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7 Team Mathfi 3 3. Scientific Foundations 3.1. Numerical methods for option pricing and hedging and calibration of financial assets models Keywords: Euler schemes, Malliavin calculus, Monte-Carlo, approximation of SDE, calibration, finite difference, quantization, tree methods. Participants: V. Bally, E. Clément, B. Jourdain, A. Kohatsu Higa, D. Lamberton, B. Lapeyre, J. Printems, D. Pommier, A. Sulem, E. Temam, A. Zanette. Efficient computations of prices and hedges for derivative products is a major issue for financial institutions. Although this research activity exists for more that fifteen years at both academy and bank levels, it remains a lot of challenging questions especially for exotic products pricing on interest rates and portfolio optimization with constraints. This activity in the Mathfi team is strongly related to the development of the Premia software. It also motivates theoretical researches both on Monte Carlo methods and numerical analysis of (integro) partial differential equations : Kolmogorov equation, Hamilton-Jacobi-Bellman equations, variational and quasi variational inequalities (see  MonteCarlo methods. The main issues concern numerical pricing and hedging of European and American derivatives and sensibility analysis. Financial modelling is generally based on diffusion processes of large dimension (greater than 10), often degenerate or on Lévy processes. Therefore, efficient numerical methods are required. Monte- Carlo simulations are widely used because of their implementation simplicity. Nevertheless, efficiency issues rely on tricky mathematical problems such as accurate approximation of functionals of Brownian motion (e.g. for exotic options), use of low discrepancy sequences for nonsmooth functions... Speeding up the algorithms is a major issue in the developement of MonteCarlo simulation (see the thesis of A. Kbaier). We develop Montecarlo algorithms based on quantization trees and Malliavin calculus. V. Bally, G. Pagès and J. Printems have developed quantization methods especially for the computation of American options , , . Recently, G. Pagès and J. Printems showed that functional quantization may be efficient for pathdependent options (such that Asian options) and for European options in some stochastic volatility models  Approximation of stochastic differential equations. In the diffusion models, the implementation of Monte-Carlo methods generally requires the approximation of a stochastic differential equation, the most common being the Euler scheme. The error can then be controlled either by the L P -norm or the probability transitions PDE-based methods. We are concerned with the numerical analysis of degenerate parabolic partial differential equations, variational and quasivariational inequalities, Hamilton-Jacobi-Bellman equations especially in the case when the discrete maximum principle is not valid and in the case of an integral term coming from possible jumps in the dynamics of the underlying processes. In large dimension, we start to investigate sparse grid methods Model calibration. While option pricing theory deals with valuation of derivative instruments given a stochastic process for the underlying asset, model calibration is about identifying the (unknown) stochastic process of the underlying asset given information about prices of options. It is generally an ill-posed inverse problem which leads to optimisation under constraints Application of Malliavin calculus in finance Keywords: Malliavin calculus, greek computations, sensibility calculus, stochastic variations calculus.
