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

• Published in: Asia-Pacific financial markets Vol. 24; no. 4; pp. 291 - 308
• Main Authors: Grossinho, Maria do Rosário, Kord Faghan, Yaser, Ševčovič, Daniel
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 01-12-2017; Springer Nature B.V
• Subjects: Black-Scholes equation; Closed form solutions; Econometrics; Economic Theory/Quantitative Economics/Mathematical Methods; Economics and Finance; Exact solutions; Finance; Free boundaries; Implicit equations; International Economics; Macroeconomics/Monetary Economics//Financial Economics; Mathematical models; Nonlinear equations; Put & call options; Securities prices; Uniqueness; Volatility; Nonlinear Black–Scholes equation; Option pricing; Early exercise boundary; 62P05; 35R35; 91B28; Perpetual American put option
• ISSN: 1387-2834; 1573-6946
• DOI: 10.1007/s10690-017-9234-1

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.