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

Asia-Pacific Financial Markets volume 24, pages 291–308 (2017)Cite this article

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.

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Acknowledgements

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.

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Authors and Affiliations

  1. CEMAPRE, Department of Mathematics, ISEG, Universidade de Lisboa, Rua do Quelhas, 6, 1200-781, Lisbon, Portugal

    Maria do Rosário Grossinho & Yaser Kord Faghan

  2. Department of Applied Mathematics and Statistics, Comenius University, 842 48, Bratislava, Slovak Republic

    Daniel Ševčovič

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  1. Maria do Rosário Grossinho
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  2. Yaser Kord Faghan
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  3. Daniel Ševčovič
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Correspondence to Daniel Ševčovič.

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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

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  • DOI: https://doi.org/10.1007/s10690-017-9234-1

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