Stability of an implicit method to evaluate option prices under local volatility with jumps

In this paper, we consider a local volatility model with jumps under which the price of a European option can be derived by a partial integro-differential equation (PIDE) with nonconstant coefficients. In order to solve numerically the PIDE, we generalize the implicit method with three time levels   which is constructed to avoid iteration at each time step. We show that the implicit method has the stability with respect to the discrete ℓ2ℓ2-norm by using an energy method. We combine the implicit method with an operator splitting method to solve a linear complementarity problem (LCP) with nonconstant coefficients that describes the price of an American option. Finally we conduct some numerical simulations to verify the analysis of the method. The proposed method leads to a tridiagonal linear system at each time step and thus the option prices can be computed in a few seconds on a computer.