A jump-diffusion model for a single-asset market is considered. Under this assumption the value of a European contingency claim satisfies a general partial integro-differential equation (PIDE). The equation is localized and discretized in space using finite differences and finite elements and in time by the second order backward differentiation formula (BDF2). The resulting system is solved by an iterative method based on a simple splitting of the matrix. Using the fast Fourier transform, the amount of work per iteration may be reduced to O(nlog2n) and only O(n) entries need to be stored for each time level. Numerical results showing the quadratic convergence of the methods are given for Merton's model and Kou's model. Keywords: Option pricing; Jump-diffusion processes; Finite differences; Finite elements; Fast Fourier transform; Integro-differential equations
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