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

• Main Authors: Grossinho, Maria do Rosario, Faghan, Yaser Kord, Sevcovic, Daniel
• Format: Journal Article
• Language: English
• Published: 03-11-2016
• Subjects: Quantitative Finance - Mathematical Finance
• DOI: 10.48550/arxiv.1611.00885

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.