TWO-GRID ALGORITHMS FOR PRICING AMERICAN OPTIONS BY A PENALTY METHOD

In this manuscript we present two-grid algorithms for the American option pricing problem with a smooth penalty method where the variational inequality, associated with the optimal stopping time problem, is approximated with a nonlinear Black-Scholes equation. In order to compute the numerical solution of the latter unconstrained problem we must solve a system of nonlinear algebraic equations resulting from the discretization by e.g. the finite difference or the finite element method. We propose two-grid algorithms as we first solve the nonlinear system on a coarse grid with mesh size h(c) and further a linearized system on a fine grid with mesh size h(f), satisfying h(f) = O((h(c))(2k)), k = 1, 2,..., where k is the number of Newton iterations. Numerical experiments illustrate the computational efficiency of the algorithms.