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.
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This research was supported by the European Union in the FP7-PEOPLE-2012-ITN Project STRIKE—Novel Methods in Computational Finance (304617), the Project CEMAPRE MULTI/00491 financed by FCT/MEC through national funds and the Slovak research Agency Project VEGA 1/0251/16.
The authors declare that they have no competing interests.
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Grossinho, M.d.R., Kord Faghan, Y. & Ševčovič, D. Pricing Perpetual Put Options by the Black–Scholes Equation with a Nonlinear Volatility Function. Asia-Pac Financ Markets 24, 291–308 (2017). https://doi.org/10.1007/s10690-017-9234-1