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Zbl 1101.47059
Cont, Rama; Voltchkova, Ekaterina
A finite difference scheme for option pricing in jump diffusion and exponential Lévy models. (English)
[J] SIAM J. Numer. Anal. 43, No. 4, 1596-1626 (2005). ISSN 0036-1429; ISSN 1095-7170/e

The authors propose a finite difference approach to finding the solution of a parabolic partial integro-differential equation (PIDE) with possible singular kernels. This problem is discussed within the framework of the option pricing of an underlying random asset that is driven by a Lévy process or a more general time-inhomogeneous jump-diffusion model. Solving the aforementioned PIDE by finite difference methods involves several approximations such as the localization of the PIDE to a bounded domain, treatment of the singularity arising from small jumps, discretization of the equation in space and iteration in time. \par The paper provides a discussion of localization errors and an estimate for such errors under an integrability condition on the Lévy measure. In this context, the authors propose an explicit-implicit finite difference scheme for which issues related to the pricing of European and barrier options, consistency, stability and convergence are discussed. By way of conclusion, numerical tests are done for smooth and nonsmooth initial conditions in order to determine the effect of various numerical parameters on the accuracy of the finite difference method.
[Mark A. Petersen (Potchefstroom)]

MSC 2000:: *47N10 Appl. of operator theory in optimization, math. programming, etc.
47G20 Integro-differential operators
65M06 Finite difference methods (IVP of PDE)
65M12 Stability and convergence of numerical methods (IVP of PDE)
49L25 Viscosity solutions
60H30 Appl. of stochastic analysis
60G51 Processes with independent increments
91B28 Finance etc.

Keywords: parabolic integro-differential equations; finite difference methods; Lévy process; jump-diffusion models; option pricing; viscosity solutions

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