The value of a contingent claim under a jump-diffusion process satisfies a partial integro-differential equation. 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 strategy 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.