A high-order deferred correction method for the solution of free boundary problems using penalty iteration, with an application to American option pricing

This paper presents a high-order deferred correction algorithm combined with penalty iteration for solving free and moving boundary problems, using a fourth-order finite difference method. Typically, when free boundary problems are solved on a fixed computational grid, the order of the solution is low due to the discontinuity in the solution at the free boundary, even if a high order method is used. Using a detailed error analysis, we observe that the order of convergence of the solution can be increased to fifth-order by solving successively corrected finite difference systems, where the corrections are derived from the previously computed lower order solutions. Since our method applies corrections to the right-hand side of the system and does not destroy the monotone property of the discretization matrix, penalty iterations converge quickly in only a few iterations given a good initial guess. We demonstrate the efficiency of our algorithm using several examples. Numerical results show that our algorithm gives exactly fifth-order convergence for both the solution and the free boundary location in the ideal case. We also test our algorithm on challenging American put option pricing problems. Our algorithm gives expected high-order convergence and compares favorably with existing methods that aim for high accuracy.