Abstract:
Systems, methods and computer program products stored on a computer readable medium or media are described for providing pricing of mortgage-backed securities and other financial instruments. Multiple cashflow paths resulting from different interest rate development scenarios are consolidated into a single cashflow path. A continuously compounded interest rate spread (“equivalent cc spread”) corresponding to the option adjusted spread prevailing at the market is determined, as is a relationship between the two. Reformulating computations of option adjusted spreads and prices using an equivalent cc spread and factoring out assumption changes approximates individual cashflow path and interest rate path calculations and allows use of consolidated cashflows calculations as a very good approximation. In some embodiments, the consolidated cashflow and the relationship between two spreads is employed to compute the value of the financial instrument, as a very good approximation, at a later time, so that the price reflects changes in the market yield curve and the option adjusted spread. For example, interest rate paths, cashflow paths, consolidated cashflow paths, discount factors and the equivalent cc spread are computed at market close, and only the consolidated cashflow paths, discount factors and the equivalent cc spread (and not the interest rate paths and cashflow paths) are stored for later on-demand use when the discount factors are adjusted to reflect the then current market environment.

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This application claims priority of U.S. Provisional Application Ser. No. 61/096,247, filed Sep. 11, 2008, which is incorporated by reference herein in its entirety. 
    
    
     COPYRIGHT NOTICE 
     A portion of the disclosure of this patent document contains material which is subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent document or the patent disclosure, as it appears in the Patent and Trademark Office patent files or records, but otherwise reserves all copyrights whatsoever. 
     BACKGROUND OF THE INVENTION 
     The invention disclosed herein relates generally to the field of securities whose value is based on future cashflows, and analysis thereof, including providing valuations thereof. One example of such a security is mortgage-backed securities (MBSs). Other securities (or derivatives, instruments, etc.) to which the disclosure herein applies are those whose value is interest path dependent. The invention relates to methods, systems and computer program products for implementing such valuation and analysis. 
     MBS instruments, which comprise pools of mortgages and collateralized mortgage obligations (CMOs), are analyzed for various reasons including computing valuations and metrics such as pricing, effective duration and key rate durations, and sensitivity to market variables. MBS instrument valuations and metrics may be provided for different interest rate models and for different market scenarios within an interest rate model. An interest rate model “predicts” what the interest rates will be in the future over a specified period of time based on model assumptions and/or parameters obtained from past market behavior. Since future interest rates are uncertain, a credible interest rate model offers many different possible “paths” for future interest rate dynamics. For example, an “exogenous” interest rate model uses the current yield curve as a starting point in its “prediction” of future interest rates. One interest rate model currently in use assumes normal dynamics of instantaneous “short” rate driven by two stochastic factors (so-called two factor Hull-White model). The specific dynamics are defined by the following parameters: current yield curve; ATM (at-the-money) swaption implied volatilities for 2-year and 10-year tenors for different maturities; mean reversion of each of the 2 stochastic factors (assumed constant); and correlation between the factors. 
     MBS analytics strive to answer various questions about the behavior of an instrument based on the change in the market. MBS valuations involve interest rate path simulations and cashflow computations. MBS instruments may have many possible interest rate paths along which cashflows may be generated dependent on interest rates, prepayments, defaults, and other market variables in general. Interest rate paths, which each can comprise a series of future interest rates on a specified schedule based on an interest rate model, can be simulated based on current interest rate term structure and ATM volatilities (A simulated interest rate path (or paths) is referred to herein as an “interest rate path” (or “paths”)). Cashflow paths are computed based on structure of MBS, interest rate paths, prepayment and default assumptions, and other conditions. 
     Methodologies for computing such valuations and metrics, and for simulating interest rate paths and for computing cashflows are known in the art, and are discussed, for example, in:  Analysis of Mortgage Backed Securities , Harvey J. Stein, Alexander Belikoff, Kirill Levin and Xusheng Tian, Jan. 5, 2007 &amp; Sep. 3, 2008 ; Mortgage OAS Analysis , Harvey Stein, Feb. 16, 2006 ; A Theoretical and Numerical Analysis of Collateralized Mortgage Obligations , Bjornar Andre Ulstein, Master Thesis in Financial Economics, The Norwegian School of Economics and Business Administration, Jun. 18, 2008 ; Fixed - Income Securities and Derivatives Handbook , Moorad Choudhry, Bloomberg Press, Princeton, N.J., 2005; and  Cash Flow Models for Pricing Mortgages , Philip Booth and Duncan Walsh, Research Paper 2000.02, City University Business School, London, November 2000, the entire disclosures of all of which are incorporated herein by reference. 
     Generally, the price, or present value, of an MBS instrument is the average of its discounted cashflows along each simulated cashflow path and along each simulated interest rate path. For example, see Expression 1 below. (The Expressions referred to herein may be found in the mathematical presentation below.) A discount factor is the present value of a unit of currency due to be paid at the end of a given period. 
     Pricing of MBS instruments involves non-linear analyses, and typically includes the use of Monte Carlo simulations, e.g., computational algorithms that rely on repeated random sampling to compute their results, which involve significant complexity, computer processing and computer storage resources. This makes such simulations very expensive for on-demand use, even when used with variance reduction techniques. 
     Because of complexity, cost and resource requirements, MBS valuations typically include two stages of processing performed at different times. In a first stage of processing (Stage 1), a number of interest rate paths are simulated and cashflow paths, e.g., a sequence of future payments made by the MBS instrument on a specified schedule, are computed for each interest rate path. 
