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

Maria do Rosário Grossinho (), Yaser Kord Faghan and Daniel Ševčovič ()
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Maria do Rosário Grossinho: Universidade de Lisboa
Yaser Kord Faghan: Universidade de Lisboa
Daniel Ševčovič: Comenius University

Asia-Pacific Financial Markets, 2017, vol. 24, issue 4, No 3, 308 pages

Abstract: 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.

Keywords: Option pricing; Nonlinear Black–Scholes equation; Perpetual American put option; Early exercise boundary; 35R35; 91B28; 62P05 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10690-017-9234-1

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