An Efficient Algorithm for Options Under Merton's Jump-Diffusion Model on Nonuniform Grids

In this paper, we consider the fast numerical valuation of European and American options under Merton's jump-diffusion model, which is given by a partial integro-differential equations. Due to the singularities and discontinuities of the model, the time-space grids are nonuniform with refinement near the strike price and expiry. On such nonuniform grids, the spatial differential operators are discretized by finite difference methods, and time stepping is performed using the discontinuous Galerkin finite element method. Owing to the nonuniform grids, algebraic multigrid method is used for solving the dense algebraical system resulting from the discretization of the integral term associated with jumps in models, which is more challenging. Numerical comparison of algebraic multigrid, the generalized minimal residual method, and the incomplete LU preconditioner shows that algebraic multigrid method is superior to and more effective than the other two methods in solving such dense algebraical system.