     In Stage 1 processing in this previous approach, for a particular interest rate model, all of the interest rate paths are generated (and stored), and used to compute the cashflow paths along each of the interest rate paths for each market scenario in the valuation. In addition, Stage 1 processing also computes the discount factors for each cashflow path. This processing is performed once for a concerned instrument or set of instruments and saved for use in Stage 2 processing. Since Stage 1 processing typically involves large numbers of interest rate paths and cashflow path computations, which consumes significant computer processing and storage resources, such processing is typically run on a scheduled basis, e.g., nightly, as opposed to on-demand, e.g., when an investor requests a price for an instrument. 
     In a second stage of processing (Stage 2), pricing is computed using the previously computed cashflow paths and discount factors, and the current market environment for the particular instrument, i.e., observable market “parameters” at a given time which may vary in time and which are not specific to any instrument being analyzed. For example, these parameters may include the interest rate model parameters discussed above. 
     Stage 2 processing is performed on demand typically hours after or the day after Stage 1 processing. This allows very fast on-demand calculation of the MBS price based on the current market environment and the interest rate paths and cashflow paths previously calculated in Stage 1 processing, while reflecting changes in the market environment since the Stage 1 processing. 
     However, while Stage 1 processing and Stage 2 processing both are based on the then current market environment, since the processing takes place at different times or on different days, the market environments on which the Stage 1 and Stage 2 processing are based are typically different. For example, Stage 1 processing is performed overnight based on the previous day&#39;s market environment and Stage 2 processing is performed on demand the next day based on the current day&#39;s market environment. Therefore, pricing of MBS instruments using this previous approach may not accurately reflect the market environment at the time of the on-demand Stage 2 processing. 
     Also, while the current two stage approach may be suitable for less complex valuations, e.g., involving lower numbers of market scenarios and interest rate paths, because of complexity and required resources, the current two stage approach may not be scalable for more complex valuations involving larger numbers of market scenarios. For example, the current two stage approach may be suitable with respect to processing and storage resources and timeliness for a valuation including, e.g., three market scenarios (the current market environment; parallel shift up from the current market environment; and parallel shift down from the current market environment) and 32 interest rate paths (which requires 96 cashflow path computations). However, the current two stage approach may not be suitable from a quantitative standpoint for more complex MBS valuations involving larger numbers of scenarios and/or interest rate paths because of the increased processing and storage requirements. 
     Additionally, different investors may have differing views as to what variables most appropriately represent the market environment, and may want price, option adjusted spread (OAS), effective duration, key rate durations and other metrics computed for an MBS instrument in such a manner that the investor can specify the market environment. For example, the investor may desire to choose a specific type and date of a yield curve, may provide a custom curve, or may change the inputs for the analysis (price or OAS). However, the current two stage processing approach does not easily accommodate user-selected variables and metrics. 
     SUMMARY OF THE INVENTION 
     Throughout, reference is made to an embodiment or embodiments of the invention. Applicants do not intend, and it is to be understood, that the invention disclosed herein is not limited to such embodiments. Embodiments of the invention described herein provide for methods, systems and computer readable media useful to provide pricing of an MBS instrument, e.g., providing a price given an OAS, or providing an OAS given a price, under various scenarios, conditions and assumptions. While various embodiments of the invention are described herein with respect to MBS instruments, it is to be understood that the invention has application to other securities, derivatives and instruments, e.g., whose value is based on cashflows and/or which are interest path dependent. 
     Various embodiments of the invention comprise methods, systems and computer program products which implement the functionality described herein. 
     According to one or more embodiments of the invention, pricing an MBS instrument for a particular interest rate model comprises: computing one or more consolidated cashflow paths, each from a plurality of simulated cashflow paths, each based on a simulated interest rate path, and computing the change in price or OAS for the instrument using only a previously generated consolidated cashflow path or paths with associated relationship parameters, e.g., computing the price of the MBS instrument comprising discounting the consolidated cashflow(s) obtained from the consolidated cashflow path(s). 
     According to one or more embodiments of the invention, pricing an MBS instrument for a particular interest rate model comprises: computing one or more consolidated cashflow paths, each from a plurality of simulated cashflow paths, each based on a simulated interest rate path; computing discount factors using a spread to a continuously compounded interest rate curve equivalent to a provided or computed OAS (referred to herein as the “equivalent cc spread”) and a relationship between the equivalent cc spread and the OAS; and discounting the consolidated cashflow(s) obtained from the consolidated path(s) using the computed discount factors. 
     In one or more embodiments, discount factors are computed based on a market environment that exists when the consolidated cashflow path(s) are computed, and the discount factors are adjusted to reflect a market environment subsequent to and different from the market environment used to generate the discount factors. Due to change in OAS, the discount factors are adjusted based on knowledge of all discount factors for all simulation paths. This is discussed below under the heading “Compute the consolidated cashflow.” However, adjusting by a spread to a continuously compounded rate is a simple multiplication that does not depend on a simulation path and hence does not require knowledge of all interest rates in order to perform such an adjustment. The change in OAS is mapped to a change in a spread to the continuously compounded rate. 
     According to one or more embodiments of the invention, computation of at least the interest rate and cashflow paths of the particular interest rate model (and scenario), and the discount factors for each path occurs at one time or interval based on a then current market environment, and computation of the adjusted discount factors and the discounted consolidated cashflows occurs at subsequent time or interval when the market environment is expected to or has changed. For example, at least the interest rate and cashflow paths, the discount factors for each path, the consolidated cashflow paths and the equivalent cc spread are computed at the end of a day based on that day&#39;s market environment, while the adjusted discount factors and the discounted consolidated cashflows are computed on demand the next day based on the then current market environment. According to some embodiments, zero coupon bond prices are used to adjust the discount factors, for example as described by the expressions below. 
