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 Lévy process or, more generally, a time-inhomogeneous jumpdiffusion process. We discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the Lévy measure. We propose an explicit-implicit time-stepping scheme to solve the equation and study stability and convergence of the schemes proposed, using the notion of viscosity solution. Numerical tests are performed with smooth and non-smooth initial conditions. Our scheme can be used for European and barrier options, applies in the case of pure-jump models or degenerate diffusion coefficients, and extends to time-dependent coefficients.