Analytical and numerical results for American style of perpetual put options through transformation into nonlinear stationary Black-Scholes equations

TL;DR

This paper investigates the nonlinear Black-Scholes equation with state-dependent volatility to determine the early exercise boundary for perpetual American put options, providing analytical transformations and numerical results.

Contribution

It introduces a transformation method for solving nonlinear stationary Black-Scholes equations with free boundary problems for perpetual American puts.

Findings

Derived an ODE and implicit equation for the free boundary.

Numerically approximated the early exercise boundary and option prices.

Analyzed the dependence of results on model parameters.

Abstract

We analyze and calculate the early exercise boundary for a class of stationary generalized Black-Scholes equations in which the volatility function depends on 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 including, e.g., non-zero transaction costs, investors preferences, feedback and illiquid markets effects and risk from unprotected portfolio. We present a method how to transform the problem of American style of perpetual put options into a solution of an ordinary differential equation and implicit equation for the free boundary position. We finally present results of numerical approximation of the early exercise boundary, option price and their dependence on model parameters.