This paper develops a numerical method for a partial integro-differential equation and a partial integro- differential complementarity problem arising from European and American options valuations respectively when the underlying assets are driven by a jump diffusion process. The method is based on a fitted finite volume scheme for the spatial discretization and the Crank-Nicolson scheme for the time discretization. The fully discretized system is solved by an iterative method coupled with an FFT for the evaluation of the discretized integral term, while the constraint in the American option model is imposed by adding a penalty term to the original partial integro- differential complementarity problem. We show that the system matrix of the discretized system is an M-matrix and propose an algorithm for solving the discretized system. Numerical experiments are implemented to show the efficiency and robustness of this method. (c) 2007 Elsevier Inc. All rights reserved.