatedly stated. Otherwise, the evidence has uncovered, the message gets lost to the audience. 
         [0078]    In macroeconomic policy, empirical evidence has shown that the single most effective tools at the disposal of governments and central banks to cool or stimulate economic activity are monetary policy through the act of raising or cutting key interest rates on one hand or fiscal policy through the act of raising or cutting a few key tax rates. 
         [0079]    In mathematics, when establishing natural or desirable algebraic properties, one often finds that what is easily proved in the introductory trivial cases, suddenly stops working beyond a few given orders, apparently inexplicably. The initiated reader may immediately think about the impossibility of algebraic factorization of polynomials of order higher than four into polynomials of order one that is also known as the Galois Theorem. One may also think in some ways about the last Fermat theorem that puzzled the mathematical community for centuries. The above mathematical parallels, while historically interesting, may not translate to any practical benefit for the issue at hand. 
         [0080]    In his book A New Kind of Science, Stephen Wolfram from a cellular automata perspective, further illustrate the Law of the few by expanding considerably on the observation that complicated phenomena may be explained by simple underlying mechanisms. His argument is supported by connections to numerous fields of science, including artificial intelligence, biology, chaos theory, computer science, consciousness, economics, extra-terrestrial intelligence, fluid dynamics, logic, mathematics and physics. 
         [0081]    In relation to our subject matter, it is perhaps less surprising to see how authors such as Sahalia 1 among others are able to report that diffusion processes and even Levy Processes conditional densities can be recovered in Hermite polynomials series using less than a handful terms. 
         [0082]    Indeed the most compelling illustration of this Law of the Few is that the reduction of BICs to merely the underlying and bonds as is done in the Black Scholes analysis under the dynamic efficiency assumption has been acceptable as replication tool for derivatives hedging for a while. 
         [0083]    Our enunciation of the Law of the few here will concentrate on a few tools whose demonstration may not seem so sophisticated but whose application effectively provide compelling solutions to the Derivatives under BICs pricing problems, problems whose intractability has often been stressed under the quote often attributed to Richard Bellman: “The curse of Dimensionality.”2 We get rid of the curse here with two tools: the change of variable exact representation techniques and the basis projection approximation tools. In Chapter XII, these will be supplemented with low discrepancy sequences for multi-dimensional integration and Principal Component Analysis techniques. 
         [0084]    The Law of the few may be given a more scientific argumentation by relating it to the second law of thermodynamics in the principle of entropy. Here the entropic connection with the law of the few may be stated as: systems tend to store information in a manner that minimizes the state of disorder. Acknowledging this rule reduces the problems to finding the right storage rule. 
         [0085]    The principle of entropy minimization has been so far used in finance to address derivatives prices calibration issues as will be further seen in Volume II. 
       3. The Law of the Few In the BICs Analysis 
     3.1. Effective Dimension Reduction By Variable Changes 
       [0086]    For substantially most of the payoff structures of derivatives contracts, it is possible to make changes of variables so that at each stage of the iterative process yielding the price of the derivatives contract, the intermediate prices are functions of at most two or three variables. The generality of this proposition can be articulated in the following: 
       Proposition 1 
       [0087]    Let us suppose there exists a sequence of functions g i  and a vector sequence (X i ) i≧k  such that for any i≧k,X i+1 =g i  (X i ,S i+1 ) and Π k,i+1,n   0,+/− (ƒ)(S k+1 ; . . . ; S i+1 )=Π k,i+1,n   0,+/− (ƒ)(X i+1 ). 
         [0088]    Let us also suppose 
         [0000]    
       
         
           
             
               ( 
               
                 
                   B 
                   
                     i 
                     + 
                     1 
                   
                 
                  
                 
                   ( 
                   
                     K 
                     , 
                     
                       S 
                       
                         i 
                         + 
                         1 
                       
                     
                   
                   ) 
                 
               
               ) 
             
             
               K 
               ∈ 
               
                 I 
                 
                   S 
                   
                     i 
                     + 
                     1 
                   
                 
               
             
           
         
       
     
         [0000]    is the list of payout payment functions of the selected BIC set at time t i+1  and there exists a sequence of functions h i  and a vector sequence (Y i ) i≧k  such that for any i≧k,Y i+1 =h i  (X i , S i+1 ) and 
         [0000]    
       
         
           
             
               
                 ( 
                 
                   
                     Π 
                     
                       k 
                       , 
                       i 
                     
                     
                       0 
                       , 
                       
                         + 
                         
                           / 
                           - 
                         
                       
                     
                   
                    
                   
                     ( 
                     
                       
                         B 
                         
                           i 
                           + 
                           1 
                         
                       
                        
                       
                         ( 
                         
                           K 
                           , 
                           
                             S 
                             
                               i 
                               + 
                               1 
                             
                           
                         
                         ) 
                       
                     
                     ) 
                   
                 
                 ) 
               
               
                 K 
                 ∈ 
                 
                   I 
                   
                     S 
                     
                       i 
                       + 
                       1 
                     
                   
                 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       h 
                       i 
                     
                      
                     
                       ( 
                       
                         K 
                         , 
                         
                           Y 
                           i 
                         
                       
                       ) 
                     
                   
                   ) 
                 
                 
                   K 
                   ∈ 
                   
                     I 
                     
                       S 
                       
                         i 
                         + 
                         1 
                       
                     
                   
                 
               
               . 
             
           
         
       
     
         [0089]    We note 
         [0000]    
       
         
           
             
               ( 
               
                 
                   ϒ 
                   
                     k 
                     , 
                     
                       i 
                       + 
                       1 
                     
                     , 
                     n 
                   
                   
                     f 
                     , 
                     0 
                     , 
                     
                       + 
                       
                         / 
                         - 
                       
                     
                   
                 
                  
                 
                   ( 
                   K 
                   ) 
                 
               
               ) 
             
             
               K 
               ∈ 
               
                 I 
                 
                   S 
                   
                     i 
                     + 
                     1 
                   
                 
               
             
           
         
       
     
         [0000]    the list of coordinates of Π k,i+1,n   0,+/− (ƒ)(S k+1 ; . . . ; S i+1 ) in the BIC set 
         [0000]    
       
         
           
             
               
                 ( 
                 
                   
                     B 
                     
                       i 
                       + 
                       1 
                     
                   
                    
                   
                     ( 
                     
                       K 
                       , 
                       
                         S 
                         
                           i 
                           + 
                           1 
                         
                       
                     
                     ) 
                   
                 
                 ) 
               
               
                 K 
                 ∈ 
                 
                   I 
                   
                     S 
                     
                       i 
                       + 
                       1 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0090]    Then Π k,i,n   0,+/− (ƒ) can be reduced to a function of X i  and Y i  or a a function of X i−1 , Y i−1  and S i . 
       Proof 
       [0091]    
       
         
           
             
               
                 
                   Π 
                   
                     k 
                     , 
                     i 
                     , 
                     n 
                   
                   
                     0 
                     , 
                     
                       + 
                       
                         / 
                         - 
                       
                     
                   
                 
                  
                 
                   ( 
                   f 
                   ) 
                 
               
                
               
                 ( 
                 
                   
                     S 
                     
                       k 
                       + 
                       1 
                     
                   
                   ; 
                   … 
                    
                   
                       
                   
                   ; 
                   
                     S 
                     i 
                   
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   K 
                   ∈ 
                   
                     I 
                     
                       S 
                       
                         i 
                         + 
                         1 
                       
                     
                   
                 
               
                
               
                 
                   
                     ϒ 
                     
                       k 
                       , 
                       
                         i 
                         + 
                         1 
                       
                       , 
                       n 
                     
                     
                       f 
                       , 
                       0 
                       , 
                       
                         + 
                         
                           / 
                           - 
                         
                       
                     
                   
                    
                   
                     ( 
                     
                       
                         S 
                         
                           k 
                           + 
                           1 
                         
                       
                       ; 
                       … 
                        
                       
                           
                       
                       ; 
                       K 
                     
                     ) 
                   
                 
                 × 
                 
                   
                     Π 
                     
                       k 
                       , 
                       i 
                     
                     
                       0 
                       , 
                       
                         + 
                         
                           / 
                           - 
                         
                       
                     
                   
                    
                   
                     ( 
                     
                       
                         B 
                         
                           i 
                           + 
                           1 
                         
                       
                        
                       
                         ( 
                         
                           K 
                           , 
                           
                             S 
                             
                               i 
                               + 
                               1 
                             
                           
                         
                         ) 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   ϒ 
                   
                     k 
                     , 
                     
                       i 
                       + 
                       1 
                     
                     , 
                     n 
                   
                   
                     f 
                     , 
                     0 
                     , 
                     
                       + 
                       
                         / 
                         - 
                       
                     
                   
                 
                  
                 
                   ( 
                   
                     
                       S 
                       
                         k 
                         + 
                         1 
                       
                     
                     ; 
                     … 
                      
                     
                         
                     
                     ; 
                     K 
                   
                   ) 
                 
               
                
               
                   
               
                
               can 
                
               
                   
               
                
               also 
                
               
                   
               
                
               be 
                
               
                   
               
                
               written 
                
               
                   
               
                
               as 
                
               
                   
               
                
               
                 
                   ϒ 
                   
                     k 
                     , 
                     
                       i 
                       + 
                       1 
                     
                     , 
                     n 
                   
                   
                     f 
                     , 
                     0 
                     , 
                     
                       + 
                       
                         / 
                         - 
                       
                     
                   
                 
                  
                 
                   ( 
                   
                     
                       g 
                       i 
                     
                      
                     
                       ( 
                       
                         
                           X 
                           i 
                         
                         , 
                         K 
                       
                       ) 
                     
                   
                   ) 
                 
               
             
             , 
             
               so 
                
               
                 : 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   
                     
                       
                         
                           Π 
                           
                             k 
                             , 
                             i 
                             , 
                             n 
                           
                           
                             0 
                             , 
                             
                               + 
                               
                                 / 
                                 - 
                               
                             
                           
                         
                          
                         
                           ( 
                           f 
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           
                             S 
                             
                               k 
                               + 
                               1 
                             
                           
                           ; 
                           … 
                            
                           
                               
                           
                           ; 
                           
                             S 
                             i 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           K 
                           ∈ 
                           
                             I 
                             
                               S 
                               
                                 i 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                        
                       
                         
                           
                             ϒ 
                             
                               k 
                               , 
                               
                                 i 
                                 + 
                                 1 
                               
                               , 
                               n 
                             
                             
                               f 
                               , 
                               0 
                               , 
                               
                                 + 
                                 
                                   / 
                                   - 
                                 
                               
                             
                           
                            
                           
                             ( 
                             
                               
                                 g 
                                 i 
                               
                                
                               
                                 ( 
                                 
                                   
                                     X 
                                     i 
                                   
                                   , 
                                   K 
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                         × 
                         
                           
                             h 
                             i 
                           
                            
                           
                             ( 
                             
                               K 
                               , 
                               
                                 Y 
                                 i 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     = 
                     
                       
                         
                           Π 
                           
                             k 
                             , 
                             i 
                             , 
                             n 
                           
                           
                             0 
                             , 
                             
                               + 
                               
                                 / 
                                 - 
                               
                             
                           
                         
                          
                         
                           ( 
                           f 
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           
                             X 
                             i 
                           
                           , 
                           
                             Y 
                             i 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   Π 
                   
                     k 
                     , 
                     i 
                     , 
                     n 
                   
                   
                     0 
                     , 
                     
                       + 
                       
                         / 
                         - 
                       
                     
                   
                 
                  
                 
                   ( 
                   f 
                   ) 
                 
               
                
               
                 ( 
                 
                   
                     S 
                     
                       k 
                       + 
                       1 
                     
                   
                   ; 
                   … 
                    
                   
                       
                   
                   ; 
                   
                     S 
                     i 
                   
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   K 
                   ∈ 
                   
                     I 
                     
                       S 
                       
                         i 
                         + 
                         1 
                       
                     
                   
                 
               
                
               
                 
                   
                     ϒ 
                     
                       k 
                       , 
                       
                         i 
                         + 
                         1 
                       
                       , 
                       n 
                     
                     
                       f 
                       , 
                       0 
                       , 
                       
                         + 
                         
                           / 
                           - 
                         
                       
                     
                   
                    
                   
                     ( 
                     
                       
                         g 
                         i 
                       
                        
                       
                         ( 
                         
                           
                             
                               g 
                               
                                 i 
                                 - 
                                 1 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   X 
                                   
                                     i 
                                     - 
                                     1 
                                   
                                 
                                 , 
                                 
                                   S 
                                   i 
                                 
                               
                               ) 
                             
                           
                           , 
                           K 
                         
                         ) 
                       
                     
                     ) 
                   
                 
                 × 
                 
                   
                     h 
                     i 
                   
                    
                   
                     ( 
                     
                       K 
                       , 
                       
                         
                           h 
                           
                             i 
                             - 
                             1 
                           
                         
                          
                         
                           ( 
                           
                             
                               Y 
                               
                                 i 
                                 - 
                                 1 
                               
                             
                             , 
                             
                               S 
                               i 
                             
                           
                           ) 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   
                     Π 
                     
                       k 
                       , 
                       i 
                       , 
                       n 
                     
                     
                       0 
                       , 
                       
                         + 
                         
                           / 
                           - 
                         
                       
                     
                   
                    
                   
                     ( 
                     f 
                     ) 
                   
                 
                  
                 
                   ( 
                   
                     
                       S 
                       
                         k 
                         + 
                         1 
                       
                     
                     ; 
                     … 
                      
                     
                         
                     
                     ; 
                     
                       S 
                       i 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   
                     Π 
                     
                       k 
                       , 
                       i 
                       , 
                       n 
                     
                     
                       0 
                       , 
                       
                         + 
                         
                           / 
                           - 
                         
                       
                     
                   
                    
                   
                     ( 
                     f 
                     ) 
                   
                 
                  
                 
                   ( 
                   
                     
                       X 
                       
                         i 
                         - 
                         1 
                       
                     
                     , 
                     
                       Y 
                       
                         i 
                         - 
                         1 
                       
                     
                     , 
                     
                       S 
                       i 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0092]    The World Series baseball illustration of chapter IV is a prime example where the use of such auxiliary variables provide a noticeable computational simplification. Indeed, we write the payoff there as: 
         [0000]    
       
         
           
             
               
                 f 
                  
                 
                   ( 
                   
                     
                       X 
                       
                         t 
                         1 
                       
                     
                     , 
                     … 
                      
                     
                         
                     
                     , 
                     
                       X 
                       
                         t 
                         7 
                       
                     
                   
                   ) 
                 
               
               = 
               
                 
                   1 
                   
                     { 
                     
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             1 
                           
                           7 
                         
                          
                         
                           X 
                           
                             t 
                             j 
                           
                         
                       
                       ≥ 
                       4 
                     
                     } 
                   
                 
                 = 
                 
                   1 
                   
                     { 
                     
                       
                         
                           Y 
                           
                             t 
                             6 
                           
                         
                         + 
                         
                           X 
                           
                             t 
                             7 
                           
                         
                       
                       ≥ 
                       4 
                     
                     } 
                   
                 
               
             
              
             
                 
             
           
         
       
       
         
           
             
               with 
                
               
                   
               
                
               
                 Y 
                 
                   t 
                   
                     n 
                     + 
                     1 
                   
                 
               
             
             = 
             
               
                 
                   g 
                   n 
                 
                  
                 
                   ( 
                   
                     
                       Y 
                       
                         t 
                         n 
                       
                     
                     , 
                     
                       X 
                       
                         t 
                         
                           n 
                           + 
                           1 
                         
                       
                     
                   
                   ) 
                 
               
               = 
               
                 
                   Y 
                   
                     t 
                     n 
                   
                 
                 + 
                 
                   X 
                   
                     t 
                     
                       n 
                       + 
                       1 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 for 
                  
                 
                     
                 
                  
                 n 
               
               = 
               0 
             
             , 
             … 
              
             
                 
             
             , 
             6 
           
         
       
     
         [0093]    A few examples of payout payment functions of some well-known European derivative contracts further illustrate how these auxiliary variables can be identified naturally and, pending specification of the BICs premium payment functions, these are: 
       EXAMPLE  
     Sample Reduced Form Derivatives Payouts 
       [0094]    1. Double Barrier Option, with lower barrier L and upper barrier H, 
         [0000]    
       
         
           
             
                 
             
              
             
               
                 
                   f 
                   0 
                   n 
                 
                  
                 
                   ( 
                   
                     
                       S 
                       0 
                     
                     ; 
                     … 
                      
                     
                         
                     
                     ; 
                     
                       S 
                       n 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   
                     ( 
                     
                       δ 
                        
                       
                         ( 
                         
                           
                             S 
                             n 
                           
                           - 
                           
                             K 
                             _ 
                           
                         
                         ) 
                       
                     
                     ) 
                   
                   + 
                 
                  
                 
                   1 
                   
                     { 
                     
                       L 
                       &lt; 
                       
                         S 
                         n 
                       
                       &lt; 
                       H 
                     
                     } 
                   
                 
                 × 
                 … 
                 × 
                 
                   1 
                   
                     { 
                     
                       L 
                       &lt; 
                       
                         S 
                         1 
                       
                       &lt; 
                       H 
                     
                     } 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   f 
                   0 
                   n 
                 
                  
                 
                   ( 
                   
                     
                       S 
                       0 
                     
                     ; 
                     … 
                      
                     
                         
                     
                     ; 
                     
                       S 
                       n 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   
                     ( 
                     
                       δ 
                        
                       
                         ( 
                         
                           
                             S 
                             n 
                           
                           - 
                           
                             K 
                             _ 
                           
                         
                         ) 
                       
                     
                     ) 
                   
                   + 
                 
                  
                 
                   1 
                   
                     { 
                     
                       L 
                       &lt; 
                       
                         S 
                         n 
                       
                       &lt; 
                       H 
                     
                     } 
                   
                 
                 × 
                 
                   1 
                   
                     { 
                     
                       
                         Max 
                          
                         
                           { 
                           
                             
                               X 
                               
                                 n 
                                 - 
                                 1 
                               
                               1 
                             
                             , 
                             
                               S 
                               n 
                             
                           
                           } 
                         
                       
                       &lt; 
                       H 
                     
                     } 
                   
                 
                  
                 
                   1 
                   
                     { 
                     
                       L 
                       &lt; 
                       
                         Min 
                          
                         
                           { 
                           
                             
                               X 
                               
                                 n 
                                 - 
                                 1 
                               
                               2 
                             
                             , 
                             
                               S 
                               n 
                             
                           
                           } 
                         
                       
                     
                     } 
                   
                 
               
             
             , 
             
               
 
             
              
             
                 
             
              
             with 
           
         
       
       
         
           
             
                 
             
              
             
               
                 X 
                 
                   i 
                   + 
                   1 
                 
               
               = 
               
                 
                   ( 
                   
                     
                       
                         
                           X 
                           
                             i 
                             + 
                             1 
                           
                           1 
                         
                       
                     
                     
                       
                         
                           X 
                           
                             i 
                             + 
                             1 
                           
                           2 
                         
                       
                     
                   
                   ) 
                 
                 = 
                 
                   
                     ( 
                     
                       
                         
                           
                             
                               Max 
                               
                                 0 
                                 ≤ 
                                 k 
                                 ≤ 
                                 n 
                               
                             
                              
                             
                               S 
                               k 
                             
                           
                         
                       
                       
                         
                           
                             
                               Min 
                               
                                 0 
                                 ≤ 
                                 k 
                                 ≤ 
                                 n 
                               
                             
                              
                             
                               S 
                               k 
                             
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             Max 
                              
                             
                               { 
                               
                                 
                                   X 
                                   i 
                                   1 
                                 
                                 , 
                                 
                                   S 
                                   
                                     i 
                                     + 
                                     1 
                                   
                                 
                               
                               } 
                             
                           
                         
                       
                       
                         
                           
                             Min 
                              
                             
                               { 
                               
                                 
                                   X 
                                   i 
                                   2 
                                 
                                 , 
                                 
                                   S 
                                   
                                     i 
                                     + 
                                     1 
                                   
                                 
                               
                               } 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0095]    2. Asian Option 
         [0000]    
       
         
           
             , 
             
               
                 
                   f 
                   0 
                   n 
                 
                  
                 
                   ( 
                   
                     
                       S 
                       0 
                     
                     ; 
                     … 
                      
                     
                         
                     
                     ; 
                     
                       S 
                       n 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   
                     ( 
                     
                       δ 
                        
                       
                         ( 
                         
                           
                             
                               1 
                               n 
                             
                              
                             
                               ( 
                               
                                 
                                   ∑ 
                                   
                                     k 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                   S 
                                   k 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             K 
                             _ 
                           
                         
                         ) 
                       
                     
                     ) 
                   
                   + 
                 
                 = 
                 
                   
                     ( 
                     
                       δ 
                        
                       
                         ( 
                         
                           
                             
                               1 
                               n 
                             
                              
                             
                               ( 
                               
                                 
                                   X 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                 
                                 + 
                                 
                                   S 
                                   n 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             K 
                             _ 
                           
                         
                         ) 
                       
                     
                     ) 
                   
                   + 
                 
               
             
             , 
           
         
       
     
         [0000]    with 
         [0000]    
       
         
           
             
               
                 X 
                 
                   i 
                   + 
                   1 
                 
               
               = 
               
                 
                   
                     ∑ 
                     
                       k 
                       = 
                       1 
                     
                     
                       i 
                       + 
                       1 
                     
                   
                    
                   
                     S 
                     k 
                   
                 
                 = 
                 
                   
                     X 
                     i 
                   
                   + 
                   
                     S 
                     
                       i 
                       + 
                       1 
                     
                   
                 
               
             
             ; 
           
         
       
     
         [0096]    3. Lookback Option 
         [0000]    
       
         
           
             , 
             
               
                 
                   f 
                   0 
                   n 
                 
                  
                 
                   ( 
                   
                     
                       S 
                       0 
                     
                     ; 
                     … 
                      
                     
                         
                     
                     ; 
                     
                       S 
                       n 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   ( 
                   
                     
                       
                         Max 
                         
                           1 
                           ≤ 
                           k 
                           ≤ 
                           ≤ 
                           n 
                         
                       
                        
                       
                         S 
                         k 
                       
                     
                     - 
                     
                       
                         K 
                         _ 
                       
                       n 
                     
                   
                   ) 
                 