     According to one or more embodiments of the invention, processing can be performed in two stages similar to the previous approach. In a first stage, for the selected interest rate model and market scenarios: the interest rate paths are simulated; the cashflow paths are calculated; the consolidated cashflow paths are calculated; the discount factors are computed; and the equivalent cc spread is computed and the relationship between the equivalent cc spread and the OAS is determined. In this first stage, the consolidated cashflows, the equivalent cc spread, the relationship between the equivalent cc spread and the OAS, and the discount factors are stored, but the interest rate paths, the cashflow paths and the discount factors need not be stored, thereby reducing the use of computer processing and computer storage resources. In a second stage of processing: the discount factors according to the current market environment are computed; the stored consolidated cashflow paths, the equivalent cc spread, and the relationship between the equivalent cc spread and the OAS are retrieved, and the consolidated cashflows are computed using the adjusted discount factors. The second stage processing is performed on demand typically hours after or the day after the first stage processing. 
     According to one or more embodiments of the invention, the equivalent cc spread and the relationship between the equivalent cc spread and the OAS are computed based on the same market environment as are computation of the interest rate model&#39;s interest rate and cashflow paths, and the discount factors for each path. According to one or more embodiments, all of the above stage 1 computations can be performed on demand prior to Stage 2 processing, i.e., under the same market environment, in a timely manner, particularly for less complex interest rate models. The discount rates are adjusted when the input OAS has changed (e.g., moved by the spread amount). However, if the OAS has not changed, adjustment of discount factors is not needed. 
     In the previous approach, the shift or spread used to compute the cashflow paths along the various interest rate paths for each scenario is relative to the continuously compounded rate. Therefore, computing the discount factors involves a parallel shift up or down, which is independent of the specific interest rate path, and therefore is not intensive or complicated. However, computing the consolidated cashflow path for each scenario relative to an interest rate that is not continuously compounded (e.g., a semi-annually compounded forward rate curve), involves a relationship between the spread and the discount factors that is not a simple parallel shift up or down and depends on the actual interest rate information on each path, which is non-trivial and requires that all interest rate paths and cashflow paths be stored to compute a price. To avoid this complexity, one or more embodiments of the invention computes pricing using consolidated cashflow paths, the equivalent cc spread, and the relationship between this equivalent cc spread and the OAS. 
     Regarding the accuracy of valuations performed in accordance with one or more embodiments of the invention, when the consolidated cashflows are calculated based on the same market environment as the interest rate paths and the cashflow paths, with or without a different input price/OAS and discount factor adjustment, then the pricing results are identical to the pricing results achieved by the prior approach (up to the numerical error of a rootfinder, or approximately about 1e-9 on 32-bit computers). For a changed market environment, e.g., the consolidated cashflows are computed based on one market environment and the discount factors are adjusted based on another market environment, the results are a very quick and close approximation and provide commercially acceptable pricing, particularly when the actual market environment is close to that when the consolidated cashflows were computed. 
     According to one or more embodiments of the invention, after closing of the relevant financial market, the market closing environment is saved in relation to the concerned instrument(s), and where the same environment applies to more than one instrument, that environment is saved for all of the instruments having the same environment. However, users can input their own price/OAS to compute OAS/price, and other sensitivity (for example, effective duration, key rate durations, etc.). 
     According to one or more embodiments of the invention, pricing of MBS instruments using consolidated cashflow paths can be performed in changing market environments and under different investor assumptions. For example, price, OAS, effective duration, key rate durations, and other metrics for MBS instruments are computed in such a manner that an investor (e.g., user, client, etc.) can specify the market environment (e.g., pick a specific type and date of a yield curve or provide a custom curve) as well as change the inputs for the analysis (price or OAS). Such pricing can be provided, e.g., on demand. Thus, the price of an MBS instrument can be computed under a specific calculation scenario, and subject to various instrument sensitivities, e.g., to OAS, by creating appropriate scenarios and pricing the instrument under those scenarios. 
     The computations referred to herein may be executed by at least one computer having access to storage device(s) and/or memory which store financial information related to the financial instrument, and programming stored on a computer readable medium or media. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The invention is illustrated in the figures of the accompanying drawings, which are meant to be exemplary and not limiting, and in which like references are intended to refer to like or corresponding structure and/or functionality, etc. 
         FIG. 1  is a block diagram depicting an exemplary computer system with which embodiments of the invention may at least partially be implemented. 
         FIG. 2  is a flow diagram of a process for generating/computing of interest rate paths, cashflow paths and discount factors which can be used in accordance with one or more embodiments of the invention. 
         FIG. 3  is a flow diagram of a process for generating/computing consolidated cashflows and other parameters for use in computing the price of an MBS instrument according to one or more embodiments of the invention. 
         FIG. 4  is a flow diagram of a process for generating/computing the price of the MBS instrument based on adjusted the discount factors, the current market environment and any user-supplied parameters, according to one or more embodiments of the invention. 
     
