A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models

We present a finite difference method for solving parabolic partial integro-differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Levy process or, more generally, a time-inhomogeneous jump-diffusion process. We discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the Levy measure. We propose an explicit-implicit finite difference scheme which can be used to price European and barrier options in such models. We study stability and convergence of the scheme proposed and, under additional conditions, provide estimates on the rate of convergence. Numerical tests are performed with smooth and nonsmooth initial conditions.

[1]  T. Chan.  Pricing contingent claims on stocks driven by Lévy processes , 1999 .

[2]  W. Schoutens.  Lévy Processes in Finance: Pricing Financial Derivatives , 2003 .

[3]  Xiaolan Zhang,et al.  Analyse numérique des options américaines dans un modèle de diffusion avec sauts , 1994 .

[4]  Steven J. Ruuth,et al.  Implicit-explicit methods for time-dependent partial differential equations , 1995 .

[5]  E. Jakobsen,et al.  A “maximum principle for semicontinuous functions” applicable to integro-partial differential equations , 2006 .

[6]  H. P..  Annales de l'Institut Henri Poincaré , 1931, Nature.

[7]  R. Cont,et al.  Non-parametric calibration of jump–diffusion option pricing models , 2004 .

[8]  P. Lions,et al.  Two approximations of solutions of Hamilton-Jacobi equations , 1984 .

[9]  Guy Barles,et al.  On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations , 2002 .

[10]  S. Levendorskii,et al.  Non-Gaussian Merton-Black-Scholes theory , 2002 .

[11]  Hélyette Geman,et al.  Pure Jump Levy Processes for Asset Price Modelling , 2002 .

[12]  Rama Cont,et al.  Integro-differential equations for option prices in exponential Lévy models , 2005, Finance Stochastics.

[13]  Christoph Schwab,et al.  Fast deterministic pricing of options on Lévy driven assets , 2002 .

[14]  R. Schilling.  Financial Modelling with Jump Processes , 2005 .

[15]  R. Cont,et al.  Financial Modelling with Jump Processes , 2003 .

[16]  Sayah Awatif.  Equqtions D'Hamilton-Jacobi Du Premier Ordre Avec Termes Intégro-Différentiels: Partie II: Unicité Des Solutions De Viscosité , 1991 .

[17]  G. Barles,et al.  Convergence of approximation schemes for fully nonlinear second order equations , 1990, 29th IEEE Conference on Decision and Control.

[18]  Jose Luis Menaldi,et al.  Second Order Elliptic Integro-Differential Problems , 2002 .

[19]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[20]  N. Jacob,et al.  Pseudo Differential Operators and Markov Processes: Volume I: Fourier Analysis and Semigroups , 2001 .

[21]  H. Soner.  Optimal Control of Jump-Markov Processes and Viscosity Solutions , 1988 .

[22]  N. Krylov.  On the rate of convergence of finite-difference approximations for Bellmans equations with variable coefficients , 2000 .

[23]  Curt Randall,et al.  Pricing Financial Instruments: The Finite Difference Method , 2000 .

[24]  G. Barles,et al.  Convergence of approximation schemes for fully nonlinear second order equations , 1991 .

[25]  E. Seneta,et al.  Simulation of estimates using the empirical characteristic function , 1987 .

[26]  Olivier Alvarez,et al.  Viscosity solutions of nonlinear integro-differential equations , 1996 .

[27]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[28]  Huy En Pham.  Optimal Stopping of Controlled Jump Diiusion Processes: a Viscosity Solution Approach , 1998 .

[29]  George Labahn,et al.  A penalty method for American options with jump diffusion processes , 2004, Numerische Mathematik.

[30]  Sayah Awatif,et al.  Equqtions D'Hamilton-Jacobi Du Premier Ordre Avec Termes Intégro-Différentiels , 2007 .

[31]  G. Barles,et al.  Backward stochastic differential equations and integral-partial differential equations , 1997 .

[32]  Rama Cont,et al.  Calibration of Jump-Diffusion Option Pricing Models: A Robust Non-Parametric Approach , 2002 .

[33]  Steven J. Ruuth,et al.  Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations , 1997 .

[34]  Ole E. Barndorff-Nielsen,et al.  Processes of normal inverse Gaussian type , 1997, Finance Stochastics.

[35]  P. Carr,et al.  Option valuation using the fast Fourier transform , 1999 .

[36]  E. Eberlein.  Application of Generalized Hyperbolic Lévy Motions to Finance , 2001 .

[37]  A. Bensoussan,et al.  Contrôle impulsionnel et inéquations quasi variationnelles , 1982 .

[38]  P. Protter.  Stochastic integration and differential equations , 1990 .

[39]  Niels Jacob,et al.  Pseudo-Differential Operators and Markov Processes , 1996 .

[40]  Dilip B. Madan,et al.  Option Pricing, Interest Rates and Risk Management: Purely Discontinuous Asset Price Processes , 2001 .