                 = 
                 
                   ( 
                   
                     
                       Max 
                        
                       
                         { 
                         
                           
                             X 
                             
                               n 
                               - 
                               1 
                             
                           
                           , 
                           
                             S 
                             n 
                           
                         
                         } 
                       
                     
                     - 
                     
                       
                         K 
                         _ 
                       
                       n 
                     
                   
                   ) 
                 
               
             
             , 
           
         
       
     
         [0000]    with X i+1 =Max {X i ,S i+1 }; or 
         [0000]    
       
         
           
             
               
                 f 
                 0 
                 n 
               
                
               
                 ( 
                 
                   
                     S 
                     0 
                   
                   ; 
                   … 
                    
                   
                       
                   
                   ; 
                   
                     S 
                     n 
                   
                 
                 ) 
               
             
             = 
             
               
                 ( 
                 
                   
                     K 
                     n 
                   
                   - 
                   
                     
                       Min 
                       
                         1 
                         ≤ 
                         k 
                         ≤ 
                         ≤ 
                         n 
                       
                     
                      
                     
                       S 
                       k 
                     
                   
                 
                 ) 
               
               = 
               
                 ( 
                 
                   
                     K 
                     n 
                   
                   - 
                   
                     Min 
                      
                     
                       { 
                       
                         
                           X 
                           
                             n 
                             - 
                             1 
                           
                         
                         , 
                         
                           S 
                           n 
                         
                       
                       } 
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               
                 with 
                  
                 
                     
                 
                  
                 
                   X 
                   
                     i 
                     + 
                     1 
                   
                 
               
               = 
               
                 Min 
                  
                 
                   { 
                   
                     
                       X 
                       i 
                     
                     , 
                     
                       S 
                       
                         i 
                         + 
                         1 
                       
                     
                   
                   } 
                 
               
             
             ; 
           
         
       
     
         [0097]    4.Volatility Swap, 
         [0000]    
       
         
           
             
               
                 
                   f 
                   0 
                   n 
                 
                  
                 
                   ( 
                   
                     
                       S 
                       0 
                     
                     ; 
                     … 
                      
                     
                         
                     
                     ; 
                     
                       S 
                       n 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   
                     
                       252 
                       
                         n 
                         - 
                         1 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         
                           ( 
                           
                             
                               Log 
                                
                               
                                 [ 
                                 
                                   
                                     S 
                                     k 
                                   
                                   
                                     S 
                                     
                                       k 
                                       - 
                                       1 
                                     
                                   
                                 
                                 ] 
                               
                             
                             - 
                             
                               
                                 1 
                                 n 
                               
                                
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     n 
                                   
                                    
                                   
                                     Log 
                                      
                                     
                                       [ 
                                       
                                         
                                           S 
                                           i 
                                         
                                         
                                           S 
                                           
                                             i 
                                             - 
                                             1 
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
                 - 
                 
                   K 
                   _ 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   f 
                   0 
                   n 
                 
                  
                 
                   ( 
                   
                     
                       X 
                       
                         n 
                         - 
                         1 
                       
                     
                     , 
                     
                       S 
                       
                         n 
                         - 
                         1 
                       
                     
                     , 
                     
                       S 
                       n 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   
                     
                       252 
                       
                         n 
                         - 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           X 
                           
                             n 
                             - 
                             1 
                           
                         
                         + 
                         
                           
                             Log 
                              
                             
                               [ 
                               
                                 
                                   S 
                                   n 
                                 
                                 
                                   S 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                 
                               
                               ] 
                             
                           
                           2 
                         
                         + 
                         
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   2 
                                    
                                   n 
                                 
                               
                               ) 
                             
                             
                               n 
                               2 
                             
                           
                            
                           
                             
                               Log 
                                
                               
                                 [ 
                                 
                                   
                                     S 
                                     n 
                                   
                                   
                                     S 
                                     0 
                                   
                                 
                                 ] 
                               
                             
                             2 
                           
                         
                       
                       ) 
                     
                   
                 
                 - 
                 
                   K 
                   _ 
                 
               
             
             , 
             
               
 
             
              
             
                 
             
              
             
               
                 
                   with 
                    
                   
                       
                   
                    
                   
                     X 
                     
                       i 
                       + 
                       1 
                     
                   
                 
                 = 
                 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       
                         i 
                         + 
                         1 
                       
                     
                      
                     
                       Log 
                        
                       
                         
                           { 
                           
                             
                               S 
                               k 
                             
                             
                               S 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                           ] 
                         
                         2 
                       
                     
                   
                   = 
                   
                     
                       X 
                       i 
                     
                     + 
                     
                       
                         Log 
                          
                         
                           [ 
                           
                             
                               S 
                               
                                 i 
                                 + 
                                 1 
                               
                             
                             
                               S 
                               i 
                             
                           
                           ] 
                         
                       
                       2 
                     
                   
                 
               
               ; 
             
           
         
       
     
         [0098]    One may also see in many of the examples reviewed in Volume II BICs 4 Derivatives Volume II: Applications how this transformation is used for derivatives pricing. This includes the pricing of American options, baseball options, CMOs, passport options, etc. In most of those examples, we see that the selected change of variable in each case reduces the number of variables to one or two. 
         [0099]    Indeed, the markovian dynamic of the underlyings as is primarily assumed in current stochastic processes based modeling assumptions implies that in the specification of the BICs premium payment functions, Y i  reduces to merely S i , providing a very desirable level of simplifications. The derivatives contract premium payment function at each trading time ti reduces to a function of X i−1  and S i . 
         [0100]    Dimension reduction by change of variables is a core principle for reducing computational costs. The Girsanov Theorem and its applications for change of measure, which provide the means for various computation reductions for derivatives pricing in Gaussian processes is indeed a change of variable to facilitate integrations. 
       3.2. General Approach For Computational Tractability By Projection On Appropriate Representation Spaces 
     3.2.1. Problem Statement 
       [0101]    Viewed with the information of time t k , k&lt;i, Π k,i,n   0,+/− (ƒ) is in general a function of S k+1 , . . . S i ,Π k,i,n   0,+/− (ƒ)(S k+1 , . . . S i ). 
         [0102]    When there are opportunities to introduce auxiliary variables as introduced in the previous section, we have Π k,i,n   0,+/− (ƒ)(S k+1 , . . . S i )=Π k,i,n   0,+/− (ƒ)(X i−1 , Y i−1 ,S i ). 
         [0103]    When the dynamic of the underlying is markovian, we have: Π k,i,n   0,+/− (ƒ)(S k+1 , . . . S i )=Π k,i,n   0,+/− (ƒ)(X i−1 ,S i ). 
         [0104]    In each of these cases, for the function to be fully described, one must have, readily available, the realized value of the function for each possible value of its variables. 
         [0105]    Hence, in the iterative relationship of chapter V, when we write, 
         [0000]    
       
         
           
             
               
                 
                   Π 
                   
                     k 
                     , 
                     i 
                     , 
                     n 
                   
                   
                     0 
                     , 
                     
                       + 
                       
                         / 
                         - 
                       
                     
                   
                 
                  
                 
                   ( 
                   f 
                   ) 
                 
               
                
               
                 ( 
                 
                   
                     S 
                     
                       k 
                       + 
                       1 
                     
                   
                   ; 
                   … 
                    
                   
                       
                   
                   ; 
                   
                     S 
                     i 
                   
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   K 
                   ∈ 
                   
                     I 
                     
                       S 
                       
                         i 
                         + 
                         1 
                       
                     
                   
                 
               
                
               
                 
                   
                     ϒ 
                     
                       k 
                       , 
                       
                         i 
                         + 
                         1 
                       
                       , 
                       n 
                     
                     
                       f 
                       , 
                       0 
                       , 
                       
                         + 
                         
                           / 
                           - 
                         
                       
                     
                   
                    
                   
                     ( 
                     
                       
                         g 
                         i 
                       
                        
                       
                         ( 
                         
                           
                             
                               g 
                               
                                 i 
                                 - 
                                 1 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   X 
                                   
                                     i 
                                     - 
                                     1 
                                   
                                 
                                 , 
                                 
                                   S 
                                   i 
                                 
                               
                               ) 
                             
                           
                           , 
                           K 
                         
                         ) 
                       
                     
                     ) 
                   
                 
                 × 
                 
                   
                     h 
                     i 
                   
                    
                   
                     ( 
                     
                       K 
                       , 
                       
                         
                           h 
                           
                             i 
                             - 
                             1 
                           
                         
                          
                         
                           ( 
                           
                             
                               Y 
                               
                                 i 
                                 - 
                                 1 
                               
                             
                             , 
                             
                               S 
                               i 
                             
                           
                           ) 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0106]    We are actually making an inner product of functions. This may actually mean that at each step, the corresponding number of computations may be multiplied by the number of possible values the variables may take. 
         [0107]    For instance, if we note Cardinal (I S     i   )=S i , Cardinal (I X     i   )=x i , Cardinal (I Y     i   )=y i  and where Cardinal ( ) represents the number of elements of the set between brackets. The inner product above actually represents s i ×x i ×y i  inner products, where an inner product is actually s i+1  multiplications and s i+1 −1 additions. 
         [0108]    We have earlier proposed methods to reduce the number of variables. This will work in many cases, but in cases such as basket options and their many flavors, it is very clear that it will quickly become unpractical to compute that quantity for each possible value of S 1 ; . . . ;S i−1  and that an appropriate change of variable may not be sufficient to reduce the problem to tractability. 
         [0109]    To address such a problem, we need to select effective basis of representations that will enable reconstructions of those functions in each case merely from a few coefficients that can be computed. 
         [0110]    The goal of this section is to leverage the abundant literature on functional representations to reduce the s i ×x i   33  y i  number of inner products to a more tractable number. 
       3.2.2. Solution 
       [0111]    The general idea of our solution approach that quantitatively articulate the law of the few is as follows: 
         [0112]    Let&#39;s suppose f is a function that can be decomposed in a functional basis B=(b k ) k∈ℑ , i.e. there exist a unique vector (δ k ) k∈ℑ  such that 
         [0000]    
       
         
           
             
               f 
               = 
               
                 
                   ∑ 
                   
                     k 
                     ∈ 
                      
                   
                 
                  
                 
                   
                     δ 
                     k 
                   
                    
                   
                     b 
                     k 
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where ℑ is a finite or infinite indexing set, such as the set of integers or non negative integers. 
         [0113]    For any given error tolerance level ε, the basis selection goal is to choose B such that there exist a finite subset of the indexing set ℑ s ⊂ℑ small enough such that for any function f belonging to the set of functions of interest, and for a given norm of reference [[]], we have: 
         [0000]    
       
         
           
             
                
               
                 f 
                 - 
                 
                   
                     ∑ 
                     
                       k 
                       ∈ 
                       
                          
                         s 
                       
                       ⋐ 
                        
                     
                   
                    
                   
                     
                       δ 
                       k 
                     
                      
                     
                       b 
                       k 
                     
                   
                 
               
                
             
             &lt; 
             ɛ 
           
         
       
     
         [0114]    We approach this solution with the restatement of a classic result of quadratic optimization. 
       Proposition 2  
       [0115]    Let us assume the function f is to be approximated by f 1 (X), . . . , f n (X) for integration purposes. 
         [0116]    Let us also suppose that none of the f i  can be obtained as a linear combination of the others, i.e. the functions are linearly independent. 
         [0117]    Let&#39;s Call Δ the definition set of X and μ a Measure on a field on X. 
         [0118]    We note 
         [0000]    
       
         
           
             
               
                 a 
                 
                   i 
                   , 
                   j 
                 
               
               = 
               
                 
                   ∫ 
                   Δ 
                 
                  
                 
                   
                     
                       f 
                       i 
                     
                      
                     
                       ( 
                       X 
                       ) 
                     
                   
                    
                   
                     
                       f 
                       j 
                     
                      
                     
                       ( 
                       X 
                       ) 
                     
                   
                    
                   
                      
                     
                       μ 
                        
                       
                         ( 
                         X 
                         ) 
                       
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 b 
                 j 
               
               = 
               
                 
                   ∫ 
                   Δ 
                 
                  
                 
                   
                     
                       f 
                       j 
                     
                      
                     
                       ( 
                       X 
                       ) 
                     
                   
                    
                   
                     f 
                      
                     
                       ( 
                       X 
                       ) 
                     
                   
                    
                   
                      
                     
                       μ 
                        
                       
                         ( 
                         X 
                         ) 
                       
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               A 
               = 
               
                 
                   ( 
                   
                     a 
                     
                       i 
                       , 
                       j 
                     
                   
                   ) 
                 
                 
                   
                     1 
                     ≤ 
                     i 
                     ≤ 
                     n 
                   
                   
                     1 
                     ≤ 
                     j 
                     ≤ 
                     n 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               B 
               = 
               
                 
                   ( 
                   
                     b 
                     j 
                   
                   ) 
                 
                 
                   1 
                   ≤ 
                   j 
                   ≤ 
                   n 
                 
               
             
             ; 
           
         
       
     
         [0119]    Let&#39;s call α f  the vector of coefficients approximating f in the basis of f 1 (X), . . . ,f n (X). 
         [0120]    Then, with respect to the norm 
         [0000]    
       
         
           
             
               
                 
                    
                   f 
                    
                 
                 2 
               
               = 
               
                 
                   
                     ∫ 
                     Δ 
                   
                    
                   
                     
                       
                         f 
                         2 
                       
                        
                       
                         ( 
                         X 
                         ) 
                       
                     
                      
                     
                        
                       
                         μ 
                          
                         
                           ( 
                           X 
                           ) 
                         
                       
                     
                   
                 
               
             
             , 
             
               
 
             
              
             
               
                 
                   
                     Arg 
                      
                     
                         
                     
                      
                     Min 
                   
                   
                     
                       ( 
                       
                         α 
                         j 
                         f 
                       
                       ) 
                     
                     
                       j 
                       = 
                       
                         1 
                          
                         
                             
                         
                          
                         … 
                          
                         
                             
                         
                          
                         n 
                       
                     
                   
                 
                  
                 
                   
                      
                     
                       f 
                       - 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           
                             α 
                             j 
                             f 
                           
                            
                           
                             f 
                             j 
                           
                         
                       
                     
                      
                   
                   2 
                 
               
               = 
               
                 
                   A 
                   
                     - 
                     1 
                   
                 
                  
                 B 
               
             
           
         
       
       
         
           
             with 
              
             
               : 
             
           
         
       
       
         
           
             
               
                 Min 
                 
                   
                     ( 
                     
                       α 
                       j 
                       f 
                     
                     ) 
                   
                   
                     j 
                     = 
                     
                       1 
                        
                       
                           
                       
                        
                       … 
                        
                       
                           
                       
                        
                       n 
                     
                   
                 
               
                
               
                 
                    
                   
                     f 
                     - 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         
                           α 
                           j 
                           f 
                         
                          
                         
                           f 
                           j 
                         
                       
                     
                   
                    
                 
                 2 
                 2 
               
             
             = 
             
               
                 
                    
                   f 
                    
                 
                 2 
                 2 
               
               - 
               
                 
                   BA 
                     
                   
                     - 
                     1 
                   
                     
                     
                   t 
                 
                  
                 B 
               
             
           
         
       
     
         [0121]    Proof. The proof is trivial by noting that the gradient vector equals zero only at the single point A −1 B. Then one verifies that the Hessian A is symmetric positive definite at that point 
         [0122]    When one needs to represent a function and compress the amount of information about the function that needs to be stored or transmitted, what comes immediately to the trained engineer&#39;s mind are Fourier Series representation, Wavelets series representation, Polynomials or Hermite Polynomials. 
         [0123]    These methods have been tried and tested in many fields of engineering and perform well in these settings. Fourier series representations are particularly effective when the functions under consideration are very smooth. In a discrete space representation, the COR-IP statement means we can extend the function in a continuous space to any level of regularity desired. Our presentation here did not limit itself to functions that form a Hilbertian Basis because we may have cases where we have an intuition on the shape of functions that may approximately reconstruct the functions at hand. Such functions may not constitute a Hilbert Basis, yet may help achieve better compression. In those instances, the formula above, where we implicitly orthogonalize the basis functions, may prove very helpful. 
         [0124]    Let&#39;s now review an example that gives us the opportunity to see change of variables and functional projections in action, the case of volatility swaps 
       3.2.3. Application—Example: Volatility And Moments Derivatives 
       [0125]    Let us suppose our goal is to apply the above functional representation approach with effective dimension reduction by variable change to price volatility derivatives. 
         [0126]    We recall that the payoff at payout payment time of the volatility swap contract is generally described with the following representation: 
         [0127]    Volatility Swap, 
         [0000]    
       
         
           
             
               
                 
                   f 
                   n 
                   0 
                 
                  
                 
                   ( 
                   
                     
                       S 
                       0 
                     
                     ; 
                     … 
                      
                     
                         
                     
                     ; 
                     
                       S 
                       n 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   
                     
                       252 
                       
                         n 
                         - 
                         1 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         
                           ( 
                           
                             
                               Log 
                                
                               
                                 [ 
                                 
                                   
                                     S 
                                     k 
                                   
                                   
                                     S 
                                     
                                       k 
                                       - 
                                       1 
                                     
                                   
                                 
                                 ] 
                               
                             
                             - 
                             
                               
                                 1 
                                 n 
                               
                                
                               
                                 ( 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     n 
                                   
                                    
                                   
                                     Log 
                                      
                                     
                                       [ 
                                       
                                         
                                           S 
                                           i 
                                         
                                         
                                           S 
                                           
                                             i 
                                             - 
                                             1 
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
                 - 
                 
                   K 
                   _ 
                 
               
             
             ; 
           
         
       
     
         [0000]    we note: 
         [0000]    
       
         
           
             
               
                 V 
                 n 
               
                
               
                   
               
                
               • 
                
               
                 
                   ∑ 
                   
                     k 
                     = 
                     1 
                   
                   n 
                 
                  
                 
                   
                     ( 
                     
                       
                         Log 
                          
                         
                           [ 
                           
                             
                               S 
                               k 
                             
                             
                               S 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                           ] 
                         
                       
                       - 
                       
                         
                           1 
                           n 
                         
                          
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               Log 
                                
                               
                                 [ 
                                 
                                   
                                     S 
                                     i 
                                   
                                   
                                     S 
                                     
                                       i 
                                       - 
                                       1 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                           ) 
                         
                       
                     
                     ) 
                   
                   2 
                 
               
             
             ; 
           
         
       
     
         [0128]    This allows us to propose the following clarifying definition 
       Definition 
       [0129]    A volatility derivatives contract is a derivatives contract whose payout payment function is a function of the realized values of underlying expressed as a function of the function Vn of the underlyings. 
         [0130]    We focus first on effective dimension reduction by variables change. 
       Lemma 1 
       [0131]    
       
         
           
             
               
                 V 
                 n 
               
               = 
               
                 
                   M 
                   n 
                   2 
                 
                 - 
                 
                   
                     1 
                     n 
                   
                    
                   
                     
                       ( 
                       
                         M 
                         n 
                         1 
                       
                       ) 
                     
                     2 
                   
                 
               
             
             , 
             
               
 
             
              
             with 
           
         
       
       
         
           
             
               
                 X 
                 i 
               
                
               
                   
               
                
               • 
                
               
                   
               
                
               
                 Log 
                  
                 
                   [ 
                   
                     
                       S 
                       i 
                     
                     
                       S 
                       
                         i 
                         - 
                         1 
                       
                     
                   
                   ] 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 M 
                 k 
                 
                   1 
                    
                   
                       
                   
                 
               
                
               
                   
               
                
               • 
                
               
                   
               
                
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   k 
                 
                  
                 
                   X 
                   i 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 M 
                 k 
                 
                   2 
                    
                   
                       
                   
                 
               
                
               
                   
               
                
               • 
                
               
                   
               
                
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   k 
                 
                  
                 
                   X 
                   i 
                   2 
                 
               
             
             ; 
           
         
       
     
       Proof 
       [0132]    
       
         
           
             
               
                 
                   
                     V 
                     n 
                   
                    
                   
                       
                   
                    
                   • 
                    
                   
                       
                   
                    
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                       
                         ( 
                         
                           
                             Log 
                              
                             
                               [ 
                               
                                 
                                   S 
                                   k 
                                 
                                 
                                   S 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                               
                               ] 
                             
                           
                           - 
                           
                             
                               1 
                               n 
                             
                              
                             
                               ( 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   n 
                                 
                                  
                                 
                                   Log 
                                    
                                   
                                     [ 
                                     
                                       
                                         S 
                                         i 
                                       
                                       
                                         S 
                                         
                                           i 
                                           - 
                                           1 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
                 ; 
               
               = 
               
                 
                   
                     ∑ 
                     
                       k 
                       = 
                       1 
                     
                     n 
                   
                    
                   
                     ( 
                     
                       
                         
                           Log 
                            
                           
                             [ 
                             
                               
                                 S 
                                 k 
                               
                               
                                 S 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                             
                             ] 
                           
                         
                         2 
                       
                       - 
                       
                         
                           2 
                           n 
                         
                          
                         
                           Log 
                            
                           
                             [ 
                             
                               
                                 S 
                                 k 
                               
                               
                                 S 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                             
                             ] 
                           
                         
                          
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               Log 
                                
                               
                                 [ 
                                 
                                   
                                     S 
                                     i 
                                   
                                   
                                     S 
                                     
                                       i 
                                       - 
                                       1 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           1 
                           
                             n 
                             2 
                           
                         
                          
                         
                           
                             ( 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                 Log 
                                  
                                 
                                   [ 
                                   
                                     
                                       S 
                                       i 
                                     
                                     
                                       S 
                                       
                                         i 
                                         - 
                                         1 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                     ) 
                   
                 
                 = 
                 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                       
                         Log 
                          
                         
                           [ 
                           
                             
                               S 
                               k 
                             
                             
                               S 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                           
                           ] 
                         
                       
                       2 
                     
                   
                   - 
                   
                     
                       1 
                       n 
                     
                      
                     
                       
                         ( 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             n 
                           
                            
                           
                             Log 
                              
                             
                               [ 
                               
                                 
                                   S 
                                   i 
                                 
                                 
                                   S 
                                   
                                     i 
                                     - 
                                     1 
                                   
                                 
                               
                               ] 
                             
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
               
             
             ; 
           
         
       
     
       3.2.3. 1.From the Computationally Impossible To the Computationally Possible Through the Use of Summarizing Variables 
       [0133]    We move from the computationally impossible to the computationally possible in the pricing and hedging of volatility derivatives contracts through the BICs analysis. This is done by showing that the variables M k   1  and M k   2  can be used as the function upon which at each stage the premium payment amounts depend, especially in the case of underlyings with independent increments such as those driven by Levy processes. 
         [0134]    Once the dependence on (M k   1 ,M k   2 k) at time tk established, we estimate how large the set of values spanned is. 
       States Spanned By (M k   1 ,M k   2 ) 
       [0135]    A few considerations, 
         [0000]    
       
