A Computational Scheme for Options under Jump Diffusion Processes

In this paper we develop two novel numerical methods for the partial integral differential equation arising from the valuation of an option whose underlying asset is governed by a jump diffusion process. These methods are based on a fitted finite volume method for the spatial discretization, an implicit-explicit time stepping scheme and the Crank-Nicolson time stepping method. We show that the discretization methods are unconditionally stable in time and the system matrices of the resulting linear systems are M-matrices. The resulting linear systems involve products of a dense matrix and vectors and an Fast Fourier Transformation (FFT) technique is used for the evaluation of these products. Furthermore, a splitting technique is proposed for the solution of the discretized system arising from the Crank-Nicolson scheme. Numerical results are presented to show the rates of convergence and the robustness of the numerical method.