On power penalty methods for linear complementarity problems arising from American option pricing

Power penalty methods for solving a linear parabolic complementarity problem arising from American option pricing have attracted much attention. These methods require us to solve a series of systems of nonlinear equations (called penalized equations). In this paper, we first study the relationships among the solutions of penalized equations under appropriate conditions. Additionally, since these penalized equations are neither smooth nor convex, some existing algorithms, such as Newton method, cannot be applied directly to solve them. We shall apply the nonlinear Jacobian method to solve penalized equations and verify that the iteration sequence generated by the method converges monotonically to the solution of the penalized equation. Some numerical results confirm the theoretical results and the efficiency of the proposed algorithm.