Transformation methods for evaluating approximations to the optimal exercise boundary for linear and nonlinear Black-Scholes equations

TL;DR

This paper introduces a transformation technique and iterative numerical scheme to analyze and approximate the early exercise boundary for both linear and nonlinear Black-Scholes equations, including applications to Asian options.

Contribution

It presents a novel transformation method converting free boundary problems into fixed domain equations and proposes an iterative scheme for approximating the early exercise boundary in nonlinear models.

Findings

Numerical results demonstrate the effectiveness of the method for various models.

The free boundary's dependence on model parameters is analyzed.

Application to Asian options showcases the method's versatility.

Abstract

The purpose of this survey chapter is to present a transformation technique that can be used in analysis and numerical computation of the early exercise boundary for an American style of vanilla options that can be modelled by class of generalized Black-Scholes equations. We analyze qualitatively and quantitatively the early exercise boundary for a linear as well as a class of nonlinear Black-Scholes equations with a volatility coefficient which can be a nonlinear function of the second derivative of the option price itself. A motivation for studying the nonlinear Black-Scholes equation with a nonlinear volatility arises from option pricing models taking into account e.g. nontrivial transaction costs, investor's preferences, feedback and illiquid markets effects and risk from a volatile (unprotected) portfolio. We present a method how to transform the free boundary problem for the early exercise boundary position into a solution of a time depending nonlinear nonlocal parabolic equation defined on a fixed domain. We furthermore propose an iterative numerical scheme that can be used in order to find an approximation of the free boundary. In the case of a linear Black-Scholes equation we are able to derive a nonlinear integral equation for the position of the free boundary. We present results of numerical approximation of the early exercise boundary for various types of linear and nonlinear Black-Scholes equations and we discuss dependence of the free boundary on model parameters. Finally, we discuss an application of the transformation method for the pricing equation for American type of Asian options.