    
    
     DETAILED DESCRIPTION 
     Embodiments of the invention may be implemented by systems using one or more programmable digital computers and computer readable storage media.  FIG. 1  depicts an example of one such computer system  100 , which includes at least one processor  110 , such as, e.g., an Intel or Advanced Micro Devices microprocessor, coupled to a communications channel or bus  112 . The computer system  100  further includes at least one input device  114  such as, e.g., a keyboard, mouse, touch pad or screen, or other selection or pointing device, at least one output device  116  such as, e.g., an electronic display device, at least one communications interface  118 , at least one computer readable medium or data storage device  120  such as a magnetic disk or an optical disk and memory  122  such as Random-Access Memory (RAM), each coupled to the communications channel  112 . The communications interface  118  may be coupled to a network  142 . 
     One skilled in the art will recognize that many variations of the system  100  are possible, e.g., the system  100  may include multiple channels or buses  112 , various arrangements of storage devices  120  and memory  122 , as different units or combined units, one or more computer-readable storage medium (CRSM) readers  136 , such as, e.g., a magnetic disk drive, magneto-optical drive, optical disk drive, or flash drive, multiple components of a given type, e.g., processors  110 , input devices  114 , communications interfaces  118 , etc. 
     In one or more embodiments, computer system  100  communicates over the network  142  with at least one another computer  144 , which may comprise one or more host computers and/or server computers and/or one or more other computers, e.g. computer system  100 , performing host and/or server functions including web server and/or application server functions. In one or more embodiments, a database  146  is accessed by the at least one other computer  144 . The at least one other computer  144  may include components as described for computer system  100 , and other components as is well known in the computer arts. Network  142  may comprise one or more LANS, WANS, intranets, the Internet, and other networks known in the art. In one or more embodiments, computer system  100  is configured as a work station that communicates with the at least one other computer  144  over the network  142 . In one or more embodiments, computer system  100  is configured as a client in a client-server system in which the at least one other computer comprises one or more servers. Additional computer systems  100 , any of which may be configured as a work station and/or client computer, may communicate with the at least one other computer  144  and/or another computer system  100  over the network  142 . 
     The terms “client” and “server” may describe programs and running processes instead of or in addition to their application to computer systems described above. Generally, a (software) client may consume information and/or computational services provided by a (software) server. 
     Various embodiments of the invention are described herein with respect to MBS instruments. However, it is to be understood that the invention has application to other securities, derivatives and instruments that are interest path dependent. 
       FIG. 2  illustrates an exemplary flow for an embodiment of process  200  for generating the interest rate paths, computing the cashflow paths along each interest rate path, and generating the discount factors along each interest rate path. In block  202  of  FIG. 2 , interest rate model parameters such as price or OAS; market environment variables (yield curve, volatilities, indices, etc.); interest rate model assumptions; and prepayment and default assumptions are input to the process. Currently, users can affect prepayment and default indirectly by either changing the parameters of the interest rate model or by changing some loan attributes (e.g. LTV), which ultimately affects modeled parameters. 
     In block  206 , interest rate paths are generated based on the interest rate model parameters. In the “do loop” represented by blocks  210  and  212 , discount factors are generated along each generated interest rate path. In the “do loop” represented by blocks  216  and  218 , a cashflow path CF(i, j) computed for each generated interest rate path. In block,  224 , the generated interest rate paths and discount factors, and the computed cashflow paths are stored. 
     The functions in block  206  may be performed by the at least one computer  144 , e.g., as an interest rate engine, and in one or more embodiments, as previously performed. (See, for example, Expression 6.) The discount factors in block  210  may be generated by the at least one computer  144 , as previously performed. The functions in block  216  may be performed by the at least one computer  144 , e.g., as a cashflow engine, and in one or more embodiments, as previously performed. Blocks  206 ,  210  and  216  may be implemented by a Monte Carlo based method, as previously performed. 
     While the flow for process  200  shows values and parameters being input to the process within one block ( 202 ), it is to be understood that they may be input at any appropriate point in the flow. Similarly, while the flow for process  200  shows values and parameters being stored in specific blocks, it is to be understood that they may be stored at any appropriate point in the flow. As mentioned above, known methodologies may be used to generate the interest rate paths and the discount factors, and to compute the cashflow paths and the OAS or price, as performed, for example, by the at least one other computer  144  and associated components. 
       FIG. 3  illustrates an exemplary flow for an embodiment of process  250  for computing a consolidated cashflow from the cashflow paths computed by the process  200  for each market scenario and the equivalent cc spread. According to one or more embodiments of the invention, process  250  may be performed approximately when the processing for process  200  is performed, e.g., in a first stage of processing and based on the same market environment. 
     Various data and variables are input or read into the process in block  252 , including: the cashflow paths and the OAS or price. While process flow  250  shows these as being read in the process within one block, it is to be understood that they may be read in at any appropriate point in the flow. 
     In the “do loop” represented by blocks  258  and  262 , a single consolidated cashflow path CF(i) (Expression 6) is computed for the set of interest rate paths of each scenario based on the discount factors D(i,j) and the cashflows CF(i,j) for each payment date i and path j computed by process  200 . (The expressions CF(i), etc., are discussed below.) In block  268 , the equivalent cc spread (s c ) is computed based on input or computed OAS or price and the consolidated cashflows CF(i), and the relationship between the equivalent cc spread and the computed or input OAS is computed, which are stored in block  274 . 
       FIG. 4  illustrates an exemplary flow for an embodiment of process  300  for computing price based on the computed consolidated cashflows and an equivalent cc spread. In one or more embodiments of the invention, this processing may performed at a time or interval when the market environment is expected to have changed from the market environment at the time of running processes  200  and  250 , i.e., in a second stage of processing. 
     In block  302 , the current market environment is input, and the consolidated cashflows, the equivalent cc spread, the OAS, the discount factors stored by processes  200  and  250  are read into process  300 . A user-supplied OAS, if any, may be input in block  302 . While the flow for process  300  shows these as being read in the process within one block, it is to be understood that they may be read in at any appropriate point in the flow. 
     In block  308 , the zero coupon bond price (ZCB(i)) is computed based on the yield curve input with the current market environment. (See, for example, Expressions 5 and 6). In block  314 , if a new user-supplied OAS was input, then in block  320  compute a new equivalent cc spread and pass this spread and the user-supplied OAS to block  326 , else in block  322  pass the read-in equivalent cc spread to block  326 . 
     In block  326 , the read-in discount factors are adjusted based on the zero coupon bond prices (ZCB(i)) and the equivalent cc spread or the new equivalent c spread. In block  332 , a price is computed for each scenario based on the adjusted discount factors, the read-in equivalent cc spread and the consolidated cashflow for the scenario. Once the user-supplied OAS is converted to the equivalent c/c spread, this spread can be applied to the discount factors derived from the current yield curve and the zero coupon bond prices. These discount factors are not stochastic as they are fully specified by the current curve, so no interest rate model is involved. The discount factors are adjusted via expressions below using the equivalent c/c spread, and these adjusted discount factors are used to discount the consolidated cashflows. 
     According to some embodiments of the invention: no interest rate modeling is needed at the time of valuation; only pre-generated consolidated cashflows are used without the need of expensive cashflow generation at the time of valuation; and/or only consolidated cashflow are stored without the need for expensive storage of many cashflow paths (one for each simulation path). 
     In other embodiments, price, option adjusted spread, effective duration, key rate durations, and other metrics for MBS instruments are computed in such a manner that an investor can specify the market environment (e.g., pick a specific type and date of a yield curve or provide a custom curve) as well as change the inputs for the analysis (price or OAS and receive results in a timely manner. In such cases, a different yield curve may be used to generate and discount cashflows. For example, a swap curve may be used to generate cashflows while a treasury curve is used to discount cashflows. In the process flow of  FIG. 4 , these parameters may be input in block  302 , and passed to blocks  308 ,  320  and  326  as appropriate. 
     Once computed, the price of the MBS and any other variables, e.g., OAS, effective duration, key rate duration, and other metrics, may be displayed on a computer screen, electronically mailed, printed to a report, or output in a similar manner. The price, along with any of the other variables described herein, may be stored in memory, a database, transmitted over a network or networks, and/or displayed on a, for example, desktop computer, monitor, or portable device, or transmitted to a printer for printing. 
     According to one or more embodiments of the invention various computations involved in pricing on MBS are performed by the at least one computer  144  in accordance with algorithms implementing one or more of the following mathematical expressions. 
     Price is computed given an OAS, or an OAS is computed given a price, by the at least one computer  144 , as follows: Price is the average of present values of cashflows of all paths, and the price based on the given OAS, PV(s), may be expressed as follows: 
                       PV   ⁡     (   s   )       =       1   N     ⁢       ∑     j   =   1     N     ⁢       ∑     i   =   1     M     ⁢       CF   ⁡     (     i   ,     ω   j       )       ⁢     D   ⁡     (     i   ,       ω   j     ;   s       )                 ,           (     Expression   ⁢           ⁢   1     )               
where,  s  is spread (OAS), and CF(i, ω j ), and D(i; ω j ;  s ) are the cashflows and discount factors at time t i  (i=1, 2, . . . , M) and path j (j=1, 2, . . . , N), respectively.
 
