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

Maria Grossinho (), Yaser Kord Faghan and Daniel Sevcovic

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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.

Date: 2016-11, Revised 2017-11
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