10 6 Activity Report INRIA 2004 affect market prices. Another application in finance concerns recursive utilities as introduced by Duffie and Epstein (1992) . Such a utility function associated with a consumption rate (c t, 0 t T ) corresponds to the solution of a BSDE with terminal condition ξ which can be a function of the terminal wealth, and a driver f(t, c t, y) depending on the consumption c t. The standard utility problem corresponds to a linear driver f of the type f(t, c, y) = u(c) β t y, where u is a deterministic, non decreasing, concave function and β is the discount factor. In the case of reflected BSDEs, introduced in , the solution Y is forced to remain above some obstacle process. It satisfies dy t = f(t, Y t, Z t )dt + dk t Z tdw t ; Y T = ξ. (2) where K is a nondecreasing process. For example the price of an American option satisfies a reflected BSDE where the obstacle is the payoff. The optimal stopping time is the first time when the prices reaches the payoff. (  et ). 4. Application Domains 4.1. Modelisation of financial assets Keywords: fractional Brownian motion, stable law, stochastic volatility. Participants: V. Genon-Catalot, T. Jeantheau, A. Sulem Statistics of stochastic volatility models. It is well known that the Black-Scholes model, which assumes a constant volatility, doesn t completly fit with empirical observations. Several authors have thus proposed a stochastic modelisation for the volatility, either in discrete time (ARCH models) or in continuous time (see Hull and White ). The price formula for derivative products depend then of the parameters which appear in the associated stochastic equations. The estimation of these parameters requires specific methods. It has been done in several asymptotic approaches, e.g. high frequency , ,  Application of stable laws in finance. Statistical studies show that market prices do not follow diffusion prices but rather discontinuous dynamics. Stable laws seem appropriate to model cracks, differences between ask and bid prices, interventions of big investors. Moreover pricing options in the framework of geometric α-stable processes lead to a significant improvement in terms of volatility smile. Statistic analysis of exchange rates lead to a value of α around A. Tisseyre has developped analytical methods in order to compute the density, the repartition function and the partial Laplace transform for α-stable laws. These results are applied for option pricing in stable markets. (see ) Fractional Brownian Motion (FBM). The Fractional Brownian Motion B H (t) with Hurst parameter H has originally been introduced by Kolmogorov for the study of turbulence. Since then many other applications have been found. If H = 1 2 then B H(t) coincides with the standard Brownian motion, which has independent increments. If H > 1 2 then B H(t) has a long memory or strong aftereffect. On the other hand, if 0 < H < 1 2, then ρ H (n) < 0 and B H (t) is anti-persistent: positive values of an increment is usually followed by negative ones and conversely. The strong aftereffect is often observed in the logarithmic returns log Yn Y n 1 for financial quantities Y n while the anti-persistence appears in turbulence and in the behavior of volatilities in finance. For all H (0, 1) the process B H (t) is self-similar, in the sense that B H (αt) has the same law as α H B H (t), for all α > 0. Nevertheless, if H 1 2, B H(t) is not a semi-martingale nor a Markov process , , , and integration with respect to a FBM requires a specific stochastic integration theory.
12 8 Activity Report INRIA 2004 time at the lower/right boundary . These results generalize the results obtained in the no jump case. The case of portfolio optimisation with partial observation is studied in , . 5. Software 5.1. Development of the software PREMIA for financial option computations Keywords: calibration, hedging, options, pricer, pricing. Participants: V. Bally, Adel Ben Haj Yedder, L. Caramellino, J. Da Fonseca, B. Jourdain, A. Kohatsu Higa, B. Lapeyre, M. Messaoud, N. Privault (University of La Rochelle, A. Sulem, E. Temam, A. Zanette. We develop a software called PREMIA designed for pricing and hedging options on assets and interest rates and for calibration of financial models. This software  contains the most recent algorithms published in the mathematical finance literature with their detailed description and comments of the numerical methods in this field. The target is to reach on the first hand the market makers who want to be informed on this field, and on the other hand the PHD students in finance or mathematical finance. Premia is thus concentrated on derivatives with rigorous numerical treatment and didactic inclination. Premia is developed in collaboration with a consortium of financial institutions or departments: It is presently composed of: IXIS CIB (Corporate & Investment Bank), CALYON, the Crédit Industriel et Commercial, EDF, GDF, Société générale and Summit. History of PREMIA: The development of Premia started in There exists now 6 releases. Premia1, 2 and 4 contain finite difference algorithms, tree methods and Monte Carlo methods for pricing and hedging European and American options on stocks in the Black-Scholes model in one and two dimension. Premia3 is dedicated to Monte Carlo methods for American options in large dimension. Moreover, it has an interface with the software Scilab . Premia5 and 6 contain more sophisticated algorithms such as quantization methods for American options ,  and methods based on Malliavin calculus both for European and American options . Pricing and hedging algorithms for some models with jumps, local volatility and stochastic volatility are implemented. These versions contains also some calibration algorithms. In 2004, the main development in the release Premia7 has consisted in implementing routines for pricing derivatives in interest rate models: (Vasicek, Hull-White, CIR, CIR++, Black-Karasinsky, Squared-Gaussian, Li, Ritchken, Sankarasubramanian HJM, Bhar Chiarella HJM, BGM). Moreover new algorithms for calibrating in various models (stochastic volatility, jumps,..) have been implemented and numerical methods based on Malliavin calculus for jump processes have been further explored. Premia1, 2, 3, 4, 5, 6 have been delivered to the members of the bank consortium in May 1999, December 1999, February 2001, February 2002, February 2003, February 2004 respectively. Premia7 will be delivered in February The next release, Premia8, under development in 2005, will be dedicated to the pricing of credit risk derivatives and pricing and calibration for interest rate derivatives. Premia3 and Premia4 can be downloaded from the web site: 6. New Results 6.1. Monte Carlo methods and stochastic algorithms Variance reduction methods in Monte Carlo simulations Participants: B. Arouna, B. Lapeyre, N. Moreni.