     For spread s c  on continuously compounded zero or forward curve, the discount factors may be represented by:
 
 D ( i,ω   j   ;s   c )= D ( i,ω   j ) e   −s     c     t     i   .  (Expression 2)
 
     The spread adjusted path-wise discount factor, D(i,ω j )e −s     c     t     i    expresses the product of path-wise discount factor and path independent spread discount factor. The price based on the spread s c , PV(s c ), may be expressed as follows: 
     From Expression 1 and Expression 2, 
                           PV   ⁡     (     s   c     )       =       ⁢       1   N     ⁢       ∑     j   =   1     N     ⁢       ∑     i   =   1     M     ⁢       CF   ⁡     (     i   ,     ω   j       )       ⁢     D   ⁡     (     i   ,     ω   j       )       ⁢     ⅇ       -     s   c       ⁢     t   i                             =       ⁢       ∑     i   =   1     M     ⁢       [       1   N     ⁢       ∑     j   =   1     N     ⁢       CF   ⁡     (     i   ,     ω   j       )       ⁢     D   ⁡     (     i   ,     ω   j       )             ]     ⁢       ⅇ       -     s   c       ⁢     t   i         .                       (     Expression   ⁢           ⁢   3     )                       ⁢     =       ∑     i   =   1     M     ⁢         CF   _     ⁡     (   i   )       ⁢     ZCB   ⁡     (   i   )       ⁢       ⅇ       -     s   c       ⁢     t   i         .                   (     Expression   ⁢           ⁢   4     )               
Where  CF (i) represents a consolidated cashflow, which may be expressed as follows:
 
                         CF   _     ⁡     (   i   )       =       1     ZCB   ⁡     (   i   )         ⁡     [       1   N     ⁢       ∑     j   =   1     N     ⁢       CF   ⁡     (     i   ,     ω   j       )       ⁢     D   ⁡     (     i   ,     ω   j       )             ]         ,           (     Expression   ⁢           ⁢   5     )               
where ZCB(i) is price of a zero coupon bond at time 0 with maturity t i , i.e.
 
     
       
         
           
             
               
                 
                   
                     ZCB 
                     ⁡ 
                     
                       ( 
                       i 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       N 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                         
                           D 
                           ⁡ 
                           
                             ( 
                             
                               i 
                               , 
                               
                                 ω 
                                 j 
                               
                             
                             ) 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     6 
                   
                   ) 
                 
               
             
           
         
       
     
     Expression 2, which decomposes spread adjusted path-wise discount factor into the product of path-wise discount factor and path independent factor, is used to obtain Expression 4 from Expression 1. For spread s f  on semi-annual compounded forward curve: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           D 
                           ⁡ 
                           
                             ( 
                             
                               i 
                               , 
                               
                                 
                                   ω 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                   
                                 
                                 ; 
                                 
                                   s 
                                   f 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               D 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     i 
                                     - 
                                     1 
                                   
                                   , 
                                   
                                     
                                       ω 
                                       j 
                                     
                                     ; 
                                     
                                       s 
                                       f 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 1 
                                 + 
                                 
                                   
                                     
                                       f 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           i 
                                           , 
                                           
                                             ω 
                                             j 
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       s 
                                       f 
                                     
                                   
                                   2 
                                 
                               
                               ] 
                             
                           
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   t 
                                   i 
                                 
                                 - 
                                 
                                   t 
                                   
                                     i 
                                     - 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             D 
                             ⁡ 
                             
                               ( 
                               
                                 0 
                                 , 
                                 
                                   
                                     ω 
                                     j 
                                   
                                   ; 
                                   
                                     s 
                                     f 
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             ∏ 
                             
                               k 
                               = 
                               1 
                             
                             i 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             [ 
                             
                               1 
                               + 
                               
                                 
                                   
