Fast numerical valuation of options with jump under Merton's model

In this paper, we consider discontinuous Galerkin (DG) finite element together with finite difference (FD) scheme for solving Merton's jump-diffusion model, which is given by a partial integro-differential equations (PIDEs). Spatial differential operators are discretized using FD on a uniform grid, and time stepping is performed using the DG finite element method. The treatment of the integral term associated with jumps in models is more challenging. The discretization of this integral term will lead to full matrices for the non-locality of the integral operator. To fast solve this model, multigrid method is used for solving such linear algebraical system. Numerical comparison of multigrid method and GMRES method shows that multigrid method is superior to and more effective than GMRES method in solving the dense algebraic systems resulting from the FD approximations of the PIDEs.