         
           
             
               
                 Δ 
                 i 
                 x 
               
                
               
                   
               
                
               • 
                
               
                   
               
                
               
                 
                   
                     h 
                     i 
                   
                   - 
                   
                     l 
                     i 
                   
                 
                 
                   2 
                    
                   
                     p 
                     i 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               s 
               i 
               x 
             
             = 
             
               
                 2 
                  
                 
                   p 
                   i 
                 
               
               + 
               1 
             
           
         
       
       
         
           
             
               
                 
                   X 
                   i 
                 
                  
                 
                     
                 
                  
                 • 
                  
                 
                     
                 
                  
                 
                   
                     
                       h 
                       i 
                     
                     + 
                     
                       l 
                       i 
                     
                   
                   2 
                 
               
               + 
               
                 
                   j 
                   i 
                 
                  
                 
                   Δ 
                   i 
                   x 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 j 
                 i 
               
               ∈ 
               
                 { 
                 
                   
                     - 
                     
                       p 
                       i 
                     
                   
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   0 
                   , 
                   1 
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     p 
                     i 
                   
                 
                 } 
               
             
             = 
             
               J 
               i 
             
           
         
       
       
         
           
             
               
                 M 
                 i 
                 1 
               
               = 
               
                 
                   
                     ∑ 
                     
                       k 
                       = 
                       1 
                     
                     i 
                   
                    
                   
                     X 
                     k 
                   
                 
                 = 
                 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       i 
                     
                      
                     
                       
                         
                           h 
                           k 
                         
                         + 
                         
                           l 
                           k 
                         
                       
                       2 
                     
                   
                   + 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       i 
                     
                      
                     
                       
                         j 
                         k 
                       
                        
                       
                         Δ 
                         k 
                         x 
                       
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   
                     M 
                     i 
                     2 
                   
                   = 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       i 
                     
                      
                     
                       X 
                       k 
                       2 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       i 
                     
                      
                     
                       
                         ( 
                         
                           
                             
                               
                                 h 
                                 i 
                               
                               + 
                               
                                 l 
                                 i 
                               
                             
                             2 
                           
                           + 
                           
                             jΔ 
                             i 
                             x 
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         i 
                       
                        
                       
                         
                           ( 
                           
                             
                               
                                 h 
                                 k 
                               
                               + 
                               
                                 l 
                                 k 
                               
                             
                             2 
                           
                           ) 
                         
                         2 
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         i 
                       
                        
                       
                         
                           ( 
                           
                             
                               j 
                               k 
                             
                              
                             
                               Δ 
                               k 
                               x 
                             
                           
                           ) 
                         
                         2 
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         i 
                       
                        
                       
                         
                           j 
                           k 
                         
                          
                         
                           
                             Δ 
                             k 
                             x 
                           
                            
                           
                             ( 
                             
                               
                                 h 
                                 k 
                               
                               + 
                               
                                 l 
                                 
                                   k 
                                    
                                   
                                       
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 If 
                  
                 
                     
                 
                  
                 for 
                  
                 
                     
                 
                  
                 k 
               
               = 
               1 
             
             , 
             … 
              
             
                 
             
             , 
             n 
             , 
             
               
                 h 
                 k 
               
               = 
               
                 
                   h 
                    
                   
                       
                   
                    
                   and 
                    
                   
                       
                   
                    
                   
                     l 
                     k 
                   
                 
                 = 
                 
                   l 
                    
                   
                       
                   
                    
                   and 
                 
               
             
           
         
       
       
         
           
             
               
                 p 
                 k 
               
               = 
               p 
             
             , 
             
               
                 Δ 
                 k 
                 x 
               
               = 
               
                 
                   Δ 
                   x 
                 
                 = 
                 
                   
                     
                       
                         h 
                         - 
                         l 
                       
                       
                         2 
                          
                         p 
                       
                     
                     . 
                     
                       
 
                     
                      
                     As 
                   
                    
                   
                       
                   
                    
                   a 
                    
                   
                       
                   
                    
                   result 
                 
               
             
             , 
             
               
                 
                   M 
                   i 
                   1 
                 
                 = 
                 
                   
                     i 
                      
                     
                         
                     
                      
                     
                       
                         h 
                         + 
                         l 
                       
                       2 
                     
                   
                   + 
                   
                     
                       Δ 
                       x 
                     
                      
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         i 
                       
                        
                       
                         j 
                         k 
                       
                     
                   
                 
               
               ; 
             
           
         
       
       
         
           
             
               
                 M 
                 i 
                 2 
               
               = 
               
                 
                   
                     i 
                      
                     
                       ( 
                       
                         
                           h 
                           + 
                           l 
                         
                         2 
                       
                       ) 
                     
                   
                   2 
                 
                 + 
                 
                   
                     ( 
                     
                       h 
                       + 
                       l 
                     
                     ) 
                   
                    
                   
                     Δ 
                     x 
                   
                    
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       i 
                     
                      
                     
                       j 
                       k 
                     
                   
                 
                 + 
                 
                   
                     
                       ( 
                       
                         Δ 
                         x 
                       
                       ) 
                     
                     2 
                   
                    
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       i 
                     
                      
                     
                       j 
                       k 
                       2 
                     
                   
                 
               
             
             ; 
           
         
       
     
       Definition 2 
       [0136]    For n, p positive integers, and y integer verifying |y|≦n×p, 
         [0000]    
       
         
           
             
               
                 
                   
                     ( 
                     i 
                     ) 
                   
                    
                   
                       
                   
                    
                   
                     
                       S 
                       2 
                     
                      
                     
                       [ 
                       
                         n 
                         , 
                         p 
                       
                       ] 
                     
                   
                    
                   
                       
                   
                    
                   • 
                    
                   
                       
                   
                    
                   
                     { 
                     
                       
                         ( 
                         
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               j 
                               k 
                             
                           
                           , 
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 1 
                               
                               n 
                             
                              
                             
                               j 
                               k 
                               2 
                             
                           
                         
                         ) 
                       
                       , 
                       
                         
                            
                           
                             j 
                             k 
                           
                            
                         
                         ≤ 
                         p 
                       
                       , 
                       
                         k 
                         = 
                         1 
                       
                       , 
                       … 
                        
                       
                           
                       
                       , 
                       p 
                     
                     } 
                   
                 
                 ; 
               
                
               
                 
 
               
                
               
                 
                   
                     ( 
                     ii 
                     ) 
                   
                    
                   
                       
                   
                    
                   
                     
                       S 
                       2 
                     
                      
                     
                       [ 
                       
                         n 
                         , 
                         p 
                         , 
                         y 
                       
                       ] 
                     
                   
                    
                   
                       
                   
                    
                   • 
                    
                   
                       
                   
                    
                   
                     { 
                     
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           j 
                           k 
                           2 
                         
                       
                       , 
                       
                         
                            
                           
                             j 
                             k 
                           
                            
                         
                         ≤ 
                         p 
                       
                       , 
                       
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               1 
                             
                             n 
                           
                            
                           
                             j 
                             k 
                           
                         
                         = 
                         y 
                       
                     
                     } 
                   
                 
                 ; 
               
                
               
                 
 
               
                
               
                 
                   
                     ( 
                     iii 
                     ) 
                   
                    
                   
                       
                   
                    
                   
                     { 
                     
                       
                         
                           S 
                           2 
                         
                          
                         
                           [ 
                           
                             n 
                             , 
                             p 
                             , 
                             y 
                           
                           ] 
                         
                       
                       + 
                       
                         j 
                         2 
                       
                     
                     } 
                   
                    
                   
                       
                   
                    
                   • 
                    
                   
                       
                   
                    
                   
                     { 
                     
                       
                         z 
                         + 
                         
                           j 
                           2 
                         
                       
                       , 
                       
                         z 
                         ∈ 
                         
                           
                             S 
                             2 
                           
                            
                           
                             [ 
                             
                               n 
                               , 
                               p 
                               , 
                               y 
                             
                             ] 
                           
                         
                       
                     
                     } 
                   
                 
                 ; 
               
             
               
           
         
       
       
         
           
             
               
                 ( 
                 iv 
                 ) 
               
                
               
                   
               
                
               
                 
                   Cs 
                   2 
                 
                  
                 
                   [ 
                   
                     n 
                     , 
                     p 
                     , 
                     y 
                   
                   ] 
                 
               
                
               
                   
               
                
               • 
                
               
                   
               
                
               Cardinal 
                
               
                   
               
                
               
                 
                   S 
                   2 
                 
                  
                 
                   [ 
                   
                     n 
                     , 
                     p 
                     , 
                     y 
                   
                   ] 
                 
               
                
               
                   
               
                
               • 
                
               
                   
               
                
               # 
                
               
                   
               
                
               
                 
                   S 
                   2 
                 
                  
                 
                   [ 
                   
                     n 
                     , 
                     p 
                     , 
                     y 
                   
                   ] 
                 
               
             
             ; 
               
           
         
       
     
       Proposition 3 
       [0137]    For n, p positive integers and y integer verifying |y|≦n×p 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       S 
                       2 
                     
                      
                     
                       [ 
                       
                         n 
                         , 
                         p 
                         , 
                         y 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       ⋃ 
                       
                         
                           j 
                           = 
                           
                             - 
                             p 
                           
                         
                         , 
                         
                           
                              
                             
                               y 
                               - 
                               j 
                             
                              
                           
                           ≤ 
                           
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                              
                             p 
                           
                         
                       
                       p 
                     
                      
                     
                       { 
                       
                         
                           
                             S 
                             2 
                           
                            
                           
                             [ 
                             
                               
                                 n 
                                 - 
                                 1 
                               
                               , 
                               p 
                               , 
                               
                                 y 
                                 - 
                                 j 
                               
                             
                             ] 
                           
                         
                         + 
                         
                           j 
                           2 
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                     
                       
                         S 
                         2 
                       
                        
                       
                         [ 
                         
                           n 
                           , 
                           p 
                           , 
                           
                             - 
                             y 
                           
                         
                         ] 
                       
                     
                   
                   ; 
                 
               
             
           
         
       
       
         
           
             
               For 
                
               
                   
               
                
               y 
             
             = 
             
               
                 - 
                 p 
               
                
               
                   
               
                
               to 
                
               
                   
               
                
               p 
             
           
         
       
       
         
           
             
               
                 
                   S 
                   2 
                 
                  
                 
                   [ 
                   
                     1 
                     , 
                     p 
                     , 
                     y 
                   
                   ] 
                 
               
               = 
               
                 
                   { 
                   
                     y 
                     2 
                   
                   } 
                 
                 = 
                 
                   
                     S 
                     2 
                   
                    
                   
                     [ 
                     
                       1 
                       , 
                       p 
                       , 
                       
                         - 
                         y 
                       
                     
                     ] 
                   
                 
               
             
             ; 
           
         
       
     
       Proof 
       [0138]    
       
         
           
             
                 
             
              
             
               
                 
                   S 
                   2 
                 
                  
                 
                   [ 
                   
                     n 
                     , 
                     p 
                   
                   ] 
                 
               
                
               
                   
               
                
               • 
                
               
                   
               
                
               
                 { 
                 
                   
                     ( 
                     
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           j 
                           k 
                         
                       
                       , 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           j 
                           k 
                           2 
                         
                       
                     
                     ) 
                   
                   , 
                   
                     
                        
                       
                         j 
                         k 
                       
                        
                     
                     ≤ 
                     p 
                   
                   , 
                   
                     k 
                     = 
                     1 
                   
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   p 
                 
                 } 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   
                     S 
                     2 
                   
                    
                   
                     [ 
                     
                       n 
                       , 
                       p 
                     
                     ] 
                   
                 
                 = 
                 
                   { 
                   
                     
                       ( 
                       
                         y 
                         , 
                         z 
                       
                       ) 
                     
                     , 
                     
                       
                          
                         y 
                          
                       
                       ≤ 
                       np 
                     
                     , 
                     
                       x 
                       ∈ 
                       
                         
                           S 
                           2 
                         
                          
                         
                           [ 
                           
                             n 
                             , 
                             p 
                             , 
                             y 
                           
                           ] 
                         
                       
                     
                   
                   } 
                 
               
               , 
               
                 
 
               
                
               
                   
               
                
               where 
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   S 
                   2 
                 
                  
                 
                   [ 
                   
                     n 
                     , 
                     p 
                     , 
                     y 
                   
                   ] 
                 
               
                
               
                   
               
                
               • 
                
               
                   
               
                
               
                 { 
                 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                       j 
                       k 
                       2 
                     
                   
                   , 
                   
                     
                        
                       
                         j 
                         k 
                       
                        
                     
                     ≤ 
                     p 
                   
                   , 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         j 
                         k 
                       
                     
                     = 
                     y 
                   
                 
                 } 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   S 
                   2 
                 
                  
                 
                   [ 
                   
                     n 
                     , 
                     p 
                     , 
                     y 
                   
                   ] 
                 
               
               = 
               
                 { 
                 
                   
                     
                       
                         
                           z 
                           + 
                           
                             
                               ( 
                               
                                 y 
                                 - 
                                 j 
                               
                               ) 
                             
                             2 
                           
                         
                         , 
                         
                           z 
                           ∈ 
                           
                             
                               S 
                               2 
                             
                              
                             
                               [ 
                               
                                 
                                   n 
                                   - 
                                   1 
                                 
                                 , 
                                 p 
                                 , 
                                 j 
                               
                               ] 
                             
                           
                         
                         , 
                       
                     
                   
                   
                     
                       
                         
                           
                              
                             j 
                              
                           
                           ≤ 
                           
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                              
                             p 
                           
                         
                         , 
                         
                           
                              
                             
                               y 
                               - 
                               j 
                             
                              
                           
                           ≤ 
                           p 
                         
                       
                     
                   
                 
                 } 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   S 
                   2 
                 
                  
                 
                   [ 
                   
                     n 
                     , 
                     p 
                     , 
                     y 
                   
                   ] 
                 
               
               = 
               
                 { 
                 
                   
                     
                       
                         
                           z 
                           + 
                           
                             j 
                             2 
                           
                         
                         , 
                         
                           z 
                           ∈ 
                           
                             
                               S 
                               2 
                             
                              
                             
                               [ 
                               
                                 
                                   n 
                                   - 
                                   1 
                                 
                                 , 
                                 p 
                                 , 
                                 
                                   y 
                                   - 
                                   j 
                                 
                               
                               ] 
                             
                           
                         
                         , 
                       
                     
                   
                   
                     
                       
                         
                           
                              
                             j 
                              
                           
                           ≤ 
                           p 
                         
                         , 
                         
                           
                              
                             
                               y 
                               - 
                               j 
                             
                              
                           
                           ≤ 
                           
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                              
                             p 
                           
                         
                       
                     
                   
                 
                 } 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               If 
                
               
                   
               
                
               we 
                
               
                   
               
                
               note 
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 { 
                 
                   
                     
                       S 
                       2 
                     
                      
                     
                       [ 
                       
                         
                           n 
                           - 
                           1 
                         
                         , 
                         p 
                         , 
                         j 
                       
                       ] 
                     
                   
                   + 
                   
                     j 
                     2 
                   
                 
                 } 
               
                
               
                   
               
                
               • 
                
               
                   
               
                
               
                 { 
                 
                   
                     z 
                     + 
                     
                       j 
                       2 
                     
                   
                   , 
                   
                     z 
                     ∈ 
                     
                       
                         S 
                         2 
                       
                        
                       
                         [ 
                         
                           
                             n 
                             - 
                             1 
                           
                           , 
                           p 
                           , 
                           
                             y 
                             - 
                             j 
                           
                         
                         ] 
                       
                     
                   
                 
                 } 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   S 
                   2 
                 
                  
                 
                   [ 
                   
                     n 
                     , 
                     p 
                     , 
                     y 
                   
                   ] 
                 
               
               = 
               
                 
                   ⋃ 
                   
                     
                       j 
                       = 
                       
                         - 
                         p 
                       
                     
                     , 
                     
                       
                          
                         
                           y 
                           - 
                           j 
                         
                          
                       
                       ≤ 
                       
                         
                           ( 
                           
                             n 
                             - 
                             1 
                           
                           ) 
                         
                          
                         p 
                       
                     
                   
                   p 
                 
                  
                 
                   { 
                   
                     
                       
                         S 
                         2 
                       
                        
                       
                         [ 
                         
                           
                             n 
                             - 
                             1 
                           
                           , 
                           p 
                           , 
                           
                             y 
                             - 
                             j 
                           
                         
                         ] 
                       
                     
                     + 
                     
                       j 
                       2 
                     
                   
                   } 
                 
               
             
           
         
       
       
         
           
             
               
                 S 
                 2 
               
                
               
                 [ 
                 
                   n 
                   , 
                   p 
                   , 
                   y 
                 
                 ] 
               
             
             = 
               
              
             
               
                 ( 
                 
                   
                     ⋃ 
                     
                       
                         j 
                         = 
                         0 
                       
                       , 
                       
                         
                            
                           
                             y 
                             - 
                             j 
                           
                            
                         
                         ≤ 
                         
                           
                             ( 
                             
                               n 
                               - 
                               1 
                             
                             ) 
                           
                            
                           p 
                         
                       
                     
                     p 
                   
                    
                   
                     { 
                     
                       
                         
                           S 
                           2 
                         
                          
                         
                           [ 
                           
                             
                               n 
                               - 
                               1 
                             
                             , 
                             p 
                             , 
                             
                               y 
                               - 
                               j 
                             
                           
                           ] 
                         
                       
                       + 
                       
                         j 
                         2 
                       
                     
                     } 
                   
                 
                 ) 
               
               ⋃ 
               
                 ( 
                 
                   
                     ⋃ 
                     
                       
                         j 
                         = 
                         0 
                       
                       , 
                       
                         
                            
                           
                             y 
                             - 
                             j 
                           
                            
                         
                         ≤ 
                         
                           
                             ( 
                             
                               n 
                               - 
                               1 
                             
                             ) 
                           
                            
                           p 
                         
                       
                     
                     p 
                   
                    
                   
                     { 
                     
                       
                         
                           S 
                           2 
                         
                          
                         
                           [ 
                           
                             
                               n 
                               - 
                               1 
                             
                             , 
                             p 
                             , 
                             
                               y 
                               + 
                               j 
                             
                           
                           ] 
                         
                       
                       + 
                       
                         j 
                         2 
                       
                     
                     } 
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   
                     
                       
                         S 
                         2 
                       
                        
                       
                         [ 
                         
                           n 
                           , 
                           p 
                           , 
                           
                             - 
                             y 
                           
                         
                         ] 
                       
                     
                     = 
                       
                      
                     
                       
                         ( 
                         
                           
                             ⋃ 
                             
                               
                                 j 
                                 = 
                                 0 
                               
                               , 
                               
                                 
                                    
                                   
                                     y 
                                     - 
                                     j 
                                   
                                    
                                 
                                 ≤ 
                                 
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                    
                                   p 
                                 
                               
                             
                             p 
                           
                            
                           
                             { 
                             
                               
                                 
                                   S 
                                   2 
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     , 
                                     p 
                                     , 
                                     
                                       
                                         - 
                                         y 
                                       
                                       - 
                                       j 
                                     
                                   
                                   ] 
                                 
                               
                               + 
                               
                                 j 
                                 2 
                               
                             
                             } 
                           
                         
                         ) 
                       
                       ⋃ 
                     
                   
                 
               
               
                 
                   
                       
                      
                     
                       ( 
                       
                         
                           ⋃ 
                           
                             
                               j 
                               = 
                               0 
                             
                             , 
                             
                               
                                  
                                 
                                   y 
                                   - 
                                   j 
                                 
                                  
                               
                               ≤ 
                               
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                  
                                 p 
                               
                             
                           
                           p 
                         
                          
                         
                           { 
                           
                             
                               
                                 S 
                                 2 
                               
                                
                               
                                 [ 
                                 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   , 
                                   p 
                                   , 
                                   
                                     
                                       - 
                                       y 
                                     
                                     + 
                                     j 
                                   
                                 
                                 ] 
                               
                             
                             + 
                             
                               j 
                               2 
                             
                           
                           } 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   
                     = 
                       
                      
                     
                       
                         ( 
                         
                           
                             ⋃ 
                             
                               
                                 j 
                                 = 
                                 0 
                               
                               , 
                               
                                 
                                    
                                   
                                     y 
                                     - 
                                     j 
                                   
                                    
                                 
                                 ≤ 
                                 
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                    
                                   p 
                                 
                               
                             
                             p 
                           
                            
                           
                             { 
                             
                               
                                 
                                   S 
                                   2 
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     , 
                                     p 
                                     , 
                                     
                                       y 
                                       + 
                                       j 
                                     
                                   
                                   ] 
                                 
                               
                               + 
                               
                                 j 
                                 2 
                               
                             
                             } 
                           
                         
                         ) 
                       
                       ⋃ 
                     
                   
                 
               
               
                 
                   
                       
                      
                     
                       ( 
                       
                         
                           ⋃ 
                           
                             
                               j 
                               = 
                               0 
                             
                             , 
                             
                               
                                  
                                 
                                   y 
                                   - 
                                   j 
                                 
                                  
                               
                               ≤ 
                               
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                  
                                 p 
                               
                             
                           
                           p 
                         
                          
                         
                           { 
                           
                             
                               
                                 S 
                                 2 
                               
                                
                               
                                 [ 
                                 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                   , 
                                   p 
                                   , 
                                   
                                     y 
                                     - 
                                     j 
                                   
                                 
                                 ] 
                               
                             
                             + 
                             
                               j 
                               2 
                             
                           
                           } 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   
                     = 
                       
                      
                     
                       
                         S 
                         2 
                       
                        
                       
                         [ 
                         
                           n 
                           , 
                           p 
                           , 
                           y 
                         
                         ] 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               Indeed 
               , 
               
                 
                   it 
                    
                   
                       
                   
                    
                   is 
                    
                   
                       
                   
                    
                   trivial 
                    
                   
                       
                   
                    
                   that 
                    
                   
                       
                   
                    
                   for 
                    
                   
                       
                   
                    
                   y 
                 
                 = 
                 
                   
                     - 
                     p 
                   
                    
                   
                       
                   
                    
                   to 
                    
                   
                       
                   
                    
                   p 
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   
                     S 
                     2 
                   
                    
                   
                     [ 
                     
                       1 
                       , 
                       p 
                       , 
                       y 
                     
                     ] 
                   
                 
                 = 
                 
                   
                     { 
                     
                       y 
                       2 
                     
                     } 
                   
                   = 
                   
                     
                       S 
                       2 
                     
                      
                     
                       [ 
                       
                         1 
                         , 
                         p 
                         , 
                         y 
                       
                       ] 
                     
                   
                 
               
               ; 
             
           
         
       
     
       Proposition 4 
       [0139]    
       
         
           
             
               
                 
                   
                     S 
                     2 
                   
                    
                   
                     [ 
                     
                       n 
                       , 
                       p 
                     
                     ] 
                   
                 
                 
                   
                     ( 
                     
                       
                         2 
                          
                         np 
                       
                       + 
                       1 
                     
                     ) 
                   
                   2 
                 
               
                
               
                   
               
                
               • 
                
               
                   
               
                
               η 
             
             ; 
             
               η 
               = 
               
                 
                   π 
                   2 
                 
                 6 
               
             
             ; 
           
         
       
     
       Proof 
       [0140]    As this is a sideshow to our argument here, we have not spent the time to try to rigorously prove this result and have no idea whether it is actually easy or difficult. This may be an interesting issue to investigate for the curious mathematical reader. 
         [0141]    Our statement is purely based on empirical observations and considerations. It seems there must be some equivalence between 
         [0000]    
       
         
           
             
               
                 
                   
                     S 
                     2 
                   
                    
                   
                     [ 
                     
                       n 
                       , 
                       p 
                     
                     ] 
                   
                 
                 
                   
                     ( 
                     
                       
                         2 
                          
                         np 
                       
                       + 
                       1 
                     
                     ) 
                   
                   2 
                 
               
                
               
                   
               
                
               and 
                
               
                   
               
                
               
                 
                   ∑ 
                   
                     k 
                     = 
                     1 
                   
                   ∞ 
                 
                  
                 
                   1 
                   
                     k 
                     2 
                   
                 
               
             
             = 
             
               
                 
                   π 
                   2 
                 
                 6 
               
               . 
             