                                     f 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         k 
                                         , 
                                         
                                           ω 
                                           j 
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     s 
                                     f 
                                   
                                 
                                 2 
                               
                             
                             ] 
                           
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   t 
                                   k 
                                 
                                 - 
                                 
                                   t 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             ∏ 
                             
                               k 
                               = 
                               1 
                             
                             i 
                           
                           ⁢ 
                           
                             
                               [ 
                               
                                 1 
                                 + 
                                 
                                   
                                     
                                       f 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           k 
                                           , 
                                           
                                             ω 
                                             j 
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       s 
                                       f 
                                     
                                   
                                   2 
                                 
                               
                               ] 
                             
                             
                               
                                 - 
                                 2 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     t 
                                     k 
                                   
                                   - 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             ∏ 
                             
                               k 
                               = 
                               1 
                             
                             i 
                           
                           ⁢ 
                           
                             
                               [ 
                               
                                 1 
                                 + 
                                 
                                   
                                     f 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         k 
                                         , 
                                         
                                           ω 
                                           j 
                                         
                                       
                                       ) 
                                     
                                   
                                   2 
                                 
                               
                               ] 
                             
                             
                               
                                 - 
                                 2 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     t 
                                     k 
                                   
                                   - 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             ∏ 
                             
                               k 
                               = 
                               1 
                             
                             i 
                           
                           ⁢ 
                           
                             
                               [ 
                               
                                 1 
                                 + 
                                 
                                   
                                     
                                       s 
                                       f 
                                     
                                     / 
                                     2 
                                   
                                   
                                     1 
                                     + 
                                     
                                       
                                         f 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             k 
                                             , 
                                             
                                               ω 
                                               j 
                                             
                                           
                                           ) 
                                         
                                       
                                       / 
                                       2 
                                     
                                   
                                 
                               
                               ] 
                             
                             
                               
                                 - 
                                 2 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     t 
                                     k 
                                   
                                   - 
                                   
                                     t 
                                     
                                       k 
                                       - 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             D 
                             ⁡ 
                             
                               ( 
                               
                                 i 
                                 , 
                                 
                                   ω 
                                   j 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               ∏ 
                               
                                 k 
                                 = 
                                 1 
                               
                               i 
                             
                             ⁢ 
                             
                               
                                 
                                   [ 
                                   
                                     1 
                                     + 
                                     
                                       
                                         
                                           s 
                                           f 
                                         
                                         / 
                                         2 
                                       
                                       
                                         1 
                                         + 
                                         
                                           
                                             f 
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 k 
                                                 , 
                                                 
                                                   ω 
                                                   j 
                                                 
                                               
                                               ) 
                                             
                                           
                                           / 
                                           2 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 
                                   
                                     - 
                                     2 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         t 
                                         k 
                                       
                                       - 
                                       
                                         t 
                                         
                                           k 
                                           - 
                                           1 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               . 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Expresion 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     7 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     since 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       D 
                       ⁡ 
                       
                         ( 
                         
                           0 
                           , 
                           
                             
                               ω 
                               j 
                             
                             ; 
                             
                               s 
                               f 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   1 
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     If f(k, ω j )=  f −   , which is path and time independent, and defining  α  as
 
  α =[1 +  f / 2] −1 ,  (Expression 8)
 
then, Expression 7 becomes:
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           D 
                           ⁡ 
                           
                             ( 
                             
                               i 
                               , 
                               
                                 
                                   ω 
                                   j 
                                 
                                 ; 
                                 
                                   s 
                                   f 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             D 
                             ⁡ 
                             
                               ( 
                               
                                 i 
                                 , 
                                 
                                   ω 
                                   j 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               ∏ 
                               
                                 k 
                                 = 
                                 1 
                               
                               i 
                             
                             ⁢ 
                             
                               
                                 [ 
                                 
                                   1 
                                   + 
                                   
                                     
                                       α 
                                       _ 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         s 
                                         f 
                                       
                                       / 
                                       2 
                                     
                                   
                                 
                                 ] 
                               
                               
                                 
                                   - 
                                   2 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       t 
                                       k 
                                     
                                     - 
                                     
                                       t 
                                       
                                         k 
                                         - 
                                         1 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               
                                 D 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     i 
                                     , 
                                     
                                       ω 
                                       j 
                                     
                                   
                                   ) 
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   1 
                                   + 
                                   
                                     
                                       α 
                                       _ 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         s 
                                         f 
                                       
                                       / 
                                       2 
                                     
                                   
                                 
                                 ] 
                               
                             
                             
                               
                                 - 
                                 2 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 t 
                                 i 
                               
                             
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     9 
                   
                   ) 
                 
               
             
           
         
       
     
     Similar to Expression 2, Expression 9 decomposes forward spread adjusted path-wise discount factors into product of two factors: one path dependent and another path-independent. Therefore, price, PV(s f ), based on the semi-annual compounded forward spread becomes: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           PV 
                           ⁡ 
                           
                             ( 
                             
                               s 
                               f 
                             
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             1 
                             N 
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 j 
                                 = 
                                 1 
                               
                               N 
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 M 
                               
                               ⁢ 
                               
                                 
                                   CF 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       i 
                                       , 
                                       
                                         ω 
                                         
                                           j 
                                           ⁢ 
                                           
                                               
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     
                                       D 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           i 
                                           , 
                                           
                                             ω 
                                             j 
                                           
                                         
                                         ) 
                                       
                                     
                                     ⁡ 
                                     
                                       [ 
                                       
                                         1 
                                         + 
                                         
                                           
                                             α 
                                             _ 
                                           
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             
                                               s 
                                               f 
                                             
                                             / 
                                             2 
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                   
                                     
                                       - 
                                       2 
                                     
                                     ⁢ 
                                     
                                       t 
                                       i 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             M 
                           
