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

TL;DR

This paper introduces a high-order deferred correction method combined with penalty iteration to accurately solve free boundary problems, achieving fourth-order convergence and demonstrating effectiveness in American option pricing.

Contribution

The paper develops a novel high-order deferred correction algorithm with penalty iteration for free boundary problems, enhancing convergence order on fixed grids.

Findings

Achieves fourth-order convergence for free boundary problems.

Demonstrates effectiveness on American option pricing.

Provides detailed error analysis and efficient convergence.

Abstract

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 fourth-order by solving successively corrected finite difference systems, where the corrections are derived from the previously computed lower order solutions. The penalty iterations converge quickly given a good initial guess. We demonstrate the accuracy and efficiency of our algorithm using several examples. Numerical results show that our algorithm gives fourth-order convergence for both the solution and the free boundary location. We also test our algorithm on the challenging American put option pricing problem. Our algorithm gives the expected high-order convergence.