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

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.