           
         
       
     
         [0142]    Another approach may be to show the equivalence with the integral of a function and compute that integral. 
       Algorithm For Computing S 2 [n,p,y] 
       [0143]    
       
         
           
             
               
                 For 
                  
                 
                     
                 
                  
                 y 
               
               = 
               
                 
                   - 
                   p 
                 
                  
                 
                     
                 
                  
                 to 
                  
                 
                     
                 
                  
                 p 
                  
                 
                     
                 
                  
                 step 
                  
                 
                     
                 
                  
                 1 
               
             
             , 
             
               
                 
                   
                     
                       S 
                       2 
                     
                      
                     
                       [ 
                       
                         1 
                         , 
                         p 
                         , 
                         y 
                       
                       ] 
                     
                   
                   = 
                   
                     { 
                     
                       y 
                       2 
                     
                     } 
                   
                 
                 ; 
               
                
               
                 
 
               
               [ 
               
                 
                   
                     
                       
                         For 
                          
                         
                             
                         
                          
                         i 
                       
                       = 
                       
                         2 
                          
                         
                             
                         
                          
                         to 
                          
                         
                             
                         
                          
                         n 
                          
                         
                             
                         
                          
                         step 
                          
                         
                             
                         
                          
                         1 
                       
                     
                   
                 
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 For 
                                  
                                 
                                     
                                 
                                  
                                 y 
                               
                               = 
                               
                                 
                                   - 
                                   i 
                                 
                                 × 
                                 p 
                                  
                                 
                                     
                                 
                                  
                                 to 
                                  
                                 
                                     
                                 
                                  
                                 i 
                                 × 
                                 p 
                                  
                                 
                                     
                                 
                                  
                                 step 
                                  
                                 
                                     
                                 
                                  
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     
                                       S 
                                       2 
                                     
                                      
                                     
                                       [ 
                                       
                                         i 
                                         , 
                                         p 
                                         , 
                                         y 
                                       
                                       ] 
                                     
                                   
                                   = 
                                   
                                     { 
                                     
                                         
                                     
                                     } 
                                   
                                 
                                 ; 
                                 
                                   
                                     
                                       Cs 
                                       2 
                                     
                                      
                                     
                                       [ 
                                       
                                         i 
                                         , 
                                         p 
                                         , 
                                         y 
                                       
                                       ] 
                                     
                                   
                                   = 
                                   0 
                                 
                                 ; 
                                 
                                   s 
                                   = 
                                   0 
                                 
                                 ; 
                               
                                
                               
                                 (* 
                                  
                                 initialization 
                                  
                                 *) 
                               
                             
                           
                         
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       
                                         
                                           For 
                                            
                                           
                                               
                                           
                                            
                                           j 
                                         
                                         = 
                                         
                                           
                                             - 
                                             p 
                                           
                                            
                                           
                                               
                                           
                                            
                                           to 
                                            
                                           
                                               
                                           
                                            
                                           p 
                                            
                                           
                                               
                                           
                                            
                                           step 
                                            
                                           
                                               
                                           
                                            
                                           1 
                                         
                                       
                                       , 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         If 
                                          
                                         
                                             
                                         
                                          
                                         
                                            
                                           
                                             y 
                                             - 
                                             j 
                                           
                                            
                                         
                                       
                                       ≤ 
                                       
                                         
                                           ( 
                                           
                                             i 
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                          
                                         
                                             
                                         
                                          
                                         p 
                                          
                                         
                                             
                                         
                                          
                                         then 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         s 
                                         = 
                                         
                                           Length 
                                            
                                           
                                             [ 
                                             
                                               
                                                 
                                                   S 
                                                   2 
                                                 
                                                  
                                                 
                                                   [ 
                                                   
                                                     i 
                                                     , 
                                                     p 
                                                     , 
                                                     y 
                                                   
                                                   ] 
                                                 
                                               
                                               ⋂ 
                                               
                                                 { 
                                                 
                                                   
                                                     
                                                       S 
                                                       2 
                                                     
                                                      
                                                     
                                                       [ 
                                                       
                                                         
                                                           i 
                                                           - 
                                                           1 
                                                         
                                                         , 
                                                         p 
                                                         , 
                                                         
                                                           y 
                                                           - 
                                                           j 
                                                         
                                                       
                                                       ] 
                                                     
                                                   
                                                   + 
                                                   
                                                     j 
                                                     2 
                                                   
                                                 
                                                 } 
                                               
                                             
                                             ] 
                                           
                                         
                                       
                                       ; 
                                     
                                   
                                 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             
                                               * 
                                               Length 
                                                
                                               
                                                   
                                               
                                                
                                               is 
                                                
                                               
                                                   
                                               
                                                
                                               the 
                                                
                                               
                                                   
                                               
                                                
                                               command 
                                                
                                               
                                                   
                                               
                                                
                                               for 
                                                
                                               
                                                   
                                               
                                                
                                               computing 
                                                
                                               
                                                   
                                               
                                                
                                               the 
                                             
                                           
                                         
                                         
                                           
                                             
                                               number 
                                                
                                               
                                                   
                                               
                                                
                                               of 
                                                
                                               
                                                   
                                               
                                                
                                               elements 
                                                
                                               
                                                   
                                               
                                                
                                               of 
                                                
                                               
                                                   
                                               
                                                
                                               a 
                                                
                                               
                                                   
                                               
                                                
                                               set 
                                                
                                               
                                                   
                                               
                                                
                                               in 
                                                
                                               
                                                   
                                               
                                                
                                               mathematica 
                                               * 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         
                                           S 
                                           2 
                                         
                                          
                                         
                                           [ 
                                           
                                             i 
                                             , 
                                             p 
                                             , 
                                             y 
                                           
                                           ] 
                                         
                                       
                                       = 
                                       
                                         
                                           
                                             S 
                                             2 
                                           
                                            
                                           
                                             [ 
                                             
                                               i 
                                               , 
                                               p 
                                               , 
                                               y 
                                             
                                             ] 
                                           
                                         
                                         ⋃ 
                                         
                                           { 
                                           
                                             
                                               
                                                 S 
                                                 2 
                                               
                                                
                                               
                                                 [ 
                                                 
                                                   
                                                     i 
                                                     - 
                                                     1 
                                                   
                                                   , 
                                                   p 
                                                   , 
                                                   
                                                     y 
                                                     - 
                                                     j 
                                                   
                                                 
                                                 ] 
                                               
                                             
                                             + 
                                             
                                               j 
                                               2 
                                             
                                           
                                           } 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         
                                           Cs 
                                           2 
                                         
                                          
                                         
                                           [ 
                                           
                                             i 
                                             , 
                                             p 
                                             , 
                                             y 
                                           
                                           ] 
                                         
                                       
                                       = 
                                       
                                         
                                           
                                             Cs 
                                             2 
                                           
                                            
                                           
                                             [ 
                                             
                                               i 
                                               , 
                                               p 
                                               , 
                                               y 
                                             
                                             ] 
                                           
                                         
                                         + 
                                         
                                           
                                             Cs 
                                             2 
                                           
                                            
                                           
                                             [ 
                                             
                                               
                                                 i 
                                                 - 
                                                 1 
                                               
                                               , 
                                               p 
                                               , 
                                               
                                                 y 
                                                 - 
                                                 j 
                                               
                                             
                                             ] 
                                           
                                         
                                         - 
                                         s 
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0144]    The following Mathematica® Code translates the algorithm 
         [0000]    
       
         
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
               
               
             
           
               
                   
               
             
             
               
                 Clear[p, n, s, Sp, S2, i, j, y, t, Cs2, ts]; 
               
               
                 p= 10; n=100; 
               
               
                 For[y= −p, y≦p, S2[1, p, y] = {y 2 }; Cs2[1, p, y] =1;y++]; 
               
               
                 For[i = 2, i≦n, 
               
             
          
           
               
                   
                 For[y= 0, y≦i*p, 
               
             
          
           
               
                   
                 S2[i, p, y] = { }; Cs2[i, p, y] = 0; s = 0; (*Initialization of variables*) 
               
               
                   
                 For[j= −p, j≦p, 
               
             
          
           
               
                   
                 If[Abs[y−j] ≦ (i−1) *p, 
               
             
          
           
               
                   
                 Sp= S2[i−1, p, y−j] + j 2 ; 
               
               
                   
                 s = Length[S2[i, p, y]∩Sp]; 
               
               
                   
                 (*Length is the function computing the Cardinal of a set in Mathematica*) 
               
               
                   
                 S2[i, p, y] = S2[i, p, y]∪Sp; 
               
               
                   
                 Cs2[i, p, y] = Cs2[i, p, y] + Cs2[i−1, p, y−j] − s 
               
             
          
           
               
                   
                 ]; 
               
               
                   
                 j++];S2[i, p, −y] = S2[i, p, y];Cs2[i, p, −y] =Cs2[i, p, y]; 
               
             
          
           
               
                   
                 y++]; 
               
             
          
           
               
                   
                 i++]; 
               
               
                   
                   
               
             
          
         
       
     
         [0145]    We can see with the code below the shape of S2[i,p,y] as a function of y looks like a bell curve and how S[i,p]/(2ip+1)̂2 asymptotically converges to the constant conjectured here as π 2 /6 
         [0000]    
       
         
               
               
             
           
               
                   
                   
               
             
             
               
                   
                 Clear[ts]; 
               
               
                   
                 t = Table[{y, Cs2[n, p, y]}, {y, −n * p, n * p}]; 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           ts 
                           = 
                           
                             Table 
                              
                             
                               [ 
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             ∑ 
                                             
                                               y 
                                               = 
                                               
                                                 
                                                   - 
                                                   i 
                                                 
                                                 * 
                                                 p 
                                               
                                             
                                             
                                               i 
                                               * 
                                               p 
                                             
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             Cs 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               2 
                                                
                                               
                                                 [ 
                                                 
                                                   i 
                                                   , 
                                                   p 
                                                   , 
                                                   y 
                                                 
                                                 ] 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                       / 
                                       
                                         
                                           ( 
                                           
                                             
                                               2 
                                               * 
                                               i 
                                               * 
                                               p 
                                             
                                             + 
                                             1 
                                           
                                           ) 
                                         
                                         
                                           
                                               
                                           
                                           ^ 
                                         
                                       
                                     
                                      
                                     2 
                                   
                                   // 
                                   N 
                                 
                                 , 
                                 
                                   { 
                                   
                                     i 
                                     , 
                                     1 
                                     , 
                                     n 
                                   
                                   } 
                                 
                               
                               ] 
                             
                           
                         
                         ; 
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                 ListPlot[t] 
               
               
                   
                 ListPlot[ts] 
               
               
                   
                   
               
             
          
         
       
     
       Pricing 
       [0146]    We note f i ≡Π k,i,n   0,+/− (ƒ)(M i   1 ,M i   2 ) and this exercise will illustrate how we represent the functions f i  in the backward iterative loop to make it tractable. For ease of presentation, we assume a frictionless market with an underlying following a Markovian dynamic. We estimate the premium payment amount for time t 0 . 
         [0147]    We thus note: f i (M i   2 ,S i )≡Π 0,i,n   0 (ƒ)(X i S i ); S i=S   0 e M)    
         [0148]    The applicable backward recursive pricing formula in the Option BIC format is: 
         [0000]    
       
         
           
             
               
                 f 
                 i 
               
                
               
                 ( 
                 
                   
                     M 
                     i 
                     2 
                   
                   , 
                   
                     S 
                     i 
                   
                 
                 ) 
               
             
             = 
             
               
 
             
              
             
               
                 
                   ( 
                   
                     
                       
                         f 
                         
                           i 
                           + 
                           1 
                         
                       
                        
                       
                         ( 
                         
                           
                             
                               M 
                               i 
                               2 
                             
                             + 
                             
                               
                                 Log 
                                  
                                 
                                   [ 
                                   
                                     
                                       F 
                                       
                                         i 
                                         + 
                                         1 
                                       
                                     
                                     
                                       S 
                                       i 
                                     
                                   
                                   ] 
                                 
                               
                               2 
                             
                           
                           , 
                           
                             F 
                             
                               i 
                               + 
                               1 
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         
                           ∂ 
                           2 
                           1 
                         
                          
                         
                           
                             f 
                             
                               i 
                               + 
                               1 
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   M 
                                   i 
                                   2 
                                 
                                 + 
                                 
                                   
                                     Log 
                                      
                                     
                                       [ 
                                       
                                         
                                           F 
                                           
                                             i 
                                             + 
                                             1 
                                           
                                         
                                         
                                           S 
                                           i 
                                         
                                       
                                       ] 
                                     
                                   
                                   2 
                                 
                               
                               , 
                               
                                 F 
                                 
                                   i 
                                   + 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                       
                        
                       
                         F 
                         
                           i 
                           + 
                           1 
                         
                       
                     
                   
                   ) 
                 
                  
                 
                   B 
                   
                     0 
                     , 
                     i 
                     , 
                     
                       i 
                       + 
                       1 
                     
                   
                   0 
                 
               
               + 
               
                 
                   
                     ∂ 
                     2 
                     1 
                   
                    
                   
                     
                       f 
                       
                         i 
                         + 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           
                             M 
                             i 
                             2 
                           
                           + 
                           
                             
                               Log 
                                
                               
                                 [ 
                                 
                                   
                                     F 
                                     
                                       i 
                                       + 
                                       1 
                                     
                                   
                                   
                                     S 
                                     i 
                                   
                                 
                                 ] 
                               
                             
                             2 
                           
                         
                         , 
                         
                           F 
                           
                             i 
                             + 
                             1 
                           
                         
                       
                       ) 
                     
                   
                 
                  
                 
                   S 
                   i 
                 
                  
                 
                   B 
                   
                     0 
                     , 
                     i 
                     , 
                     
                       i 
                       + 
                       1 
                     
                   
                   1 
                 
               
               + 
               
                 
 
               
                
               
                 
                   F 
                   
                     i 
                     + 
                     1 
                   
                 
                 ( 
                 
                     
                 
                  
                 
                   
                     
                       ∫ 
                       0 
                       1 
                     
                      
                     
                       
                         
                           ∂ 
                           2 
                           2 
                         
                          
                         
                           
                             f 
                             
                               i 
                               + 
                               1 
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   M 
                                   i 
                                   2 
                                 
                                 + 
                                 
                                   
                                     Log 
                                      
                                     
                                       [ 
                                       
                                         
                                           F 
                                           
                                             i 
                                             + 
                                             1 
                                           
                                         
                                         
                                           xS 
                                           i 
                                         
                                       
                                       ] 
                                     
                                   
                                   2 
                                 
                               
                               , 
                               
                                 
                                   F 
                                   
                                     i 
                                     + 
                                     1 
                                   
                                 
                                 / 
                                 x 
                               
                             
                             ) 
                           
                         
                       
                        
                       
                         
                           B 
                           
                             0 
                             , 
                             i 
                             , 
                             
                               i 
                               + 
                               1 
                             
                           
                           0 
                         
                          
                         
                           ( 
                           
                             1 
                             , 
                             
                               
                                 F 
                                 
                                   i 
                                   + 
                                   1 
                                 
                               
                               / 
                               
                                 x 
                                 - 
                               
                             
                           
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           S 
                           i 
                         
                         ) 
                       
                        
                       
                         
                            
                           x 
                         
                         
                           xx 
                           - 
                         
                       
                     
                   
                   + 
                   
                     
                       ∫ 
                       0 
                       1 
                     
                      
                     
                       
                         
                           ∂ 
                           2 
                           2 
                         
                          
                         
                           
                             f 
                             
                               i 
                               + 
                               1 
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   M 
                                   i 
                                   2 
                                 
                                 + 
                                 
                                   
                                     Log 
                                      
                                     
                                       [ 
                                       
                                         
                                           xF 
                                           
                                             i 
                                             + 
                                             1 
                                           
                                         
                                         
                                           S 
                                           i 
                                         
                                       
                                       ] 
                                     
                                   
                                   2 
                                 
                               
                               , 
                               
                                 
                                   F 
                                   
                                     i 
                                     + 
                                     1 
                                   
                                 
                                  
                                 x 
                               
                             
                             ) 
                           
                         
                       
                        
                       
                         
                           B 
                           
                             0 
                             , 
                             i 
                             , 
                             
                               i 
                               + 
                               1 
                             
                           
                           0 
                         
                          
                         
                           ( 
                           
                             
                               - 
                               1 
                             
                             , 
                             
                               
                                 F 
                                 
                                   i 
                                   + 
                                   1 
                                 
                               
                                
                               
                                 x 
                                 + 
                               
                             
                           
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           S 
                           i 
                         
                         ) 
                       
                        
                       
                          
                         x 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
         [0149]    The derivatives with respect to S n  are indeed meant in the discrete sense subordinated to the applicable space partition: 
         [0000]    
       
         
           
             
               
                 
                   f 
                   ′ 
                 
                  
                 
                   ( 
                   x 
                   ) 
                 
               
               = 
               
                 
                   
                     f 
                      
                     
                       ( 
                       
                         x 
                         + 
                       
                       ) 
                     
                   
                   - 
                   
                     f 
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                 
                 
                   
                     x 
                     + 
                   
                   - 
                   x 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 f 
                 ″ 
               
                
               
                 ( 
                 x 
                 ) 
               
             
             = 
             
               
                 
                   
                     
                       f 
                       ′ 
                     
                      
                     
                       ( 
                       
                         x 
                         + 
                       
                       ) 
                     
                   
                   - 
                   
                     
                       f 
                       ′ 
                     
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                 
                 
                   
                     x 
                     + 
                   
                   - 
                   x 
                 
               
               = 
               
                 
                   
                     
                       
                         f 
                          
                         
                           ( 
                           
                             x 
                             ++ 
                           
                           ) 
                         
                       
                       - 
                       
                         f 
                          
                         
                           ( 
                           
                             x 
                             + 
                           
                           ) 
                         
                       
                     
                     
                       
                         x 
                         ++ 
                       
                       - 
                       
                         x 
                         + 
                       
                     
                   
                   - 
                   
                     
                       
                         f 
                          
                         
                           ( 
                           
                             x 
                             + 
                           
                           ) 
                         
                       
                       - 
                       
                         f 
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                     
                     
                       
                         x 
                         + 
                       
                       - 
                       x 
                     
                   
                 
                 
                   
                     x 
                     + 
                   
                   - 
                   x 
                 
               
             
           
         
       
     
         [0150]    The applicable backward recursive pricing formula in the Fourier BIC format is: 
         [0000]    
       
         
           
             
                 
             
              
             
               
                 
                   
                     f 
                     i 
                   
                    
                   
                     ( 
                     
                       
                         M 
                         i 
                         2 
                       
                       , 
                       
                         S 
                         i 
                       
                     
                     ) 
                   
                 
                 = 
                 
                   
                     B 
                     
                       0 
                       , 
                       i 
                       , 
                       
                         i 
                         + 
                         1 
                       
                     
                     0 
                   
                    
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         
                           - 
                           ∞ 
                         
                       
                       
                         + 
                         ∞ 
                       
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           c 
                           n 
                         
                          
                         
                           ( 
                           
                             
                               M 
                               i 
                               2 
                             
                             , 
                             
                               S 
                               i 
                             
                           
                           ) 
                         
                       
                        
                       
                         E 
                         ( 
                         
                           
                              
                             
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 
                                   nS 
                                   
                                     i 
                                     + 
                                     1 
                                   
                                 
                               
                               
                                 
                                   H 
                                   
                                     i 
                                     + 
                                     1 
                                   
                                 
                                 - 
                                 
                                   L 
                                   
                                     i 
                                     + 
                                     1 
                                   
                                 
                               
                             
                           
                           / 
                           
                             S 
                             i 
                           
                         
                         ) 
                       
                     
                   
                 
               
               , 
               
                 
 
               
                
               
                   
               
                
               with 
             
           
         
       
       
         
           
             
               
                 c 
                 n 
               
                
               
                 ( 
                 
                   
                     M 
                     i 
                     2 
                   
                   , 
                   
                     S 
                     i 
                   
                 
                 ) 
               
             
             = 
             
               
                 1 
                 
                   
                     H 
                     
                       i 
                       + 
                       1 
                     
                   
                   - 
                   
                     L 
                     
                       i 
                       + 
                       1 
                     
                   
                 
               
                
               
                 
                   ∫ 
                   
                     L 
                     
                       i 
                       + 
                       1 
                     
                   
                   
                     H 
                     
                       i 
                       + 
                       1 
                     
                   
                 
                  
                 
