Perpetual American Put Option: an Error Estimator for Non-Standard Finite Difference Scheme

TL;DR

This paper introduces a MATLAB implementation of a non-standard finite difference scheme for perpetual American put options, including an a posteriori error estimator based on Richardson's extrapolation, tested on problems with known solutions.

Contribution

It presents a novel MATLAB-based non-standard finite difference scheme with an exact boundary condition at infinity and an error estimator for perpetual American put options.

Findings

Effective boundary condition implementation at infinity.

Accurate error estimation using Richardson's extrapolation.

Validated scheme with known analytical solutions.

Abstract

In this paper we present a MATLAB version of a non-standard finite difference scheme for the numerical solution of the perpetual American put option models of financial markets. These models can be derived from the celebrated Black-Scholes models letting the time goes to infinity. The considered problem is a free boundary problem defined on a semi-infinite interval, so that it is a non-linear problem complicated by a boundary condition at infinity. By using non-uniform maps, we show how it is possible to apply the boundary condition at infinity exactly. Moreover, we define a posteriori error estimator that is based on Richardson's classical extrapolation theory. Our finite difference scheme and error estimator are favourably tested for a simple problem with a known exact analytical solution.