13 Team Mathfi 9 Under the supervision of Bernard Lapeyre, B. Arouna has defended his PHD thesis in which he shown that stochastics algorithms can be efficiently used in order to decrease variance. He provides tractable methods of variance reduction in Monte Carlo estimation of expectations (integrals) and proves associated theoretical results. His work has been published in  and . Nicola Moreni is studying variance reduction techniques for option pricing based on Monte Carlo simulation. In particular, in a joint project with the University of Pavia (Italy), he applies path integral techniques to the pricing of path-dependent European options. He has also deals with a variance reduction technique for the Longstaff-Schwartz algorithm for American option pricing. This technique is based on extension of the work of B. Arouna to the case of American options Monte Carlo methods for American options in high dimension. Participants: L. Caramellino (University of Rome II), A. Zanette. We have done numerical comparaison between some recent Monte Carlo algorithms for pricing and hedging American options in high dimension, in particular between the quantization method of Barraquand-Martineau and an algorithm based on Malliavin calculus  Discretization of stochastic differential equations Participants: E. Clément, A. Kohatsu Higa, D. Lamberton, J. Guyon, A. Alfonsi, B. Jourdain. E.Clément, A. Kohatsu Higa and D.Lamberton develop a new approach for the error analysis of weak convergence of the Euler scheme, which enables them to obtain new results on the approximation of stochastic differential equations with memory. Their approach uses the properties of the linear equation satisfied by the error process instead of the partial differential equation derived from the Markov property of the process. It seems to be more general than the usual approach and gives results for the weak approximation of stochastic delay equations. A paper has been submitted and extensions are studied. In his thesis, A. Kbaier develops a "statistic Romberg method" for weak approximation of stochastic differential equations. This method is especially efficient for the computation of Asian options. J. Guyon, PhD student of B. Lapeyre and J.F. Delmas has studied how fast the Euler scheme XT n with timestep t = T/n converges in law to the original random variable X T. More precisely, he has looked for which class of functions f the approximate expectation E [f(xt n)] converges to E [f(x T )] with speed t. So far, (X t, t 0) has been a smooth R d -valued diffusion. When f is smooth, it is known from D. TALAY and L. TUBARO that E [f(x n T )] E [f(x T )] = C(f) t + O ( t 2). (3) Using Malliavin calculus, V. BALLY and D. TALAY have shown that this development remains true when f is only mesurable and bounded, in the case when the diffusion X is uniformly hypoelliptic. When X is uniformly elliptic, J. Guyon has extended this result to the general class of tempered distributions. When f is a tempered distribution, E [f(x T )] (resp. E [f(xt n)]) has to be understood as f, p (resp. f, p n ) where p (resp. p n ) is the density of X T (resp. XT n ). In particular, (3) is valid when f is a measurable function with polynomial growth, a Dirac distribution or a derivative of a Dirac distribution. The proof consists in controlling the linear mapping f C(f) and the remainder. It can be used to show that (3) remains valid when f is a measurable function with exponential growth, or when the tempered distribution f acts on the deterministic initial value x of the diffusion X. An article is being written, underlying applications to option pricing and hedging. Under the supervision of Benjamin Jourdain, Aurélien Alfonsi is studying the weak and strong rates of convergence of various explicit and implicit discretization schemes for Cox-Ingersoll-Ross processes, both from a theoretical point of view and by numerical experiments Approximation of the invariant measure of a diffusion Keywords: invariant measure.