                   
                      
                     
                       
                         
                           - 
                           2 
                         
                          
                          
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         nx 
                       
                       
                         
                           H 
                           
                             i 
                             + 
                             1 
                           
                         
                         - 
                         
                           L 
                           
                             i 
                             + 
                             1 
                           
                         
                       
                     
                   
                    
                   
                       
                   
                    
                   
                     
                       f 
                       
                         i 
                         + 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           
                             M 
                             i 
                             2 
                           
                           + 
                           
                             
                               Log 
                                
                               
                                 [ 
                                 
                                   x 
                                   
                                     S 
                                     i 
                                   
                                 
                                 ] 
                               
                             
                             2 
                           
                         
                         , 
                         x 
                       
                       ) 
                     
                   
                    
                   
                      
                     x 
                   
                 
               
             
           
         
       
     
         [0151]    In each of the BIC format representations above, when use the naïve representation of functions, we must make the computations over the span of S 2 [n,p] as we move recursively backwards through time representing about O((2ip+1) 2 (2p+1)) additions and multiplications at each time step. This is indeed possible and doable in a BICs trading system, where every derivatives contract is recomposed from the price of actually quoted BICs. However, if one is merely interested in pricing in the traditional frictionless framework for instance, substantial computational savings become possible. 
       3.2.3.2.From the Computationally Possible To the Computationally Optimal For Pricing 
       [0152]    In fact, by selecting the right functional representation basis functions of premium payment amounts coupled with the summarizing variables selected above, one can come as close to a close form solution for the premium payment amount at time 0 as one can hope for. Indeed, 
         [0000]    
       
         
           
             
               V 
               n 
             
             = 
             
               
                 M 
                 n 
                 2 
               
               - 
               
                 
                   1 
                   n 
                 
                  
                 
                   
                     ( 
                     
                       M 
                       n 
                       1 
                     
                     ) 
                   
                   2 
                 
               
             
           
         
       
       
         
           with 
         
       
       
         
           
             
               
                 X 
                 i 
               
               = 
               
                 Log 
                  
                 
                   [ 
                   
                     
                       S 
                       i 
                     
                     
                       S 
                       
                         i 
                         - 
                         1 
                       
                     
                   
                   ] 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 M 
                 k 
                 1 
               
               = 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   k 
                 
                  
                 
                     
                 
                  
                 
                   X 
                   i 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 M 
                 k 
                 2 
               
               = 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   k 
                 
                  
                 
                     
                 
                  
                 
                   X 
                   i 
                   2 
                 
               
             
             ; 
           
         
       
     
         [0153]    We use the Fourier space functional representation. 
         [0154]    For any function ƒ of two variables integrable, 
         [0000]    
       
         
           
             
               
                 f 
                  
                 
                   ( 
                   
                     
                       x 
                       1 
                     
                     , 
                     
                       x 
                       2 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     - 
                     ∞ 
                   
                   
                     + 
                     ∞ 
                   
                 
                  
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     
                       + 
                       ∞ 
                     
                   
                    
                   
                     
                       
                         f 
                         ~ 
                       
                        
                       
                         ( 
                         
                           
                             t 
                             1 
                           
                           , 
                           
                             t 
                             2 
                           
                         
                         ) 
                       
                     
                      
                     
                        
                       
                         2 
                          
                          
                          
                         
                             
                         
                          
                         
                           π 
                            
                           
                             ( 
                             
                               
                                 
                                   t 
                                   1 
                                 
                                  
                                 
                                   x 
                                   1 
                                 
                               
                               + 
                               
                                 
                                   t 
                                   2 
                                 
                                  
                                 
                                   x 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
             
              
             
                 
             
           
         
       
     
         [0000]    dt 1 dt 2 ; with 
         [0000]    
       
         
           
             
               
                 
                   f 
                   ~ 
                 
                  
                 
                   ( 
                   
                     
                       t 
                       1 
                     
                     , 
                     
                       t 
                       2 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     - 
                     ∞ 
                   
                   
                     + 
                     ∞ 
                   
                 
                  
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     
                       + 
                       ∞ 
                     
                   
                    
                   
                     
                       f 
                        
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             x 
                             2 
                           
                         
                         ) 
                       
                     
                      
                     
                        
                       
                         
                           - 
                           2 
                         
                          
                          
                          
                         
                             
                         
                          
                         
                           π 
                            
                           
                             ( 
                             
                               
                                 
                                   t 
                                   1 
                                 
                                  
                                 
                                   x 
                                   1 
                                 
                               
                               + 
                               
                                 
                                   t 
                                   2 
                                 
                                  
                                 
                                   x 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                      
                     
                         
                     
                      
                     
                        
                       
                         x 
                         1 
                       
                     
                      
                     
                        
                       
                         x 
                         2 
                       
                     
                   
                 
               
             
             ; 
           
         
       
     
         [0155]    A volatility derivatives contract is a contract whose payout is a function g of V n . 
         [0156]    So if ƒ is such that 
         [0000]    
       
         
           
             
               
                 f 
                  
                 
                   ( 
                   
                     
                       x 
                       1 
                     
                     , 
                     
                       x 
                       2 
                     
                   
                   ) 
                 
               
                
               • 
                
               
                   
               
                
               
                 g 
                  
                 
                   ( 
                   
                     
                       x 
                       2 
                     
                     - 
                     
                       
                         1 
                         n 
                       
                        
                       
                         
                           ( 
                           
                             x 
                             1 
                           
                           ) 
                         
                         2 
                       
                     
                   
                   ) 
                 
               
             
             , 
           
         
       
     
         [0000]    in a frictionless market where the X i  are independent is: Π 0,n (g(V n ))=B o,n   0 E(ƒ(M n   1 ,M n   2 )); 
         [0000]    
       
         
           
             
               
                 E 
                  
                 
                   ( 
                   
                     f 
                      
                     
                       ( 
                       
                         
                           M 
                           n 
                           1 
                         
                         , 
                         
                           M 
                           n 
                           2 
                         
                       
                       ) 
                     
                   
                   ) 
                 
               
               = 
               
                 E 
                  
                 
                   ( 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       
                         + 
                         ∞ 
                       
                     
                      
                     
                       
                         
                           f 
                           ~ 
                         
                          
                         
                           ( 
                           
                             
                               t 
                               1 
                             
                             , 
                             
                               t 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                          
                         
                           2 
                            
                           
                             π 
                              
                             
                               ( 
                               
                                 
                                   
                                     t 
                                     1 
                                   
                                    
                                   
                                     M 
                                     n 
                                     1 
                                   
                                 
                                 + 
                                 
                                   
                                     t 
                                     2 
                                   
                                    
                                   
                                     M 
                                     n 
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                        
                       
                           
                       
                        
                       
                          
                         
                           t 
                           1 
                         
                       
                        
                       
                          
                         
                           t 
                           2 
                         
                       
                     
                   
                   ) 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 E 
                  
                 
                   ( 
                   
                     f 
                      
                     
                       ( 
                       
                         
                           M 
                           n 
                           1 
                         
                         , 
                         
                           M 
                           n 
                           2 
                         
                       
                       ) 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     - 
                     ∞ 
                   
                   
                     + 
                     ∞ 
                   
                 
                  
                 
                   
                     
                       f 
                       ~ 
                     
                      
                     
                       ( 
                       
                         
                           t 
                           1 
                         
                         , 
                         
                           t 
                           2 
                         
                       
                       ) 
                     
                   
                    
                   
                     E 
                      
                     
                       ( 
                       
                          
                         
                           2 
                            
                           
                             π 
                              
                             
                               ( 
                               
                                 
                                   
                                     t 
                                     1 
                                   
                                    
                                   
                                     M 
                                     n 
                                     1 
                                   
                                 
                                 + 
                                 
                                   
                                     t 
                                     2 
                                   
                                    
                                   
                                     M 
                                     n 
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ) 
                     
                   
                    
                   
                       
                   
                    
                   
                      
                     
                       t 
                       1 
                     
                   
                    
                   
                      
                     
                       t 
                       2 
                     
                   
                 
               
             
             ; 
           
         
       
     
         [0157]    If the (X i ) i=1, . . . ,n  are independent. With a BICs frame of mind with elicitation of summarizing variables, it is easy to see how to compute the conditional expectations iteratively backwards, yielding, 
         [0000]    
       
         
           
             
                 
             
              
             
               
                 
                   E 
                    
                   
                     ( 
                     
                       f 
                        
                       
                         ( 
                         
                           
                             M 
                             n 
                             1 
                           
                           , 
                           
                             M 
                             n 
                             2 
                           
                         
                         ) 
                       
                     
                     ) 
                   
                 
                 = 
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     
                       + 
                       ∞ 
                     
                   
                    
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       
                         + 
                         ∞ 
                       
                     
                      
                     
                       
                         
                           f 
                           ~ 
                         
                          
                         
                           ( 
                           
                             
                               t 
                               1 
                             
                             , 
                             
                               t 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           
                             ∏ 
                             
                               k 
                               = 
                               1 
                             
                             n 
                           
                            
                           
                               
                           
                            
                           
                             E 
                              
                             
                               ( 
                               
                                 h 
                                  
                                 
                                   ( 
                                   
                                     
                                       X 
                                       k 
                                     
                                     , 
                                     
                                       t 
                                       1 
                                     
                                     , 
                                     
                                       t 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                        
                       
                          
                         
                           t 
                           1 
                         
                       
                        
                       
                          
                         
                           t 
                           2 
                         
                       
                     
                   
                 
               
               ; 
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   h 
                    
                   
                     ( 
                     
                       
                         X 
                         k 
                       
                       , 
                       
                         t 
                         1 
                       
                       , 
                       
                         t 
                         2 
                       
                     
                     ) 
                   
                 
                  
                 
                   • 
                   
                     2 
                      
                     
                       π 
                        
                       
                         ( 
                         
                           
                             
                               t 
                               1 
                             
                              
                             
                               X 
                               k 
                             
                           
                           + 
                           
                             
                               t 
                               2 
                             
                              
                             
                               X 
                               k 
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               ; 
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   h 
                    
                   
                     ( 
                     
                       
                         X 
                         k 
                       
                       , 
                       
                         t 
                         1 
                       
                       , 
                       
                         t 
                         2 
                       
                     
                     ) 
                   
                 
                 = 
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     
                       + 
                       ∞ 
                     
                   
                    
                   
                     
                       
                         h 
                         ~ 
                       
                        
                       
                         ( 
                         
                           t 
                           , 
                           
                             t 
                             1 
                           
                           , 
                           
                             t 
                             2 
                           
                         
                         ) 
                       
                     
                      
                     
                        
                       
                         2 
                          
                          
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         
                           tX 
                           k 
                         
                       
                     
                      
                     
                         
                     
                      
                     
                        
                       t 
                     
                   
                 
               
               ; 
             
           
         
       
       
         
           
             
               
                 
                   h 
                   ~ 
                 
                  
                 
                   ( 
                   
                     t 
                     , 
                     
                       t 
                       1 
                     
                     , 
                     
                       t 
                       2 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     
                       + 
                       ∞ 
                     
                   
                    
                   
                     
                        
                       
                         2 
                          
                          
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         
                           t 
                           2 
                         
                          
                         
                           x 
                           2 
                         
                       
                     
                      
                     
                        
                       
                         
                           - 
                           2 
                         
                          
                          
                          
                         
                             
                         
                          
                         
                           π 
                            
                           
                             ( 
                             
                               t 
                               - 
                               
                                 t 
                                 1 
                               
                             
                             ) 
                           
                         
                          
                         x 
                       
                     
                      
                     
                         
                     
                      
                     
                        
                       x 
                     
                   
                 
                  
                 
                   = 
                   
                     Mathematica 
                      
                     
                         
                     
                      
                     Formula 
                   
                 
                  
                 
                   
                      
                     
                       - 
                       
                         
                           
                             π 
                              
                             
                               ( 
                               
                                 t 
                                 - 
                                 
                                   t 
                                   1 
                                 
                               
                               ) 
                             
                           
                           2 
                         
                         
                           2 
                            
                           
                             t 
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         - 
                         2 
                       
                        
                        
                        
                       
                           
                       
                        
                       
                         t 
                         2 
                       
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   E 
                    
                   
                     ( 
                     
                       h 
                        
                       
                         ( 
                         
                           
                             X 
                             k 
                           
                           , 
                           
                             t 
                             1 
                           
                           , 
                           
                             t 
                             2 
                           
                         
                         ) 
                       
                     
                     ) 
                   
                 
                 = 
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     
                       + 
                       ∞ 
                     
                   
                    
                   
                     
                       
                         h 
                         ~ 
                       
                        
                       
                         ( 
                         
                           t 
                           , 
                           
                             t 
                             1 
                           
                           , 
                           
                             t 
                             2 
                           
                         
                         ) 
                       
                     
                      
                     
                       E 
                        
                       
                         ( 
                         
                            
                           
                             2 
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               tX 
                               k 
                             
                           
                         
                         ) 
                       
                     
                      
                     
                         
                     
                      
                     
                        
                       t 
                     
                   
                 
               
               ; 
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 If 
                  
                 
                     
                 
                  
                 
                   X 
                   k 
                 
                  
                 
                     
                 
                  
                 is 
                  
                 
                     
                 
                  
                 a 
                  
                 
                   
                       
                   
                    
                   
                       
                   
                 
                  
                 Levy 
                  
                 
                     
                 
                  
                 process 
               
               , 
               
                 
                   
                     E 
                      
                     
                       ( 
                       
                          
                         
                           2 
                            
                           π 
                            
                           
                               
                           
                            
                            
                            
                           
                               
                           
                            
                           
                             tX 
                             k 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                      
                     
                       Δ 
                        
                       
                           
                       
                        
                       
                         t 
                         k 
                       
                        
                       
                         
                           Ψ 
                           k 
                         
                          
                         
                           ( 
                           
                             2 
                              
                             π 
                              
                             
                                 
                             
                              
                             t 
                           
                           ) 
                         
                       
                     
                   
                 
                 ; 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   E 
                    
                   
                     ( 
                     
                       h 
                        
                       
                         ( 
                         
                           
                             X 
                             k 
                           
                           , 
                           
                             t 
                             1 
                           
                           , 
                           
                             t 
                             2 
                           
                         
                         ) 
                       
                     
                     ) 
                   
                 
                 = 
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     
                       + 
                       ∞ 
                     
                   
                    
                   
                     
                       
                         
                            
                           
                             - 
                             
                               
                                 
                                   π 
                                    
                                   
                                     ( 
                                     
                                       t 
                                       - 
                                       
                                         t 
                                         1 
                                       
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               
                                 2 
                                  
                                 
                                   t 
                                   2 
                                 
                               
                             
                           
                         
                          
                         
                           E 
                            
                           
                             ( 
                             
                                
                               
                                 2 
                                  
                                  
                                  
                                 
                                     
                                 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 
                                   tX 
                                   k 
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             - 
                             2 
                           
                            
                            
                            
                           
                               
                           
                            
                           
                             t 
                             2 
                           
                         
                       
                     
                      
                     
                         
                     
                      
                     
                        
                       t 
                     
                   
                 
               
               ; 
             
           
         
       
       
         
           
             
               
                 E 
                  
                 
                   ( 
                   
                     f 
                      
                     
                       ( 
                       
                         
                           M 
                           n 
                           1 
                         
                         , 
                         
                           M 
                           n 
                           2 
                         
                       
                       ) 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     - 
                     ∞ 
                   
                   
                     + 
                     ∞ 
                   
                 
                  
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     
                       + 
                       ∞ 
                     
                   
                    
                   
                     
                       
                         f 
                         ~ 
                       
                        
                       
                         ( 
                         
                           
                             t 
                             1 
                           
                           , 
                           
                             t 
                             2 
                           
                         
                         ) 
                       
                     
                      
                     
                       ( 
                       
                         
                           ∏ 
                           
                             k 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                             
                         
                          
                         
                           ( 
                           
                               
                           
                            
                           
                             
                               ∫ 
                               
                                 - 
                                 ∞ 
                               
                               
                                 + 
                                 ∞ 
                               
                             
                              
                             
                               
                                 
                                   
                                      
                                     
                                       - 
                                       
                                         
                                           
                                             π 
                                              
                                             
                                               ( 
                                               
                                                 t 
                                                 - 
                                                 
                                                   t 
                                                   1 
                                                 
                                               
                                               ) 
                                             
                                           
                                           2 
                                         
                                         
                                           2 
                                            
                                           
                                             t 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                    
                                   
                                     E 
                                      
                                     
                                       ( 
                                       
                                          
                                         
                                           2 
                                            
                                            
                                            
                                           
                                               
                                           
                                            
                                           π 
                                            
                                           
                                               
                                           
                                            
                                           
                                             tX 
                                             k 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                   
                                     
                                       - 
                                       2 
                                     
                                      
                                      
                                      
                                     
                                         
                                     
                                      
                                     
                                       t 
                                       2 
                                     
                                   
                                 
                               
                                
                               
                                   
                               
                                
                               
                                  
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                      
                     
                        
                       
                         t 
                         1 
                       
                     
                      
                     
                        
                       
                         t 
                         2 
                       
                     
                   
                 
               
             
             ; 
           
         
       
     
         [0158]    If time increments are equal (Δt k =Δt) and the laws are identical (X k ≡X), 
         [0000]    
       
         
           
             
               E 
                
               
                 ( 
                 
                   f 
                    
                   
                     ( 
                     
                       
                         M 
                         n 
                         1 
                       
                       , 
                       
                         M 
                         n 
                         2 
                       
                     
                     ) 
                   
                 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   - 
                   ∞ 
                 
                 
                   - 
                   ∞ 
                 
               
                
               
                 
                   ∫ 
                   
                     - 
                     ∞ 
                   
                   
                     + 
                     ∞ 
                   
                 
                  
                 
                   
                     
                       f 
                       ~ 
                     
                      
                     
                       ( 
                       
                         
                           t 
                           1 
                         
                         , 
                         
                           t 
                           2 
                         
                       
                       ) 
                     
                   
                    
                   
                     
                       ( 
                       
                         
                           ∫ 
                           
                             - 
                             ∞ 
                           
                           
                             + 
                             ∞ 
                           
                         
                          
                         
                           
                             
                               
                                  
                                 
                                   - 
                                   
                                     
                                        
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           π 
                                            
                                           
                                             ( 
                                             
                                               t 
                                               - 
                                               
                                                 t 
                                                 1 
                                               
                                             
                                             ) 
                                           
                                         
                                         2 
                                       
                                     
                                     
                                       2 
                                        
                                       
                                         t 
                                         2 
                                       
                                     
                                   
                                 
                               
                                
                               
                                 E 
                                  
                                 
                                   ( 
                                   
                                      
                                     
                                       2 
                                        
                                        
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       tX 
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   2 
                                 
                                  
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   t 
                                   2 
                                 
                               
                             
                           
                            
                           
                               
                           
                            
                           
                              
                             t 
                           
                         
                       
                       ) 
                     
                     n 
                   
                    
                   
                       
                   
                    
                   
                      
                     
                       t 
                       1 
                     
                   
                    
                   
                      
                     
                       t 
                       2 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   ∏ 
                   
                     0 
                     , 
                     n 
                   
                 
                  
                 
                     
                 
                  
                 
                   ( 
                   
                     g 
                      
                     
                       ( 
                       
                         V 
                         n 
                       
                       ) 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   B 
                   
                     o 
                     , 
                     n 
                   
                   0 
                 
                  
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     
                       + 
                       ∞ 
                     
                   
                    
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       
                         + 
                         ∞ 
                       
                     
                      
                     
                       
                         
                           f 
                           ~ 
                         
                          
                         
                           ( 
                           
                             
                               t 
                               1 
                             
                             , 
                             
                               t 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         
                           ( 
                           
                             
                               ∫ 
                               
                                 - 
                                 ∞ 
                               
                               
                                 + 
                                 ∞ 
                               
                             
                              
                             
                               
                                 
                                   
                                      
                                     
                                       - 
                                       
                                         
                                            
                                            
                                           
                                               
                                           
                                            
                                           
                                             
                                               π 
                                                
                                               
                                                 ( 
                                                 
                                                   t 
                                                   - 
                                                   
                                                     t 
                                                     1 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                             2 
                                           
                                         
                                         
                                           2 
                                            
                                           
                                             t 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                    
                                   
                                     E 
                                      
                                     
                                       ( 
                                       
                                          
                                         
                                           2 
                                            
                                            
                                            
                                           
                                               
                                           
                                            
                                           π 
                                            
                                           
                                               
                                           
                                            
                                           tX 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                   
                                     
                                       - 
                                       2 
                                     
                                      
                                      
                                      
                                     
                                         
                                     
                                      
                                     
                                       t 
                                       2 
                                     
                                   
                                 
                               
                                
                               
                                   
                               
                                
                               
                                  
                                 t 
                               
                             
                           
                           ) 
                         
                         n 
                       
                        
                       
                           
                       
                        
                       
                          
                         
                           t 
                           1 
                         
                       
                        
                       
                          
                         
                           t 
                           2 
                         
                       
                     
                   
                 
               
             
              
             
                 
             
             ; 
           
         
       
     
       Illustration: Case of A Geometric Brownian Motion With Time Dependent Parameters 
       [0159]    
       
         
           
             
               
                 ∏ 
                 
                   0 
                   , 
                   n 
                 
               
                
               
                   
               
                
               
                 ( 
                 
                   g 
                    
                   
                     ( 
                     
                       V 
                       n 
                     
                     ) 
                   
                 
                 ) 
               
             
             = 
             
               
                 B 
                 
                   o 
                   , 
                   n 
                 
                 0 
               
                
               
                 
                   ∫ 
                   
                     - 
                     ∞ 
                   
                   
                     + 
                     ∞ 
                   
                 
                  
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     
                       + 
                       ∞ 
                     
                   
                    
                   
                     
                       
                         
                           f 
                           ~ 
                         
                          
                         
                           ( 
                           
                             
                               t 
                               1 
                             
                             , 
                             
                               t 
                               2 
                             
                           
                           ) 
                         
                       
                       
                         
                           ( 
                           
                             
                               
                                 - 
                                 2 
                               
                                
                                
                                
                               
                                   
                               
                                
                               
                                 t 
                                 2 
                               
                             
                           
                           ) 
                         
                         n 
                       
                     
                      
                     
                       
                         ∏ 
                         
                           k 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                           
                       
                        
                       
                         
                           ( 
                           
                             
                               ∫ 
                               
                                 - 
                                 ∞ 
                               
                               
                                 + 
                                 ∞ 
                               
                             
                              
