OPTION PRICING MODELS WITH JUMPS: INTEGRO-DIFFERENTIAL EQUATIONS AND INVERSE PROBLEMS.

Observation of sudden, large movements in the prices of financial assets has led to the use of stochastic processes with discontinuous trajectories - jump processes - as models for financial assets. Exponential Levy models provide an analytically tractable subclass of models with jumps and the flexibility in choice of the Levy process allows to calibrate the model to market prices of options and reproduce a wide variety of implied volatility skews/smiles. We discuss the characterization of prices of European and barrier options in exponential Levy models in terms of solutions of partial integro-differential equations (PIDEs). These equations involve, in addition to a second-order differential operator, a non-local integral term which requires specific treatment both at the theoretical and numerical level. The study of regularity of option prices in such models shows that, unlike the diffusion case, option price can exhibit lack of smoothness. The proper relation between option prices and PIDEs is then expressed using the notion of viscosity solution. Numerical solution of the PIDE allows efficient computation of option prices. The identification of exponential Levy models from option prices leads to an inverse problem for such PIDEs. We describe a regularization method based on relative entropy and its numerical implementation. This inversion algorithm, which allows to extract an implied Levy measure from a set of option prices, is illustrated by numerical examples.

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

[2]  D. Madan,et al.  1option Pricing with V. G. Martingale Components , 1991 .

[3]  V. Morozov.  On the solution of functional equations by the method of regularization , 1966 .

[4]  Situ Rong.  On solutions of backward stochastic differential equations with jumps and applications , 1997 .

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

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

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

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

[9]  D. Nualart,et al.  Backward stochastic differential equations and Feynman-Kac formula for Levy processes, with applications in finance , 2001 .

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

[11]  Koponen,et al.  Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[12]  David S. Bates.  Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options , 1998 .

[13]  R. C. Merton,et al.  Option pricing when underlying stock returns are discontinuous , 1976 .

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

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

[16]  N. Shephard,et al.  Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics , 2001 .

[17]  Jesper Andreasen,et al.  Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Pricing , 1999 .

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

[19]  E. Eberlein,et al.  New Insights into Smile, Mispricing, and Value at Risk: The Hyperbolic Model , 1998 .

[20]  David S. Bates.  Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Thephlx Deutschemark Options , 1993 .

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

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

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

[24]  M. Yor,et al.  The Fine Structure of Asset Retums : An Empirical Investigation ' , 2006 .

[25]  Steven Kou,et al.  A Jump Diffusion Model for Option Pricing , 2001, Manag. Sci..

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

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

[28]  E. Nicolato,et al.  Option Pricing in Stochastic Volatility Models of the Ornstein‐Uhlenbeck type , 2003 .

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

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

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