A computational scheme for options under jump diffusion processes

In this paper we develop two novel numerical methods for thepartial integral differential equation arising from the valuation of an optionwhose underlying asset is governed by a jump diffusion process. These methodsare based on a fitted finite volume method for the spatial discretization, animplicit-explicit time stepping scheme and the Crank-Nicolson time steppingmethod. We show that the discretization methods are unconditionally stablein 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 andan Fast Fourier Transformation (FFT) technique is used for the evaluation ofthese products. Furthermore, a splitting technique is proposed for the solutionof the discretized system arising from the Crank-Nicolson scheme. Numericalresults are presented to show the rates of convergence and the robustness ofthe numerical method.