                           ⁢ 
                           
                             
                               
                                 [ 
                                 
                                   
                                     1 
                                     N 
                                   
                                   ⁢ 
                                   
                                     
                                       ∑ 
                                       
                                         j 
                                         = 
                                         1 
                                       
                                       N 
                                     
                                     ⁢ 
                                     
                                       
                                         CF 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             i 
                                             , 
                                             
                                               ω 
                                               j 
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         D 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             i 
                                             , 
                                             
                                               ω 
                                               j 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   1 
                                   + 
                                   
                                     
                                       α 
                                       _ 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         s 
                                         f 
                                       
                                       / 
                                       2 
                                     
                                   
                                 
                                 ] 
                               
                             
                             
                               
                                 - 
                                 2 
                               
                               ⁢ 
                               
                                 t 
                                 i 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             M 
                           
                           ⁢ 
                           
                             
                               
                                 CF 
                                 _ 
                               
                               ⁡ 
                               
                                 ( 
                                 i 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   ZCB 
                                   ⁡ 
                                   
                                     ( 
                                     i 
                                     ) 
                                   
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     1 
                                     + 
                                     
                                       
                                         α 
                                         _ 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         
                                           s 
                                           f 
                                         
                                         / 
                                         2 
                                       
                                     
                                   
                                   ] 
                                 
                               
                               
                                 
                                   - 
                                   2 
                                 
                                 ⁢ 
                                 
                                   t 
                                   i 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     10 
                   
                   ) 
                 
               
             
           
         
       
     
     Based on Expressions 4 and 10: 
                     ⅇ       -     s   c       ⁢     t   i         =         [     1   +       α   _     ⁢           ⁢       s   f     /   2         ]         -   2     ⁢     t   i         .             (     Expression   ⁢           ⁢   11     )                   α   _     1     =             s   f     2     ⁡     [       ⅇ       s   c     /   2       -   1     ]         -   1       .             (     Expression   ⁢           ⁢   12     )                 f   _     =       ⁢     2   ⁡     [         α   _       -   1       -   1     ]                                       ⁢     =       ⁢         s   f     ⁢       ⌈       ⅇ       s   c     /   2       -   1     ⌉       -   1         -   2               (     by   ⁢           ⁢   Expression   ⁢           ⁢   8     )               
When s f  is small:
 
   f ≈ 2( s   f   /s   c −1).  (Expression 13)
 
     Price/OAS based on consolidated cashflow is computed as follows according to one or more embodiments of the invention. 
     In accordance with one or more embodiments of the invention, all of the cashflow paths of a scenario are converted to a single, consolidated cashflow, which is used as an approximation to compute price/OAS when the yield curve is unchanged or changed slightly. This may be expressed according to one embodiment, as follows: 
     Compute the Consolidated Cashflow.
         1. Based on type of calculation, compute price (PV (s f )) or OAS (s f ) as usual.   2. Compute consolidated cashflow  CF (i) based on Expressions 5 and 6.   3. Let PV (s f )=PV (s c ), and solve for continuously compounded OAS (s c ) based on Expression 4.   4. Discount the consolidated cashflow with a new yield curve that is close to the original one.       

     Compute Price PV (ŝ f ) with OAS ŝ f :
         1. convert ŝ f  to ŝ c  based on Expression 12 or 13 for approximation.   2. PV (ŝ f )=PV (ŝ c ), compute price based on Expression 4, where D(i) is the discount factors of the new yield curve.       

     Compute OAS ŝ f  price PV (ŝ f ):
         1. In PV (ŝ f )=PV (ŝ c ), solve for based on Expression 4, where D(i) is the discount factors of the new yield curve.   2. convert ŝ c  to ŝ f  based on Expression 12 or 13 for approximation.
 
Using the Consolidated Cashflow on Other Yield Curves
       

     The discussion above assumes that the same yield curve is used to both generate and discount cashflows. However, cashflows may be generated by one yield curve (for example, a swap curve S 23 ), and discounted by another yield curve (for example, a treasury curve I 111 ). The first yield curve, that generates the interest rates dynamics and cashflows, is referred to as the dynamic curve, and the second yield curve, that discounts the cashflows, is referred to as the discount curve. 
     The path-wise discount factors of the discount curve are computed so that the path-wise cashflows can be discounted accordingly. The previously computed (or “precanned”) cashflows (based on the dynamic curve) and discount curve are used to compute price/OAS. According to one or more embodiments, these are carried out as follows. Assume that path-wise discount factors of the discount curve D (2)  (i, ωj) is obtained by adjusting path-wise discount factors of the dynamic curve
 
 D   (2) ( i,ω   j )= D ( i,ω   j ) A ( i ),  (Expression 14)
 
where D(i, ω j ) is the path-wise discount factor of the first yield curve, and A(i) is the per time-step adjustment. Then
 
     
       
         
           
             
               
                 
                   
                     
                       ZCB 
                       
                         ( 
                         2 
                         ) 
                       
                     
                     ⁡ 
                     
                       ( 
                       i 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       N 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                         
                           D 
                           
                             ( 
                             2 
                             ) 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             i 
                             , 
                             
                               ω 
                               j 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   by 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   6 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     = 
                     
                       
                         1 
                         N 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             1 
                           
                           N 
                         
                         ⁢ 
                         
                           
                             D 
                             ⁡ 
                             
                               ( 
                               
                                 i 
                                 , 
                                 
                                   ω 
                                   j 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             A 
                             ⁡ 
                             
                               ( 
                               i 
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   by 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   Expression 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   14 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     = 
                     
                       
                         ZCB 
                         ⁡ 
                         
                           ( 
                           i 
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           A 
                           ⁡ 
                           
                             ( 
                             i 
                             ) 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   Therefore 
                   , 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     A 
                     ⁡ 
                     
                       ( 
                       i 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ZCB 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       ⁡ 
                       