                             
                               
                                  
                                 
                                   - 
                                   
                                     
                                        
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           π 
                                            
                                           
                                             ( 
                                             
                                               t 
                                               - 
                                               
                                                 t 
                                                 1 
                                               
                                             
                                             ) 
                                           
                                         
                                         2 
                                       
                                     
                                     
                                       2 
                                        
                                       
                                         t 
                                         2 
                                       
                                     
                                   
                                 
                               
                                
                               
                                 E 
                                  
                                 
                                   ( 
                                   
                                      
                                     
                                       2 
                                        
                                        
                                        
                                       
                                           
                                       
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       
                                         tX 
                                         k 
                                       
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                  
                                 t 
                               
                             
                           
                           ) 
                         
                          
                         
                            
                           
                             t 
                             1 
                           
                         
                          
                         
                            
                           
                             t 
                             2 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 If 
                  
                 
                     
                 
                  
                 
                   X 
                   t 
                 
                  
                 
                   
                       
                   
                    
                   
                       
                   
                 
                  
                 follows 
                  
                 
                     
                 
                  
                 a 
                  
                 
                     
                 
                  
                 normal 
                  
                 
                     
                 
                  
                 law 
               
               , 
               
                 
 
               
                
               
                 
                   E 
                    
                   
                     ( 
                     
                        
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         
                           tX 
                           k 
                         
                       
                     
                     ) 
                   
                 
                 = 
                 
                   
                     
                        
                       
                         Δ 
                          
                         
                             
                         
                          
                         
                           
                             t 
                             k 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   - 
                                   2 
                                 
                                  
                                 
                                   π 
                                   2 
                                 
                                  
                                 
                                   σ 
                                   2 
                                 
                                  
                                 
                                   t 
                                   2 
                                 
                               
                               + 
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                  
                                  
                                 
                                     
                                 
                                  
                                 bt 
                               
                             
                             ) 
                           
                         
                       
                     
                     - 
                     
                       
                         i 
                          
                         
                             
                         
                          
                         
                           
                             π 
                              
                             
                               ( 
                               
                                 t 
                                 - 
                                 
                                   t 
                                   1 
                                 
                               
                               ) 
                             
                           
                           2 
                         
                       
                       
                         2 
                          
                         
                           t 
                           2 
                         
                       
                     
                     + 
                     
                       Δ 
                        
                       
                           
                       
                        
                       
                         
                           t 
                           k 
                         
                          
                         
                           ( 
                           
                             
                               
                                 - 
                                 2 
                               
                                
                               
                                 π 
                                 2 
                               
                                
                               
                                 σ 
                                 2 
                               
                                
                               
                                 t 
                                 2 
                               
                             
                             + 
                             
                               2 
                                
                               π 
                                
                               
                                   
                               
                                
                                
                                
                               
                                   
                               
                                
                               bt 
                             
                           
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   π 
                                 
                                 
                                   2 
                                    
                                   
                                     t 
                                     2 
                                   
                                 
                               
                               + 
                               
                                 2 
                                  
                                 
                                   π 
                                   2 
                                 
                                  
                                 
                                   σ 
                                   2 
                                 
                                  
                                 Δ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   t 
                                   k 
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                           t 
                           2 
                         
                       
                       + 
                       
                         2 
                          
                         
                           π 
                            
                           
                             ( 
                             
                               
                                 b 
                                  
                                 
                                     
                                 
                                  
                                 Δ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   t 
                                   k 
                                 
                               
                               + 
                               
                                 
                                   t 
                                   1 
                                 
                                 
                                   2 
                                    
                                   
                                     t 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                         
                          
                         t 
                       
                       - 
                       
                         π 
                          
                         
                           
                             t 
                             1 
                             2 
                           
                           
                             2 
                              
                             
                               t 
                               2 
                             
                           
                         
                       
                     
                     = 
                     
                       
                         
                           α 
                           k 
                         
                          
                         
                           t 
                           2 
                         
                       
                       + 
                       
                         2 
                          
                         
                           πβ 
                           k 
                         
                          
                         t 
                       
                       + 
                       
                         γ 
                         k 
                       
                     
                   
                 
               
               , 
               
                 
 
               
                
               
                   
               
                
               with 
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   α 
                   k 
                 
                 = 
                 
                   ( 
                   
                     
                       π 
                       
                         2 
                          
                         
                           t 
                           2 
                         
                       
                     
                     + 
                     
                       2 
                        
                       
                         π 
                         2 
                       
                        
                       
                         σ 
                         2 
                       
                        
                       Δ 
                        
                       
                           
                       
                        
                       
                         t 
                         k 
                       
                     
                   
                   ) 
                 
               
               ; 
               
                 
                   β 
                   k 
                 
                 = 
                 
                   ( 
                   
                     
                       b 
                        
                       
                           
                       
                        
                       Δ 
                        
                       
                           
                       
                        
                       
                         t 
                         k 
                       
                     
                     + 
                     
                       
                         t 
                         1 
                       
                       
                         2 
                          
                         
                           t 
                           2 
                         
                       
                     
                   
                   ) 
                 
               
               ; 
               
                 
                   γ 
                   k 
                 
                 = 
                 
                   
                     - 
                     π 
                   
                    
                   
                     
                       t 
                       1 
                       2 
                     
                     
                       2 
                        
                       
                         t 
                         2 
                       
                     
                   
                 
               
               ; 
             
           
         
       
       
         
           
             
               
                 
                   ∫ 
                   
                     - 
                     ∞ 
                   
                   
                     + 
                     ∞ 
                   
                 
                  
                 
                   
                      
                     
                       
                         
                           α 
                           k 
                         
                          
                         
                           t 
                           2 
                         
                       
                       + 
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         
                           β 
                           k 
                         
                          
                         t 
                       
                       + 
                       
                         γ 
                         k 
                       
                     
                   
                    
                   
                       
                   
                    
                   
                      
                     t 
                   
                 
               
               = 
               
                 
                   
                      
                     
                       γ 
                       k 
                     
                   
                    
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       
                         + 
                         ∞ 
                       
                     
                      
                     
                       
                          
                         
                           
                             
                               α 
                               k 
                             
                              
                             
                               t 
                               2 
                             
                           
                           + 
                           
                             2 
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               β 
                               k 
                             
                              
                             t 
                           
                         
                       
                        
                       
                           
                       
                        
                       
                          
                         t 
                       
                     
                   
                 
                 = 
                 
                   
                     
                       π 
                       
                         - 
                         
                           α 
                           k 
                         
                       
                     
                   
                    
                   
                      
                     
                       
                         
                           
                             π 
                             2 
                           
                            
                           
                             β 
                             k 
                             2 
                           
                         
                         
                           α 
                           k 
                         
                       
                       + 
                       
                         γ 
                         k 
                       
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   ∏ 
                   
                     k 
                     = 
                     1 
                   
                   n 
                 
                  
                 
                     
                 
                  
                 
                   ( 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       
                         + 
                         ∞ 
                       
                     
                      
                     
                       
                          
                         
                           
                             - 
                             
                               
                                 
                                   π 
                                    
                                   
                                     ( 
                                     
                                       t 
                                       - 
                                       
                                         t 
                                         1 
                                       
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                               
                                 2 
                                  
                                 
                                   t 
                                   2 
                                 
                               
                             
                           
                           + 
                           
                             Δ 
                              
                             
                                 
                             
                              
                             
                               t 
                               k 
                             
                              
                             
                               
                                 Ψ 
                                 k 
                               
                                
                               
                                 ( 
                                 
                                   2 
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   t 
                                 
                                 ) 
                               
                             
                           
                         
                       
                        
                       
                          
                         t 
                       
                     
                   
                   ) 
                 
               
               = 
               
                 
                   ∏ 
                   
                     k 
                     = 
                     1 
                   
                   n 
                 
                  
                 
                     
                 
                  
                 
                   ( 
                   
                     
                       
                         π 
                         
                           - 
                           
                             α 
                             k 
                           
                         
                       
                     
                      
                     
                        
                       
                         
                           
                             
                               π 
                               2 
                             
                              
                             
                               β 
                               k 
                               2 
                             
                           
                           
                             α 
                             k 
                           
                         
                         + 
                         
                           γ 
                           k 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   ∏ 
                   
                     0 
                     , 
                     n 
                   
                 
                  
                 
                     
                 
                  
                 
                   ( 
                   
                     g 
                      
                     
                       ( 
                       
                         V 
                         n 
                       
                       ) 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   B 
                   
                     o 
                     , 
                     n 
                   
                   0 
                 
                  
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     
                       + 
                       ∞ 
                     
                   
                    
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       
                         + 
                         ∞ 
                       
                     
                      
                     
                       
                         
                           f 
                           ~ 
                         
                          
                         
                           ( 
                           
                             
                               t 
                               1 
                             
                             , 
                             
                               t 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         
                           
                             ( 
                             
                               π 
                               
                                 
                                   - 
                                   2 
                                 
                                  
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   t 
                                   2 
                                 
                               
                             
                             ) 
                           
                           
                             n 
                             / 
                             2 
                           
                         
                         
                           
                             
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               n 
                             
                              
                             
                               
                                 ∏ 
                                 
                                   k 
                                   = 
                                   1 
                                 
                                 n 
                               
                                
                               
                                   
                               
                                
                               
                                 α 
                                 k 
                               
                             
                           
                         
                       
                        
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           n 
                            
                           
                               
                           
                            
                           π 
                            
                           
                             
                               t 
                               1 
                               2 
                             
                             
                               2 
                                
                               
                                 t 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                  
                 
                   
                     π 
                     2 
                   
                    
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                         
                     
                      
                     
                       
                         β 
                         k 
                         2 
                       
                       
                         α 
                         k 
                       
                     
                   
                 
               
                
               
                   
               
                
               
                  
                 
                   t 
                   1 
                 
               
                
               
                   
               
                
               
                  
                 
                   t 
                   2 
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   α 
                   k 
                 
                 = 
                 
                   ( 
                   
                     
                       π 
                       
                         2 
                          
                         
                           t 
                           2 
                         
                       
                     
                     + 
                     
                       2 
                        
                       
                         π 
                         2 
                       
                        
                       
                         σ 
                         k 
                         2 
                       
                        
                       Δ 
                        
                       
                           
                       
                        
                       
                         t 
                         k 
                       
                     
                   
                   ) 
                 
               
               ; 
               
                 
                   β 
                   k 
                 
                 = 
                 
                   ( 
                   
                     
                       b 
                        
                       
                           
                       
                        
                       Δ 
                        
                       
                           
                       
                        
                       
                         t 
                         k 
                       
                     
                     + 
                     
                       
                         t 
                         1 
                       
                       
                         2 
                          
                         
                           t 
                           2 
                         
                       
                     
                   
                   ) 
                 
               
               ; 
             
           
         
       
     
       Remark 1  
     (i) Case of BICs Pricing Information In the Options Format 
       [0160]    When computing E(h(X k ,t 1 t 2 )), we do not always have to the Fourier Transform of the density. Indeed, the BICs pricing information may often be provided in the options format. We would handle this case for instance as 
         [0000]    
       
         
           
             
               E 
               ( 
               
                  
                 2 
                 
                   2 
                    
                   
                     π 
                      
                     
                       ( 
                       
                         
                           
                             t 
                             1 
                           
                            
                           
                             X 
                             k 
                           
                         
                         + 
                         
                           
                             t 
                             2 
                           
                            
                           
                             X 
                             k 
                             2 
                           
                         
                       
                       ) 
                     
                   
                 
               
               ) 
             
             = 
             
               
                 ∫ 
                 
                   - 
                   ∞ 
                 
                 
                   + 
                   ∞ 
                 
               
                
               
                 
                    
                   
                     2 
                      
                     
                       π 
                        
                       
                         ( 
                         
                           
                             
                               t 
                               1 
                             
                              
                             x 
                           
                           + 
                           
                             
                               t 
                               2 
                             
                              
                             
                               x 
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                 
                  
                 
                   
                     p 
                     
                       Δ 
                        
                       
                           
                       
                        
                       
                         t 
                         k 
                       
                     
                     
                       X 
                       k 
                     
                   
                    
                   
                     ( 
                     x 
                     ) 
                   
                 
                  
                 
                    
                   x 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   p 
                   
                     Δ 
                      
                     
                         
                     
                      
                     
                       t 
                       k 
                     
                   
                   
                     S 
                     k 
                   
                 
                  
                 
                   ( 
                   
                     
                       S 
                       k 
                     
                     = 
                     s 
                   
                   ) 
                 
               
                
               ds 
             
             = 
             
               
                 
                   
                     p 
                     
                       Δ 
                        
                       
                           
                       
                        
                       
                         t 
                         k 
                       
                     
                     
                       S 
                       k 
                     
                   
                    
                   
                     ( 
                     
                       
                          
                         
                           X 
                           k 
                         
                       
                       = 
                       
                          
                         x 
                       
                     
                     ) 
                   
                 
                  
                 
                    
                   x 
                 
                  
                 dx 
               
               = 
               
                 
                   
                     p 
                     
                       Δ 
                        
                       
                           
                       
                        
                       
                         t 
                         k 
                       
                     
                     
                       X 
                       k 
                     
                   
                    
                   
                     ( 
                     
                       
                         X 
                         k 
                       
                       = 
                       x 
                     
                     ) 
                   
                 
                  
                 dx 
               
             
           
         
       
       
         
           
             
               
                 
                   p 
                   
                     Δ 
                      
                     
                         
                     
                      
                     
                       t 
                       k 
                     
                   
                   
                     X 
                     k 
                   
                 
                  
                 
                   ( 
                   
                     
                       X 
                       k 
                     
                     = 
                     x 
                   
                   ) 
                 
               
               = 
               
                 
                   
                     
                       p 
                       
                         Δ 
                          
                         
                             
                         
                          
                         
                           t 
                           k 
                         
                       
                       
                         S 
                         k 
                       
                     
                      
                     
                       ( 
                       
                         
                           S 
                           k 
                         
                         = 
                         
                            
                           x 
                         
                       
                       ) 
                     
                   
                    
                   
                      
                     x 
                   
                 
                 = 
                 
                   
                     1 
                     
                       B 
                       
                         
                           t 
                           
                             k 
                             - 
                             1 
                           
                         
                         , 
                         
                           t 
                           k 
                         
                       
                       0 
                     
                   
                    
                   
                     
                       
                         ∂ 
                         2 
                       
                        
                       
                         C 
                         
                           
                             t 
                             
                               k 
                               - 
                               1 
                             
                           
                           , 
                           
                             t 
                             k 
                           
                         
                         0 
                       
                     
                     
                       dK 
                       2 
                     
                   
                    
                   
                     ( 
                     
                        
                       x 
                     
                     ) 
                   
                    
                   
                      
                     x 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               x 
               = 
               
                 Log 
                  
                 
                   [ 
                   s 
                   ] 
                 
               
             
             , 
             
               dx 
               = 
               
                 
                   
                     ds 
                     s 
                   
                    
                   
                     
 
                   
                    
                   
                     B 
                     
                       
                         t 
                         
                           k 
                           - 
                           1 
                         
                       
                       , 
                       
                         t 
                         k 
                       
                     
                     0 
                   
                    
                   
                     E 
                     ( 
                     
                        
                       
                         2 
                          
                         
                           π 
                            
                           
                             ( 
                             
                               
                                 
                                   t 
                                   1 
                                 
                                  
                                 
                                   X 
                                   k 
                                 
                               
                               + 
                               
                                 
                                   t 
                                   2 
                                 
                                  
                                 
                                   X 
                                   k 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     ) 
                   
                 
                 = 
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     
                       + 
                       ∞ 
                     
                   
                    
                   
                     
                        
                       
                         2 
                          
                         
                           π 
                            
                           
                             ( 
                             
                               
                                 
                                   t 
                                   1 
                                 
                                  
                                 x 
                               
                               + 
                               
                                 
                                   t 
                                   2 
                                 
                                  
                                 
                                   x 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                      
                     
                       
                         
                           ∂ 
                           2 
                         
                          
                         
                           C 
                           
                             
                               t 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                             , 
                             
                               t 
                               k 
                             
                           
                           0 
                         
                       
                       
                          
                         
                           K 
                           2 
                         
                       
                     
                      
                     
                       ( 
                       
                          
                         x 
                       
                       ) 
                     
                      
                     
                        
                       x 
                     
                      
                     
                        
                       x 
                     
                   
                 
               
             
           
         
       
     
         [0161]    (ii) Fourier Transform in Mathematica® 
         [0162]    □ FourierTransform[expr, t, ω] gives the symbolic Fourier transform of expr. 
         [0163]    The Fourier transform of a function f(t) is by default defined to be 
         [0000]    
       
         
           
             
               1 
               
                 
                   2 
                    
                   π 
                 
               
             
              
             
               
                 ∫ 
                 
                   - 
                   ∞ 
                 
                 ∞ 
               
                
               
                 
                   f 
                    
                   
                     ( 
                     t 
                     ) 
                   
                 
                  
                 
                    
                   
                      
                      
                     
                         
                     
                      
                     ω 
                      
                     
                         
                     
                      
                     t 
                   
                 
                  
                 
                    
                   t 
                 
               
             
           
         
       
     
         [0164]    For example, 
         [0000]    
       
         
           
             
               FourierTransform 
                
               
                   
               
               [ 
               
                 
                   Exp 
                    
                   
                     [ 
                     
                       2 
                       * 
                       j 
                       * 
                       Pi 
                       * 
                       
                         x 
                         ⋀ 
                         2 
                       
                     
                     ] 
                   
                 
                 , 
                 x 
                 , 
                 
                   2 
                   * 
                   Pi 
                   * 
                   k 
                 
               
               ] 
             
             = 
             
               
                  
                 
                   
                     
                       k 
                       2 
                     
                      
                     π 
                   
                   
                     2 
                      
                     j 
                   
                 
               
               
                 2 
                  
                 
                   
                     - 
                     j 
                   
                 
                  
                 
                   π 
                 
               
             
           
         
       
     
         [0165]    □ FourierTransform[expr, {t 1 , t 2 , . . . }, {ω 1 , ω 2 , . . . }] gives the multidimensional Fourier transform of expr. 
       3.2.3.3—General Moments Derivatives Pricing 
       [0166]    It is now trivial to see that more generally, if we wish to price a moments based derivatives contract that depends on the statistical moments of the underlyings up to the moment of p-th order, the pricing formula can be obtained as 
         [0000]    
       
         
           
             
                 
             
              
             
               
                 
                   m 
                   n 
                   p 
                 
                 = 
                 
                   
                     m 
                     n 
                     p 
                   
                    
                   
                     ( 
                     
                       
                         M 
                         n 
                         1 
                       
                       , 
                       … 
                        
                       
                           
                       
                       , 
                       
                         M 
                         n 
                         p 
                       
                     
                     ) 
                   
                 
               
               ; 
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   f 
                    
                   
                     ( 
                     
                       
                         x 
                         1 
                       
                       , 
                       … 
                        
                       
                           
                       
                       , 
                       
                         x 
                         p 
                       
                     
                     ) 
                   
                 
                  
                 
                     
                 
                  
                 • 
                  
                 
                     
                 
                  
                 
                   g 
                    
                   
                     ( 
                     
                       
                         m 
                         n 
                         p 
                       
                        
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           … 
                            
                           
                               
                           
                           , 
                           
                             x 
                             p 
                           
                         
                         ) 
                       
                     
                     ) 
                   
                 
               
               ; 
             
           
         
       
       
         
           
             
               
                 
                   f 
                   ~ 
                 
                  
                 
                   ( 
                   
                     
                       t 
                       1 
                     
                     , 
                     … 
                      
                     
                         
                     
                     , 
                     
                       t 
                       p 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     - 
                     ∞ 
                   
                   
                     + 
                     ∞ 
                   
                 
                  
                 
                     
                 
                  
                 
                   … 
                    
                   
                       
                   
                    
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       
                         + 
                         ∞ 
                       
                     
                      
                     
                       
                         f 
                          
                         
                           ( 
                           
                             
                               x 
                               1 
                             
                             , 
                             … 
                              
                             
                                 
                             
                             , 
                             
                               x 
                               p 
                             
                           
                           ) 
                         
                       
                        
                       
                          
                         
                           
                             - 
                             2 
                           
                            
                            
                            
                           
                               
                           
                            
                           
                             π 
                              
                             
                               ( 
                               
                                 
                                   
                                     t 
                                     1 
                                   
                                    
                                   
                                     x 
                                     1 
                                   
                                 
                                 + 
                                 … 
                                 + 
                                 
                                   
                                     t 
                                     p 
                                   
                                    
                                   
                                     x 
                                     p 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                        
                       
                          
                         
                           x 
                           1 
                         
                       
                        
                       
                           
                       
                        
                       … 
                        
                       
                           
                       
                        
                       
                          
                         
                           x 
                           p 
                         
                       
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   h 
                    
                   
                     ( 
                     
                       
                         X 
                         k 
                       
                       , 
                       
                         t 
                         1 
                       
                       , 
                       … 
                        
                       
                           
                       
                       , 
                       
                         t 
                         p 
                       
                     
                     ) 
                   
                 
                  
                 
                     
                 
                  
                 • 
                  
                 
                     
                 
                  
                 
                    
                   
                     2 
                      
                     
                       π 
                        
                       
                         ( 
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               1 
                             
                             p 
                           
                            
                           
                             
                               
                                 t 
                                 j 
                               
                                
                               
                                 ( 
                                 
                                   X 
                                   k 
                                 
                                 ) 
                               
                             
                             j 
                           
                         
                         ) 
                       
                     
                   
                 
               
               ; 
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   h 
                   ~ 
                 
                  
                 
                   ( 
                   
                     t 
                     , 
                     
                       t 
                       1 
                     
                     , 
                     … 
                      
                     
                         
                     
                     , 
                     
                       t 
                       p 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     - 
                     ∞ 
                   
                   
                     + 
                     ∞ 
                   
                 
                  
                 
                   
                      
                     
                       2 
                        
                       
                         π 
                          
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 j 
                                 = 
                                 2 
                               
                               p 
                             
                              
                             
                               
                                 
                                   t 
                                   j 
                                 
                                  
                                 
                                   ( 
                                   
                                     X 
                                     k 
                                   
                                   ) 
                                 
                               
                               j 
                             
                           
                           ) 
                         
                       
                     
                   
                    
                   
                      
                     
                       
                         - 
                         2 
                       
                        
                       
                         π 
                          
                         
                           ( 
                           
                             t 
                             - 
                             
                               t 
                               1 
                             
                           
                           ) 
                         
