The value of a contingent claim under a jump-diffusion process satisfies a partial integro-differential equa-

tion. A fourth-order compact finite difference scheme is applied to discretize the spatial variable of this

equation. It is discretized in time by an implicit-explicit method. Meanwhile, a local mesh refinement strat-

egy is used for handling the nonsmooth payoff condition. Moreover, the numerical quadrature method is

exploited to evaluate the jump integral term. It guarantees a Toeplitz-like structure of the integral operator

such that a fast algorithm is feasible. Numerical results show that this approach gives fourth-order accuracy

in space.