                         ( 
                         i 
                         ) 
                       
                     
                     
                       ZCB 
                       ⁡ 
                       
                         ( 
                         i 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     15 
                   
                   ) 
                 
               
             
           
         
       
     
     To discount the consolidated cashflow generated from the dynamic curve 
     
       
         
           
             
                 
             
             ⁢ 
             
               ( 
               
                 Expression 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 16 
               
               ) 
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         PV 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           s 
                           c 
                           
                             ( 
                             2 
                             ) 
                           
                         
                         ) 
                       
                     
                     = 
                       
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         M 
                       
                       ⁢ 
                       
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     1 
                                     N 
                                   
                                   ⁢ 
                                   
                                     
                                       ∑ 
                                       
                                         j 
                                         = 
                                         1 
                                       
                                       N 
                                     
                                     ⁢ 
                                     
                                       CF 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           i 
                                           , 
                                           
                                             w 
                                             j 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     D 
                                     
                                       ( 
                                       2 
                                       ) 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       i 
                                       , 
                                       
                                         w 
                                         j 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             
                               - 
                               
                                 s 
                                 c 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                               t 
                               i 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     by 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     3 
                   
                 
               
               
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       = 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           M 
                         
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       1 
                                       N 
                                     
                                     ⁢ 
                                     
                                       
                                         ∑ 
                                         
                                           j 
                                           = 
                                           1 
                                         
                                         N 
                                       
                                       ⁢ 
                                       
                                         CF 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             i 
                                             , 
                                             
                                               w 
                                               j 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     D 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         i 
                                         , 
                                         
                                           w 
                                           j 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             A 
                             ⁡ 
                             
                               ( 
                               i 
                               ) 
                             
                           
                           ⁢ 
                           
                             ⅇ 
                             
                               
                                 - 
                                 
                                   s 
                                   c 
                                   
                                     ( 
                                     2 
                                     ) 
                                   
                                 
                               
                               ⁢ 
                               
                                 t 
                                 i 
                               
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     by 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     14 
                   
                 
               
               
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       = 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           M 
                         
                         ⁢ 
                         
                           
                             
                               CF 
                               _ 
                             
                             ⁡ 
                             
                               ( 
                               i 
                               ) 
                             
                           
                           ⁢ 
                           
                             ZCB 
                             ⁡ 
                             
                               ( 
                               i 
                               ) 
                             
                           
                           ⁢ 
                           
                             A 
                             ⁡ 
                             
                               ( 
                               i 
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               ⅇ 
                               
                                 
                                   - 
                                   
                                     s 
                                     c 
                                     
                                       ( 
                                       2 
                                       ) 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   t 
                                   i 
                                 
                               
                             
                             . 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       by 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Expressions 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       5 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       6 
                     
                     ⁢ 
                     
                         
                     
                   
                 
               
               
                 
                   
                     
                         
                     
                     ⁢ 
                     
                       = 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           M 
                         
                         ⁢ 
                         
                           
                             
                               CF 
                               _ 
                             
                             ⁡ 
                             
                               ( 
                               i 
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               ZCB 
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                             ⁡ 
                             
                               ( 
                               i 
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               ⅇ 
                               
                                 
                                   - 
                                   
                                     s 
                                     c 
                                     
                                       ( 
                                       2 
                                       ) 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   t 
                                   i 
                                 
                               
                             
                             . 
                           
                         
                       
                     
                   
                 
                 
                   
                     by 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     14 
                   
                 
               
             
             ⁢ 
             
               ( 
               
                 Expression 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 17 
               
               ) 
             
           
         
       
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     Similarly 
                     , 
                   
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         PV 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           s 
                           f 
                           
                             ( 
                             2 
                             ) 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         M 
                       
                       ⁢ 
                       
                         
                           
                             CF 
                             _ 
                           
                           ⁡ 
                           
                             ( 
                             i 
                             ) 
                           
                         
                         ⁢ 
                         
                           
                             
                               
                                 ZCB 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 i 
                                 ) 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     
                                       1 
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         
                                           α 
                                           _ 
                                         
                                         
                                           ( 
                                           2 
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           s 
                                           f 
                                           
                                             ( 
                                             2 
                                             ) 
                                           
                                         
                                         / 
                                         2 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           
                             
                               - 
                               2 
                             
                             ⁢ 
                             
                               t 
                               i 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     Compute adjusted path-wise discount factors D (2) (i, ω), and compute price: (PV (s f   (2) ) or OAS (s f   (2) ), as usual. 
     Compute: s c   (2)  by solving PV (s f   (2) )=PV (s c   (2) ) 
     The remaining computations are the same as those for a dynamic yield curve case, as above under the “Compute Price” and “Compute OAS” headings, depending on whether price or OAS is desired. 
     Approximated MBS instrument prices computed as described herein drastically reduces the storage capacity required to store cashflow data. Instead of storing as many cashflows as the number of simulation interest rate paths, only one consolidated cashflow per scenario is stored. Also, the speed of the analysis is significantly improved, even under different user input scenarios, since expensive simulation of interest rate paths is replaced by adjustment of the discount factors based on the specified yield curve (which can be calculated very quickly) 
     Tests have demonstrated that the accuracy of approximated MBS instrument prices computed as described herein is suitable for commercial use and more accurate than the two-stage prior approached described above. 
     While the invention has been described and illustrated in connection with embodiments, many variations and modifications as will be evident to those skilled in this art may be made without departing from the spirit and scope of the invention as defined by the claims, and the invention is thus not to be limited to the precise details of methodology or construction set forth above. For example, the invention has application to securities, derivatives, instruments, etc., other than MBS instruments. Also, interest rate models other than those referenced herein may be used, and interest rate paths, cashflow paths and discount factors may be computed in different ways not described herein. It is to be understood that such variations and modifications are intended to be included within the scope of the invention as defined by the claims.