                       
                        
                       x 
                     
                   
                    
                   
                      
                     x 
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   E 
                    
                   
                     ( 
                     
                       h 
                        
                       
                         ( 
                         
                           
                             X 
                             k 
                           
                           , 
                           
                             t 
                             1 
                           
                           , 
                           … 
                            
                           
                               
                           
                           , 
                           
                             t 
                             p 
                           
                         
                         ) 
                       
                     
                     ) 
                   
                 
                 = 
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     
                       + 
                       ∞ 
                     
                   
                    
                   
                     
                       
                         h 
                         ~ 
                       
                        
                       
                         ( 
                         
                           t 
                           , 
                           
                             t 
                             1 
                           
                           , 
                           … 
                            
                           
                               
                           
                           , 
                           
                             t 
                             p 
                           
                         
                         ) 
                       
                     
                      
                     
                       E 
                        
                       
                         ( 
                         
                            
                           
                             2 
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               tX 
                               k 
                             
                           
                         
                         ) 
                       
                     
                      
                     
                        
                       t 
                     
                   
                 
               
               ; 
             
           
         
       
     
         [0167]    As the X k  are independent, 
         [0000]    
       
         
           
             
               
                 E 
                  
                 
                   ( 
                   
                     f 
                      
                     
                       ( 
                       
                         
                           M 
                           n 
                           1 
                         
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           M 
                           n 
                           p 
                         
                       
                       ) 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     - 
                     ∞ 
                   
                   
                     + 
                     ∞ 
                   
                 
                  
                 
                     
                 
                  
                 
                   … 
                    
                   
                       
                   
                    
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       
                         + 
                         ∞ 
                       
                     
                      
                     
                       
                         
                           f 
                           ~ 
                         
                          
                         
                           ( 
                           
                             
                               t 
                               1 
                             
                             , 
                             … 
                              
                             
                                 
                             
                             , 
                             
                               t 
                               p 
                             
                           
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           
                             ∏ 
                             
                               k 
                               = 
                               1 
                             
                             n 
                           
                            
                           
                               
                           
                            
                           
                             ( 
                             
                               
                                 ∫ 
                                 
                                   - 
                                   ∞ 
                                 
                                 
                                   + 
                                   ∞ 
                                 
                               
                                
                               
                                 
                                   
                                     h 
                                     ~ 
                                   
                                    
                                   
                                     ( 
                                     
                                       t 
                                       , 
                                       
                                         t 
                                         1 
                                       
                                       , 
                                       … 
                                        
                                       
                                           
                                       
                                       , 
                                       
                                         t 
                                         p 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   E 
                                    
                                   
                                     ( 
                                     
                                        
                                       
                                         2 
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         
                                           tX 
                                           k 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   t 
                                 
                               
                             
                             ) 
                           
                         
                         ) 
                       
                        
                       
                           
                       
                        
                       
                          
                         
                           t 
                           1 
                         
                       
                        
                       … 
                        
                       
                           
                       
                        
                       
                          
                         
                           t 
                           p 
                         
                       
                     
                   
                 
               
             
             ; 
           
         
       
     
         [0168]    If time increments are equal (Δt k =Δt) and the laws are identical (X k ≡X), 
         [0000]    
       
         
           
             
               E 
                
               
                 ( 
                 
                   f 
                    
                   
                     ( 
                     
                       
                         M 
                         n 
                         1 
                       
                       , 
                       … 
                        
                       
                           
                       
                       , 
                       
                         M 
                         n 
                         p 
                       
                     
                     ) 
                   
                 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   - 
                   ∞ 
                 
                 
                   + 
                   ∞ 
                 
               
                
               
                   
               
                
               
                 … 
                  
                 
                     
                 
                  
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     
                       + 
                       ∞ 
                     
                   
                    
                   
                     
                       
                         f 
                         ~ 
                       
                        
                       
                         ( 
                         
                           
                             t 
                             1 
                           
                           , 
                           … 
                            
                           
                               
                           
                           , 
                           
                             t 
                             p 
                           
                         
                         ) 
                       
                     
                      
                     
                       
                         ( 
                         
                           
                             ∫ 
                             
                               - 
                               ∞ 
                             
                             
                               + 
                               ∞ 
                             
                           
                            
                           
                             
                               
                                 h 
                                 ~ 
                               
                                
                               
                                 ( 
                                 
                                   t 
                                   , 
                                   
                                     t 
                                     1 
                                   
                                   , 
                                   … 
                                    
                                   
                                       
                                   
                                   , 
                                   
                                     t 
                                     p 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               E 
                                
                               
                                 ( 
                                 
                                    
                                   
                                     2 
                                      
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     tX 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                                
                               t 
                             
                           
                         
                         ) 
                       
                       n 
                     
                      
                     
                        
                       
                         t 
                         1 
                       
                     
                      
                     … 
                      
                     
                         
                     
                      
                     
                        
                       
                         t 
                         p 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   ∏ 
                   
                     0 
                     , 
                     n 
                   
                 
                  
                 
                     
                 
                  
                 
                   ( 
                   
                     g 
                      
                     
                       ( 
                       
                         m 
                         n 
                         p 
                       
                       ) 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   B 
                   
                     o 
                     , 
                     n 
                   
                   0 
                 
                  
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     
                       + 
                       ∞ 
                     
                   
                    
                   
                     … 
                      
                     
                         
                     
                      
                     
                       
                         ∫ 
                         
                           - 
                           ∞ 
                         
                         
                           + 
                           ∞ 
                         
                       
                        
                       
                         
                           
                             f 
                             ~ 
                           
                            
                           
                             ( 
                             
                               
                                 t 
                                 1 
                               
                               , 
                               … 
                                
                               
                                   
                               
                               , 
                               
                                 t 
                                 p 
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             ( 
                             
                               
                                 ∫ 
                                 
                                   - 
                                   ∞ 
                                 
                                 
                                   + 
                                   ∞ 
                                 
                               
                                
                               
                                 
                                   
                                     h 
                                     ~ 
                                   
                                    
                                   
                                     ( 
                                     
                                       t 
                                       , 
                                       
                                         t 
                                         1 
                                       
                                       , 
                                       … 
                                        
                                       
                                           
                                       
                                       , 
                                       
                                         t 
                                         p 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   E 
                                    
                                   
                                     ( 
                                     
                                        
                                       
                                         2 
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         tX 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   t 
                                 
                               
                             
                             ) 
                           
                           n 
                         
                          
                         
                            
                           
                             t 
                             1 
                           
                         
                          
                         … 
                          
                         
                             
                         
                          
                         
                            
                           
                             t 
                             p 
                           
                         
                       
                     
                   
                 
               
             
             ; 
           
         
       
     
         [0169]    It is very clear that if m n   p  is merely a linear function of M n   1 , . . . , M n   p , the integrations on □ p  can be further shrunk into integrals on □. 
         [0170]    The states span for higher order moments can also be readily guessed on the model of states span for the first two moments. This example decisively illustrates how the broad BICs argument on principle is applied to a very current and practical quantitative finance research issue with measurable edge. 
         [0171]    It becomes easier to guess the pricing algorithms and the optimized forms of the various derivatives contracts that will be reviewed in Volume II, the Asian Options case below is further illustrative 
       3.2. 4. Asian Options Derivatives Pricing 
       [0172]    
       
         
           
             
                 
             
              
             
               
                 
                   f 
                    
                   
                     ( 
                     
                       Z 
                       n 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     
                       + 
                       ∞ 
                     
                   
                    
                   
                     
                       
                         f 
                         ~ 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                      
                     
                        
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         
                           tZ 
                           n 
                         
                       
                     
                      
                     
                         
                     
                      
                     
                        
                       t 
                     
                   
                 
               
               ; 
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   with 
                    
                   
                       
                   
                    
                   
                     
                       f 
                       ~ 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                 
                 = 
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     
                       + 
                       ∞ 
                     
                   
                    
                   
                     
                       f 
                        
                       
                         ( 
                         x 
                         ) 
                       
                     
                      
                     
                        
                       
                         
                           - 
                           2 
                         
                          
                          
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         tx 
                       
                     
                      
                     
                        
                       x 
                     
                   
                 
               
               ; 
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   
                     Z 
                     n 
                   
                   = 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         0 
                       
                       n 
                     
                      
                     
                         
                     
                      
                     
                       
                         α 
                         k 
                       
                        
                       
                         S 
                         k 
                       
                     
                   
                 
                 ; 
                 
                   1 
                   ≤ 
                   k 
                 
               
               , 
               
                 
 
               
                
               
                   
               
                
               
                 
                   
                     S 
                     k 
                   
                   = 
                   
                     
                       S 
                       0 
                     
                      
                     
                        
                       
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             k 
                           
                            
                           
                               
                           
                            
                           
                             Log 
                              
                             
                               [ 
                               
                                 
                                   S 
                                   i 
                                 
                                 
                                   S 
                                   
                                     i 
                                     - 
                                     1 
                                   
                                 
                               
                               ] 
                             
                           
                         
                          
                         
                             
                         
                       
                     
                      
                     • 
                      
                     
                         
                     
                      
                     
                       S 
                       0 
                     
                      
                     
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           k 
                         
                          
                         
                             
                         
                          
                         
                           X 
                           i 
                         
                       
                     
                   
                 
                 ; 
               
             
           
         
       
       
         
           
             
               Z 
               n 
             
             = 
             
               
                 S 
                 0 
               
                
               
                 ( 
                 
                   
                     α 
                     0 
                   
                   + 
                   
                     
                        
                       
                         X 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           α 
                           1 
                         
                         + 
                         
                           
                              
                             
                               X 
                               2 
                             
                           
                            
                           
                             ( 
                             
                               
                                 α 
                                 2 
                               
                               + 
                               
                                 
                                    
                                   
                                     X 
                                     3 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       α 
                                       3 
                                     
                                     + 
                                     
                                       … 
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                            
                                           
                                             X 
                                             
                                               n 
                                               - 
                                               1 
                                             
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               α 
                                               
                                                 n 
                                                 - 
                                                 1 
                                               
                                             
                                             + 
                                             
                                               
                                                 α 
                                                 n 
                                               
                                                
                                               
                                                  
                                                 
                                                   X 
                                                   n 
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                  
                 
                   2 
                    
                   π 
                    
                   
                       
                   
                    
                   
                     tZ 
                     n 
                   
                 
               
               = 
               
                 
                   
                     ∏ 
                     
                       k 
                       = 
                       0 
                     
                     n 
                   
                    
                   
                       
                   
                    
                   
                      
                     
                       2 
                        
                       π 
                        
                       
                           
                       
                        
                       t 
                        
                       
                           
                       
                        
                       
                         α 
                         k 
                       
                        
                       
                         S 
                         0 
                       
                        
                       
                         
                           ∏ 
                           
                             i 
                             = 
                             1 
                           
                           k 
                         
                          
                         
                             
                         
                          
                         
                            
                           
                             X 
                             i 
                           
                         
                       
                     
                   
                 
                 = 
                 
                   
                     ∏ 
                     
                       k 
                       = 
                       0 
                     
                     n 
                   
                    
                   
                       
                   
                    
                   
                      
                     
                       2 
                        
                       π 
                        
                       
                           
                       
                        
                       t 
                        
                       
                           
                       
                        
                       
                         α 
                         k 
                       
                        
                       
                         S 
                         0 
                       
                        
                       
                          
                         
                           
                             ( 
                             
                               n 
                               - 
                               k 
                               + 
                               1 
                             
                             ) 
                           
                            
                           
                             X 
                             k 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               Since 
               , 
               
                 the 
                  
                 
                     
                 
                  
                 
                   
                     ( 
                     
                       X 
                       k 
                     
                     ) 
                   
                   
                     0 
                     ≤ 
                     k 
                     ≤ 
                     n 
                   
                 
                  
                 
                   
                       
                   
                    
                   
                       
                   
                 
                  
                 are 
                  
                 
                     
                 
                  
                 independent 
               
               , 
               
                 
                   X 
                   0 
                 
                 ≡ 
                 0 
               
               , 
               
                 
 
               
                
               
                   
               
                
               
                 
                   E 
                    
                   
                     ( 
                     
                        
                       
                         2 
                          
                          
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         
                           tZ 
                           n 
                         
                       
                     
                     ) 
                   
                 
                 = 
                 
                   
                      
                     
                       2 
                        
                        
                        
                       
                           
                       
                        
                       π 
                        
                       
                           
                       
                        
                       t 
                        
                       
                           
                       
                        
                       
                         α 
                         0 
                       
                        
                       
                         S 
                         0 
                       
                     
                   
                    
                   
                     
                       ∏ 
                       
                         k 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                         
                     
                      
                     
                       E 
                       ( 
                       
                          
                         
                           2 
                            
                            
                            
                           
                               
                           
                            
                           π 
                            
                           
                               
                           
                            
                           t 
                            
                           
                               
                           
                            
                           
                             α 
                             k 
                           
                            
                           
                             S 
                             0 
                           
                            
                           
                              
                             
                               
                                 ( 
                                 
                                   n 
                                   - 
                                   k 
                                   + 
                                   1 
                                 
                                 ) 
                               
                                
                               
                                 X 
                                 k 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                    
                   
                     2 
                      
                      
                      
                     
                         
                     
                      
                     π 
                      
                     
                         
                     
                      
                     t 
                      
                     
                         
                     
                      
                     
                       α 
                       k 
                     
                      
                     
                       S 
                       0 
                     
                      
                     
                        
                       
                         
                           ( 
                           
                             n 
                             - 
                             k 
                             + 
                             1 
                           
                           ) 
                         
                          
                         
                           X 
                           k 
                         
                       
                     
                   
                 
                 = 
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     
                       + 
                       ∞ 
                     
                   
                    
                   
                     
                       
                         h 
                         k 
                       
                        
                       
                         ( 
                         s 
                         ) 
                       
                     
                      
                     
                        
                       
                         2 
                          
                          
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         
                           sX 
                           k 
                         
                       
                     
                      
                     
                        
                       s 
                     
                   
                 
               
               ; 
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   h 
                   
                     a 
                     , 
                     b 
                   
                 
                  
                 
                   ( 
                   k 
                   ) 
                 
               
               = 
               
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     
                       + 
                       ∞ 
                     
                   
                    
                   
                     
                        
                       
                         2 
                          
                         b 
                          
                         
                             
                         
                          
                         
                           π 
                           ax 
                         
                       
                     
                      
                     
                        
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         kx 
                       
                     
                      
                     
                         
                     
                      
                     
                        
                       x 
                     
                   
                 
                 = 
                 
                   
                     - 
                     
                       
                         
                           ( 
                           
                             
                               - 
                               2 
                             
                              
                             π 
                              
                             
                                 
                             
                              
                             b 
                           
                           ) 
                         
                         
                           
                             2 
                              
                              
                              
                             
                                 
                             
                              
                             k 
                              
                             
                                 
                             
                              
                             π 
                           
                           a 
                         
                       
                       a 
                     
                   
                    
                   
                     Γ 
                      
                     
                       ( 
                       
                         
                           2 
                            
                            
                            
                           
                               
                           
                            
                           k 
                            
                           
                               
                           
                            
                           π 
                         
                         a 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 h 
                 k 
               
                
               
                 ( 
                 s 
                 ) 
               
             
             = 
             
               
                 
                   ∫ 
                   
                     - 
                     ∞ 
                   
                   
                     + 
                     ∞ 
                   
                 
                  
                 
                   
                      
                     
                       2 
                        
                       π 
                        
                       
                           
                       
                        
                       t 
                        
                       
                           
                       
                        
                       
                         α 
                         k 
                       
                        
                       
                         S 
                         0 
                       
                        
                       
                          
                         
                           
                             ( 
                             
                               n 
                               - 
                               k 
                               + 
                               1 
                             
                             ) 
                           
                            
                           x 
                         
                       
                     
                   
                    
                   
                      
                     
                       
                         - 
                         2 
                       
                        
                        
                        
                       
                           
                       
                        
                       π 
                        
                       
                           
                       
                        
                       sx 
                     
                   
                    
                   
                      
                     x 
                   
                 
               
               = 
               
                 
                   
                     h 
                     
                       
                         ( 
                         
                           n 
                           - 
                           k 
                           + 
                           1 
                         
                         ) 
                       
                       , 
                       
                         it 
                          
                         
                             
                         
                          
                         
                           α 
                           k 
                         
                          
                         
                           S 
                           0 
                         
                       
                     
                   
                    
                   
                     ( 
                     
                       - 
                       S 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     
                       
                         ( 
                         
                           
                             - 
                             2 
                           
                            
                           π 
                            
                           
                               
                           
                            
                            
                            
                           
                               
                           
                            
                           t 
                            
                           
                               
                           
                            
                           
                             α 
                             k 
                           
                            
                           
                             S 
                             0 
                           
                         
                         ) 
                       
                       
                         
                           
                             - 
                             2 
                           
                            
                           is 
                            
                           
                               
                           
                            
                           π 
                         
                         
                           ( 
                           
                             n 
                             - 
                             k 
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                     
                       ( 
                       
                         n 
                         - 
                         k 
                         + 
                         1 
                       
                       ) 
                     
                   
                    
                   
                     Γ 
                      
                     
                       ( 
                       
                         
                           
                             - 
                             2 
                           
                            
                            
                            
                           
                               
                           
                            
                           s 
                            
                           
                               
                           
                            
                           π 
                         
                         
                           ( 
                           
                             n 
                             - 
                             k 
                             + 
                             1 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 E 
                 ( 
                 
                    
                   
                     2 
                      
                      
                      
                     
                         
                     
                      
                     πt 
                      
                     
                         
                     
                      
                     
                       α 
                       k 
                     
                      
                     
                       S 
                       0 
                     
                      
                     
                        
                       
                         
                           ( 
                           
                             n 
                             - 
                             k 
                             + 
                             1 
                           
                           ) 
                         
                          
                         
                           X 
                           k 
                         
                       
                     
                   
                 
                 ) 
               
               = 
               
                 
                   ∫ 
                   
                     - 
                     ∞ 
                   
                   
                     + 
                     ∞ 
                   
                 
                  
                 
                   
                     
                       h 
                       k 
                     
                      
                     
                       ( 
                       s 
                       ) 
                     
                   
                    
                   
                     E 
                      
                     
                       ( 
                       
                          
                         
                           2 
                            
                            
                            
                           
                               
                           
                            
                           π 
                            
                           
                               
                           
                            
                           
                             sX 
                             k 
                           
                         
                       
                       ) 
                     
                   
                    
                   
                       
                   
                    
                   
                      
                     s 
                   
                 
               
             
           
         
       
       
         
           
             
               E 
               ( 
               
                  
                 
                   2 
                    
                   π 
                    
                   
                       
                   
                    
                   t 
                    
                   
                       
                   
                    
                   
                     α 
                     k 
                   
                    
                   
                     S 
                     0 
                   
                    
                   
                      
                     
                       
                         ( 
                         
                           n 
                           - 
                           k 
                           + 
                           1 
                         
                         ) 
                       
                        
                       
                         X 
                         k 
                       
                     
                   
                 
               
               ) 
             
             = 
             
               
                 ∫ 
                 
                   - 
                   ∞ 
                 
                 
                   + 
                   ∞ 
                 
               
                
               
                 
                   
                     
                       ( 
                       
                         
                           - 
                           2 
                         
                          
                         π 
                          
                         
                             
                         
                          
                         t 
                          
                         
                             
                         
                          
                         
                           α 
                           k 
                         
                          
                         
                           S 
                           0 
                         
                       
                       ) 
                     
                     
                       
                         
                           - 
                           2 
                         
                          
                         is 
                          
                         
                             
                         
                          
                         π 
                       
                       
                         ( 
                         
                           n 
                           - 
                           k 
                           + 
                           1 
                         
                         ) 
                       
                     
                   
                   
                     ( 
                     
                       n 
                       - 
                       k 
                       + 
                       1 
                     
                     ) 
                   
                 
                  
                 
                   Γ 
                    
                   
                     ( 
                     
                       
                         
                           - 
                           2 
                         
                          
                          
                          
                         
                             
                         
                          
                         s 
                          
                         
                             
                         
                          
                         π 
                       
                       
                         ( 
                         
                           n 
                           - 
                           k 
                           + 
                           1 
                         
                         ) 
                       
                     
                     ) 
                   
                 
                  
                 
                   E 
                    
                   
                     ( 
                     
                        
                       
                         2 
                          
                          
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         
                           sX 
                           k 
                         
                       
                     
                     ) 
                   
                 
                  
                 
                    
                   s 
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 f 
                  
                 
                   ( 
                   
                     Z 
                     n 
                   
                   ) 
                 
               
               = 
               
                 
                   ∫ 
                   
                     - 
                     ∞ 
                   
                   
                     + 
                     ∞ 
                   
                 
                  
                 
                   
                     
                       f 
                       ~ 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                    
                   
                      
                     
                       2 
                        
                       π 
                        
                       
                           
                       
                        
                       
                         tZ 
                         n 
                       
                     
                   
                    
                   
                      
                     t 
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 E 
                  
                 
                   ( 
                   
                      
                     
                       2 
                        
                       π 
                        
                       
                           
                       
                        
                       
                         tZ 
                         n 
                       
                     
                   
                   ) 
                 
               
               = 
               
                 
                    
                   
                     2 
                      
                     π 
                      
                     
                         
                     
                      
                     t 
                      
                     
                         
                     
                      
                     
                       α 
                       0 
                     
                      
                     
                       S 
                       0 
                     
                   
                 
                  
                 
                   
                     ∏ 
                     
                       k 
                       = 
                       1 
                     
                     n 
                   
                    
                   
                       
                   
                    
                   
                     E 
                     ( 
                     
                        
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         t 
                          
                         
                             
                         
                          
                         
                           α 
                           k 
                         
                          
                         
                           S 
                           0 
                         
                          
                         
                            
                           
                             
                               ( 
                               
                                 n 
                                 - 
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                              
                             
                               X 
                               k 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               E 
                
               
                 ( 
                 
                   f 
                    
                   
                     ( 
                     
                       Z 
                       n 
                     
                     ) 
                   
                 
                 ) 
               
             
             = 
             
               
                 ∫ 
                 
                   - 
                   ∞ 
                 
                 
                   + 
                   ∞ 
                 
               
                
               
                 
                   
                     f 
                     ~ 
                   
                    
                   
                     ( 
                     t 
                     ) 
                   
                 
                  
                 
                    
                   
                     2 
                      
                      
                      
                     
                         
                     
                      
