Pricing Perpetual Put Options by the Black-Scholes Equation with a Nonlinear Volatility Function

TL;DR

This paper extends the classical Black-Scholes model for perpetual put options by incorporating a nonlinear volatility function, providing existence, uniqueness, and explicit formulas for the solution and free boundary, along with numerical analysis.

Contribution

It introduces a generalized model with nonlinear volatility, deriving explicit formulas and proving fundamental properties like existence and uniqueness of solutions.

Findings

Derived a single implicit equation for the free boundary.

Obtained a closed-form formula for the option price.

Numerical results show parameter dependence of the free boundary and option price.

Abstract

We investigate qualitative and quantitative behavior of a solution of the mathematical model for pricing American style of perpetual put options. We assume the option price is a solution to the stationary generalized Black-Scholes equation in which the volatility function may depend on the second derivative of the option price itself. We prove existence and uniqueness of a solution to the free boundary problem. We derive a single implicit equation for the free boundary position and the closed form formula for the option price. It is a generalization of the well-known explicit closed form solution derived by Merton for the case of a constant volatility. We also present results of numerical computations of the free boundary position, option price and their dependence on model parameters.