14 10 Activity Report INRIA 2004 Participants: E. Clément, D. Lamberton, V. Lemaire, G. Pagès. Our work focused on the characterization of invariant measures and on processes with jumps. We are working on the approximation of the invariant probability measure for SDEs with locally lipschitz coefficients and for SDEs driven by Levy processes. We also investigate the numerical analysis of the long run behaviour of dynamical systems (invariant measure of diffusions, recursive learning algorithm) with F. Panloup, PhD student of G. Pagès (see ) Malliavin calculus for jump diffusions Keywords: Malliavin calculus, jump diffusions. Participants: V. Bally, M.P. Bavouzet, M. Messaoud. One of the financial numerical applications of the Malliavin calculus is the computation of the sensitivities (the Greeks) and the conditional expectations. In the Wiener case (when the asset follows a log-normal type diffusion for example), Fournié, Lasry, Lebouchoux, Lions and Touzi have developped a methodology based on the Malliavin calculus. The main tool is an integration by parts formula which is strongly related to the Gaussian law (since the diffusion process is a functional of the Brownian motion). In a first stage, V. Bally has worked in collaboration with Lucia Caramellino (University Rome 2) and Antonino Zanette (University of Udine, Italy) on pricing and hedging American options in a local Black Scholes model driven by a Brownian motion, by using classical Malliavin calculus . The results of this work gave rise to algorithms which have been implemented in PREMIA. V. Bally, M.P.Bavouzet and M. Messaoud use the Malliavin calculus for Poisson processes in order to compute the Greeks (the Delta for example) of European options with underlying following a jump type diffusion. Imitating the methodology of the Wiener case, the key point is to settle, under some appropriate non-degenerency condition, an integration by parts formula for general random variables. Actually, the random variables on which the calculus is based may be the amplitudes of the jumps, the jump times and the Brownian increments. On the one hand, M.P. Bavouzet and M. Messaoud deal with pure jump diffusion models and Merton model, where the law of the jump amplitudes has smooth density. One differentiates with respect to the amplitudes of the jumps only (pure jump diffusion) or to both jump amplitudes and Wiener increments (Merton model). Under some non-degenerency condition, one defines all the differential operators involved in the integration by parts formula. Numerical results show that using Malliavin approach becomes extremely efficient for a discontinuous payoff. Moreover, some localization techniques may be used to reduce the variance of the Malliavin estimator. In the case of the Merton model, it is better to use the two sources of randomness, especially when there are more jumps. On the other hand, V. Bally, MP. Bavouzet and M. Messaoud deal with pure jump diffusion models but differentiate with respect to the jump times. This case is more difficult because the law of the jump times has not smooth density, so that some border terms appear in the integration by parts formula. Thus, one introduces some weight functions in the definition of the differential operators in order to cancel these border terms. But, in this case, the non-degenerency condition is more difficult to obtain. Another application of the Malliavin s integration by parts formula is to prove that, under appropriate hypothesis, a large variety of functionals on the Wiener space (like solutions of Stochastic Partial Differential Equations) have absolute continuous laws with smooth density. Under uniform ellipticity assumption, A. Kohatsu-Higa developped a methodology which permits to compute lower bounds of the density. Then, V. Bally relaxed this hypothesis, replacing the uniform ellipticity by only local ellipticity around a deterministic curve. Following the work of V. Bally, M.P. Bavouzet is working on an extension of his results to jump diffusion case (driven by a Brownian motion and a Poisson process).
Tangent Lévy Models. Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford.
Finance & Stochastic. Contents. Rossano Giandomenico. Independent Research Scientist, Chieti, Italy.
Module 10:Application of stochastic processes in areas like finance Lecture 36:Black-Scholes Model. Stochastic Differential Equation.
Are stylized facts irrelevant in option-pricing?
Slides for DN2281, KTH 1 January 28, 2014 1 Based on the lecture notes Stochastic and Partial Differential Equations with Adapted Numerics, by J. Carlsson, K.-S. Moon, A. Szepessy, R. Tempone, G. Zouraris.
Approximating a multifactor di usion on a tree.
M.S. in Quantitative Finance & Risk Analytics (QFRA) Fall 2017 & Spring 2018 2 - Required Professional Development &Career Workshops MGMT 7770 Prof. Development Workshop 1/Career Workshops (Fall) Wed.

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