                     π 
                      
                     
                         
                     
                      
                     t 
                      
                     
                         
                     
                      
                     
                       α 
                       0 
                     
                      
                     
                       S 
                       0 
                     
                   
                 
                  
                 
                     
                 
                  
                 
                   
                     ∏ 
                     
                       k 
                       = 
                       1 
                     
                     n 
                   
                    
                   
                       
                   
                    
                   
                     
                       ( 
                       
                         
                           ∫ 
                           
                             - 
                             ∞ 
                           
                           
                             + 
                             ∞ 
                           
                         
                          
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     - 
                                     2 
                                   
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                    
                                    
                                   
                                       
                                   
                                    
                                   t 
                                    
                                   
                                       
                                   
                                    
                                   
                                     α 
                                     k 
                                   
                                    
                                   
                                     S 
                                     0 
                                   
                                 
                                 ) 
                               
                               
                                 
                                   
                                     - 
                                     2 
                                   
                                    
                                   is 
                                    
                                   
                                       
                                   
                                    
                                   π 
                                 
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     k 
                                     + 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                             
                               ( 
                               
                                 n 
                                 - 
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                            
                           
                             Γ 
                              
                             
                                 
                             
                             ( 
                             
                               
                                 
                                   - 
                                   2 
                                 
                                  
                                  
                                  
                                 
                                     
                                 
                                  
                                 s 
                                  
                                 
                                     
                                 
                                  
                                 π 
                               
                               
                                 ( 
                                 
                                   n 
                                   - 
                                   k 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                             ) 
                           
                            
                           
                             E 
                              
                             
                               ( 
                               
                                  
                                 
                                   2 
                                    
                                    
                                    
                                   
                                       
                                   
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   s 
                                    
                                   
                                       
                                   
                                    
                                   
                                     X 
                                     k 
                                   
                                 
                               
                               ) 
                             
                           
                            
                           
                              
                             s 
                           
                         
                       
                       ) 
                     
                      
                     
                        
                       t 
                     
                   
                 
               
             
           
         
       
     
         [0173]    Depending on the specification of Ψ k  the integral 
         [0000]    
       
         
           
             
               ∫ 
               
                 - 
                 ∞ 
               
               
                 + 
                 ∞ 
               
             
              
             
               
                 
                   
                     ( 
                     
                       
                         - 
                         2 
                       
                        
                       π 
                        
                       
                           
                       
                        
                       it 
                        
                       
                           
                       
                        
                       
                         α 
                         k 
                       
                        
                       
                         S 
                         0 
                       
                     
                     ) 
                   
                   
                     
                       
                         - 
                         2 
                       
                        
                       i 
                        
                       
                           
                       
                        
                       s 
                        
                       
                           
                       
                        
                       π 
                     
                     
                       ( 
                       
                         n 
                         - 
                         k 
                         + 
                         1 
                       
                       ) 
                     
                   
                 
                 
                   ( 
                   
                     n 
                     - 
                     k 
                     + 
                     1 
                   
                   ) 
                 
               
                
               
                 Γ 
                  
                 
                   ( 
                   
                     
                       
                         - 
                         2 
                       
                        
                       i 
                        
                       
                           
                       
                        
                       s 
                        
                       
                           
                       
                        
                       π 
                     
                     
                       ( 
                       
                         n 
                         - 
                         k 
                         + 
                         1 
                       
                       ) 
                     
                   
                   ) 
                 
               
                
               
                 E 
                  
                 
                   ( 
                   
                      
                     
                       2 
                        
                        
                        
                       
                           
                       
                        
                       π 
                        
                       
                           
                       
                        
                       
                         sX 
                         k 
                       
                     
                   
                   ) 
                 
               
                
               
                  
                 s 
               
             
           
         
       
     
         [0000]    may actually be analytically computable with a closed form. 
         [0174]    The price of the Asian derivatives contract hence becomes 
         [0000]    
       
         
           
             
               
                 Π 
                 
                   0 
                   , 
                   n 
                 
               
                
               
                 ( 
                 
                   f 
                    
                   
                     ( 
                     
                       Z 
                       n 
                     
                     ) 
                   
                 
                 ) 
               
             
             = 
             
               
                 B 
                 
                   o 
                   , 
                   n 
                 
                 0 
               
               × 
               
                 
                   ∫ 
                   
                     - 
                     ∞ 
                   
                   
                     + 
                     ∞ 
                   
                 
                  
                 
                   
                     
                       f 
                       ~ 
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                    
                   
                      
                     
                       2 
                        
                        
                        
                       
                           
                       
                        
                       π 
                        
                       
                           
                       
                        
                       t 
                        
                       
                           
                       
                        
                       
                         α 
                         0 
                       
                        
                       
                         S 
                         0 
                       
                     
                   
                    
                   
                       
                   
                    
                   
                     
                       ∏ 
                       
                         k 
                         = 
                         1 
                       
                       n 
                     
                      
                     
                       
                         ( 
                         
                           
                             ∫ 
                             
                               - 
                               ∞ 
                             
                             
                               + 
                               ∞ 
                             
                           
                            
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       - 
                                       2 
                                     
                                      
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     it 
                                      
                                     
                                         
                                     
                                      
                                     
                                       α 
                                       k 
                                     
                                      
                                     
                                       S 
                                       0 
                                     
                                   
                                   ) 
                                 
                                 
                                   
                                     
                                       - 
                                       2 
                                     
                                      
                                     i 
                                      
                                     
                                         
                                     
                                      
                                     s 
                                      
                                     
                                         
                                     
                                      
                                     π 
                                   
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       k 
                                       + 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                               
                                 ( 
                                 
                                   n 
                                   - 
                                   k 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                              
                             
                               Γ 
                                
                               
                                 ( 
                                 
                                   
                                     
                                       - 
                                       2 
                                     
                                      
                                     i 
                                      
                                     
                                         
                                     
                                      
                                     s 
                                      
                                     
                                         
                                     
                                      
                                     π 
                                   
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       k 
                                       + 
                                       1 
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               E 
                                
                               
                                 ( 
                                 
                                    
                                   
                                     2 
                                      
                                      
                                      
                                     
                                         
                                     
                                      
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     
                                       sX 
                                       k 
                                     
                                   
                                 
                                 ) 
                               
                             
                              
                             
                                
                               s 
                             
                           
                         
                         ) 
                       
                        
                       
                          
                         t 
                       
                     
                   
                 
               
             
           
         
       
     
       Remark 
     (i) A Few Examples of Relevant Functions f And Their Fourier Transforms 
       [0175]    For Asian Derivatives 
         [0000]    
       
         
           
             
               
                 f 
                  
                 
                   ( 
                   x 
                   ) 
                 
               
               = 
               
                 Max 
                  
                 
                   ( 
                   
                     x 
                     , 
                     0 
                   
                   ) 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   f 
                   ~ 
                 
                  
                 
                   ( 
                   k 
                   ) 
                 
               
               = 
               
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     
                       + 
                       ∞ 
                     
                   
                    
                   
                     
                       f 
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                      
                     
                        
                       
                         
                           - 
                           2 
                         
                          
                          
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         kt 
                       
                     
                      
                     
                        
                       t 
                     
                   
                 
                 = 
                 
                   - 
                   
                     1 
                     
                       4 
                        
                       
                         π 
                         2 
                       
                        
                       
                         k 
                         2 
                       
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 f 
                  
                 
                   ( 
                   x 
                   ) 
                 
               
               = 
               
                 Min 
                  
                 
                   ( 
                   
                     x 
                     , 
                     0 
                   
                   ) 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   f 
                   ~ 
                 
                  
                 
                   ( 
                   k 
                   ) 
                 
               
               = 
               
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     
                       + 
                       ∞ 
                     
                   
                    
                   
                     
                       f 
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                      
                     
                        
                       
                         
                           - 
                           2 
                         
                          
                          
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         kt 
                       
                     
                      
                     
                        
                       t 
                     
                   
                 
                 = 
                 
                   1 
                   
                     4 
                      
                     
                       π 
                       2 
                     
                      
                     
                       k 
                       2 
                     
                   
                 
               
             
             ; 
           
         
       
     
         [0176]    For Volatility Derivatives 
         [0000]    
       
         
           
             
               
                 f 
                  
                 
                   ( 
                   x 
                   ) 
                 
               
               = 
               
                 Max 
                  
                 
                   ( 
                   
                     
                       
                         x 
                       
                       - 
                       a 
                     
                     , 
                     0 
                   
                   ) 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 f 
                 ~ 
               
                
               
                 ( 
                 k 
                 ) 
               
             
             = 
             
               - 
               
                 1 
                 
                   120 
                    
                   
                       
                   
                    
                   k 
                    
                   
                       
                   
                    
                   π 
                    
                   
                       
                   
                    
                   k 
                    
                   
                     
                        
                       k 
                        
                     
                     
                       3 
                       / 
                       2 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 i 
                  
                 
                   ( 
                   
                     
                       
                         
                           
                             - 
                             15 
                           
                            
                           
                             k 
                              
                             
                               ( 
                               
                                 i 
                                 + 
                                 
                                   Sign 
                                    
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             + 
                             4 
                           
                            
                           a 
                            
                           
                             
                                
                               k 
                                
                             
                             
                               3 
                               / 
                               2 
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     15 
                                     + 
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               
                                                 
                                                   - 
                                                   20 
                                                 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   ia 
                                                   2 
                                                 
                                                  
                                                 k 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 π 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   HypergeometricPFQ 
                                                    
                                                   
                                                     [ 
                                                     
                                                       
                                                         { 
                                                         
                                                           3 
                                                           4 
                                                         
                                                         } 
                                                       
                                                       , 
                                                       
                                                         { 
                                                         
                                                           
                                                             1 
                                                             2 
                                                           
                                                           , 
                                                           
                                                             7 
                                                             4 
                                                           
                                                         
                                                         } 
                                                       
                                                       , 
                                                       
                                                         
                                                           - 
                                                           
                                                             a 
                                                             4 
                                                           
                                                         
                                                          
                                                         
                                                           k 
                                                           2 
                                                         
                                                          
                                                         
                                                           π 
                                                           2 
                                                         
                                                       
                                                     
                                                     ] 
                                                   
                                                 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   + 
                                                   3 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     
                                                       
                                                         
                                                           
                                                             - 
                                                             5 
                                                           
                                                           + 
                                                           
                                                             5 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             Cos 
                                                              
                                                             
                                                             ( 
                                                             
                                                             2 
                                                              
                                                             
                                                             a 
                                                             2 
                                                             
                                                              
                                                             k 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             π 
                                                             
                                                             ) 
                                                             
                                                             
                                                           
                                                         
                                                       
                                                     
                                                     
                                                       
                                                         
                                                           
                                                             + 
                                                             8 
                                                           
                                                            
                                                           
                                                             a 
                                                             4 
                                                           
                                                            
                                                           
                                                             k 
                                                             2 
                                                           
                                                            
                                                           
                                                             π 
                                                             2 
                                                           
                                                            
                                                           
                                                             HypergeometricPFQ 
                                                              
                                                             
                                                             [ 
                                                             
                                                             
                                                             { 
                                                             
                                                             5 
                                                             4 
                                                             
                                                             } 
                                                             
                                                             , 
                                                             
                                                             { 
                                                             
                                                             
                                                             3 
                                                             2 
                                                             
                                                             , 
                                                             
                                                             9 
                                                             4 
                                                             
                                                             
                                                             } 
                                                             
                                                             , 
                                                             
                                                             
                                                             - 
                                                             
                                                             a 
                                                             4 
                                                             
                                                             
                                                              
                                                             
                                                             k 
                                                             2 
                                                             
                                                              
                                                             
                                                             π 
                                                             2 
                                                             
                                                             
                                                             
                                                             ] 
                                                             
                                                           
                                                         
                                                       
                                                     
                                                     
                                                       
                                                         
                                                           
                                                             + 
                                                             5 
                                                           
                                                            
                                                           i 
                                                            
                                                           
                                                               
                                                           
                                                            
                                                           
                                                             Sin 
                                                              
                                                             
                                                             ( 
                                                             
                                                             2 
                                                              
                                                             
                                                             a 
                                                             2 
                                                             
                                                              
                                                             k 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             π 
                                                             
                                                             ) 
                                                             
                                                           
                                                         
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                       × 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     UnitStep 
                                      
                                     
                                       [ 
                                       a 
                                       ] 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   ) 
                 
               
               ) 
             
             ; 
           
         
       
       
         
           
             
               
                 f 
                  
                 
                   ( 
                   x 
                   ) 
                 
               
               = 
               
                 Min 
                  
                 
                   ( 
                   
                     
                       
                         x 
                       
                       - 
                       a 
                     
                     , 
                     0 
                   
                   ) 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 
                   f 
                   ~ 
                 
                  
                 
                   ( 
                   k 
                   ) 
                 
               
               = 
               
                 
                   ( 
                   
                     
                       2 
                        
                       
                           
                       
                        
                       ia 
                        
                       
                         k 
                       
                     
                     - 
                     
                       i 
                       × 
                       
                         FresnelC 
                          
                         
                           [ 
                           
                             2 
                              
                             a 
                              
                             
                               k 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       FresnelS 
                        
                       
                         [ 
                         
                           2 
                            
                           a 
                            
                           
                             k 
                           
                         
                         ] 
                       
                     
                   
                   ) 
                 
                 × 
                 
                   
                     ( 
                     
                       
                         - 
                         1 
                       
                       + 
                       
                         UnitStep 
                          
                         
                           [ 
                           
                             - 
                             a 
                           
                           ] 
                         
                       
                     
                     ) 
                   
                   
                     4 
                      
                     π 
                      
                     
                         
                     
                      
                     
                       k 
                       
                         3 
                         / 
                         2 
                       
                     
                   
                 
               
             
             ; 
           
         
       
     
         [0000]    with the following notations 
         [0177]    HypergeometricPFQ[{a 1 , . . . , a p }, {b 1 , . . . , b q }, z] is the generalized hypergeometric function p F q (a;b;z)    
         [0178]    p F q (a; b; z)  has series expansion 
         [0000]    
       
         
           
             
               ∑ 
               
                 k 
                 = 
                 0 
               
               ∞ 
             
              
             
               
                 
                   ( 
                   
                     a 
                     1 
                   
                   ) 
                 
                 k 
               
                
               
                   
               
                
               … 
                
               
                   
               
                
               
                 
                   
                     ( 
                     
                       a 
                       p 
                     
                     ) 
                   
                   k 
                 
                 / 
                 
                   
                     ( 
                     
                       b 
                       1 
                     
                     ) 
                   
                   k 
                 
               
                
               
                   
               
                
               … 
                
               
                   
               
                
               
                 
                   ( 
                   
                     b 
                     q 
                   
                   ) 
                 
                 k 
               
                
               
                 
                   z 
                   k 
                 
                 / 
                 
                   
                     k 
                     ! 
                   
                   . 
                 
               
             
           
         
       
     
         [0179]    FresnelC[z] gives the Fresnel integral C(z) and is given by ∫ 0   z cos(πt 2 /2)dt 
         [0180]    FresnelS[z] gives the Fresnel integral S(z) and is given by ∫ 0   z cos(πt 2 /2)dt 
         [0181]    UnitStep[x] represents the unit step function, equal to 0 for x&lt;0 and 1 for x□0. 
         [0182]    These functions are all predefined in many scientific software packages, for instance Mathematica® whose notation is adopted here. 
       (ii) How the Products of Integrals Inside the Integral Help Reduce the Dimension of Integrals 
       [0183]    Each integral is a function g of t and k. We represent g in a desirable functional basis such as the Fourier basis or a polynomial basis. Computing each term in the Fourier basis for instance is a triple integral. With the Taylor decomposition in a polynomial basis, we may need to perform derivations. In each case, we may need to actually compute only a handful of those. 
         [0184]    The use of the Fourier Basis here is not restrictive but merely an appropriate illustration that fits well with the use of Levy processes to model the dynamic of underlyings and the ready availability of tools to speed computations such as the Fast Fourier Transform. 
         [0185]    Indeed the general projection procedure works for any selected basis in a quadratic approximation representation. As such using the generic result, and having pre-computed the a ij  once the basis of f j , is selected, we simply need to compute the coefficients 
         [0000]    
       
         
           
             
               b 
               j 
             
             = 
             
               
                 ∫ 
                 Δ 
               
                
               
                 
                   
                     f 
                     j 
                   
                    
                   
                     ( 
                     X 
                     ) 
                   
                 
                  
                 
                   f 
                    
                   
                     ( 
                     X 
                     ) 
                   
                 
                  
                 
                   
                      
                     
                       μ 
                        
                       
                         ( 
                         X 
                         ) 
                       
                     
                   
                   . 
                 
               
             
           
         
       
     
         [0186]    As said earlier usually the functions constitute a Hilbert Basis so that the matrix A is the identity, simplifying computations. In the engineering sciences, the other basis to have been successful in this type of endeavor, is the family of wavelets. 
       (iii) How to Compute the Integrals In General 
       [0187]    In General the integrals would be computed by resealing the real set into a open set whose closure is an interval. If one is inclined to use the residue theorem, it is important to ensure that the integrand with the variables extended to the complex set is a meromorphic function. As dimensions grow, low discrepancy sequences such as those reviewed in chapter XII of the Book BICs 4 Derivatives Volume I may come into play. 
       3.3. The Law of the Few In Other Derivatives Pricing Innovations 
       [0188]    Current simplifying methods for derivatives pricing can be viewed as possibly unconscious mere applications of the Law of the Few. This can be seen in the major numerical methods for derivatives pricing currently in use, PDEs, Monte Carlo and Trees. 
       3.3.1. PDEs: Galerkin Methods 
       [0189]    The projection method as described here provides benefits similar to Galerkin methods used in the resolution of Partial Differential Equations (PDE), arguably in our use from amore general and easier to implement perspective, and thus help very effectively counter some of the benefits that could be claimed in a numerical resolution of PDEs. 
         [0190]    The essence of the Galerkin methods is as follows. Suppose we have a problem of the form: Lu=f where L is a linear operator, f is a given function, u is an unknown function. One tries to solve u by selecting a set of basis functions u 1 , . . ., u n  and try to find the best approximation of the solution to the equation that is a combination of the basis functions,
       The goal is to find       
 
         [0000]    
       
         
           
             
               
                 Arg 
                  
                 
                     
                 
                  
                 Min 
               
               
                 
                   α 
                   i 
                 
                 , 
                 
                   i 
                   = 
                   1 
                 
                 , 
                 
                     
                 
                  
                 … 
                  
                 
                     
                 
                 , 
                 n 
               
             
              
             
               
                 
                    
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         n 
                       
                        
                       
                         
                           α 
                           i 
                         
                          
                         
                           Lu 
                           i 
                         
                       
                     
                     - 
                     f 
                   
                    
                 
                 2 
                 2 
               
               . 
             
           
         
       
       
         
           
             We note: 
           
         
       
     
         [0000]    
       
         
           
             
               
                 a 
                 
                   i 
                   , 
                   j 
                 
               
               = 
               
                 &lt; 
                 
                   u 
                   i 
                 
               
             
             , 
             
               
                 
                   Lu 
                   j 
                 
                 &gt; 
               
               ; 
             
           
         
       
       
         
           
             
               A 
               = 
               
                 
                   ( 
                   
                     a 
                     
                       i 
                       , 
                       j 
                     
                   
                   ) 
                 
                 
                   i 
                   , 
                   j 
                 
               
             
             ; 
           
         
       
       
         
           
             
               
                 b 
                 i 
               
               = 
               
                 &lt; 
                 
                   u 
                   i 
                 
               
             
             , 
             
               
                 f 
                 &gt; 
               
               ; 
             
           
         
       
       
         
           
             
               B 
               = 
               
                 
                   ( 
                   
                     b 
                     i 
                   
                   ) 
                 
                 i 
               
             
             ; 
           
         
       
       
         
           
             
               
                 α 
                 i 
               
               = 
               
                 &lt; 
                 
                   u 
                   i 
                 
               
             
             , 
             
               
                 f 
                 &gt; 
               
               ; 
             
           
         
       
       
         
           
             If A is symmetric inversible, 
           
         
       
     
         [0000]    
       
         
           
             
               
                 
                   
                     Arg 
                      
                     
                         
                     
                      
                     Min 
                   
                   
                     
                       α 
                       i 
                     
                     , 
                     
                       i 
                       = 
                       1 
                     
                     , 
                     
                         
                     
                      
                     … 
                      
                     
                         
                     
                     , 
                     n 
                   
                 
                  
                 
                   
                      
                     
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                          
                         
                           
                             α 
                             i 
                           
                            
                           
                             Lu 
                             i 
                           
                         
                       
                       - 
                       f 
                     
                      
                   
                   2 
                   2 
                 
               
               = 
               
                 
                   A 
                   
                     - 
                     1 
                   
                 
                  
                 B 
               
             
             ; 
           
         
       
     
         [0000]    i.e.: 
       3.3.2. Monte Carlo 
       [0194]    The state of the art propose a least square regression method on certain polynomial functions for pricing American options in a Monte Carlo simulation framework which can be related to the projection approach described here. State of the art tests indicate that the method is robust to the choice of alternative polynomial choices and require very few polynomials. 
       3.3.3. Trees 
       [0195]    Readers familiar with binomial or trinomial pricing methods are well aware that with non-recombining trees, the number of nodes grows exponentially with the number of time steps. That is if at step 1 we have two nodes, at step 2 we have 22=4 nodes, at step n we have 2n nodes. In practice one uses the recombination assumption in trees that reflects Markov assumptions on the dynamic of the underlying. This allows the size of trees to grow linearly. The Markov assumption can be viewed here as an illustration of the law of the few where a dependence on several underlyings is reduced to a single underlying. In subsequent recent uses of trees to price path dependent options, the sate of the art proposes a path bundling method in the pricing of Asian options. Such a method can be related to the change of variable approaches. 
       4. Conclusion 
       [0196]    The perspective provided here is not meant to be exhaustive but to help the interested reader now more easily identify numerous other instances where this law is at play, facilitating the practical resolution of the problems under investigation. 
         [0197]    In any case, we have shown here how the BICs analysis, for its practical use, relies on this general law or principle to become an operational and practical tool of derivatives analysis. 
         [0198]    All patents cited in this application are hereby included by reference. While the present invention has been described in connection with preferred embodiments, it is not intended to limit the scope of the invention to the particular form set forth, but, on the contrary, it is intended to cover such alternatives, modifications, equivalents as may be included within the spirit and scope of the invention defined in the appended claims.

Technology Category: g