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


  • Maria do Rosário Grossinho

    () (Universidade de Lisboa)

  • Yaser Kord Faghan

    (Universidade de Lisboa)

  • Daniel Ševčovič

    () (Comenius University)


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.

Suggested Citation

  • Maria do Rosário Grossinho & Yaser Kord Faghan & Daniel Ševčovič, 2017. "Pricing Perpetual Put Options by the Black–Scholes Equation with a Nonlinear Volatility Function," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 24(4), pages 291-308, December.
  • Handle: RePEc:kap:apfinm:v:24:y:2017:i:4:d:10.1007_s10690-017-9234-1
    DOI: 10.1007/s10690-017-9234-1

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    References listed on IDEAS

    1. J. D. Evans & R. Kuske & Joseph B. Keller, 2002. "American options on assets with dividends near expiry," Mathematical Finance, Wiley Blackwell, vol. 12(3), pages 219-237, July.
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    6. Umut Çetin & Robert A. Jarrow & Philip Protter, 2008. "Liquidity risk and arbitrage pricing theory," World Scientific Book Chapters,in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 8, pages 153-183 World Scientific Publishing Co. Pte. Ltd..
    7. Maria do Rosario Grossinho & Yaser Faghan Kord & Daniel Sevcovic, 2017. "Analytical and numerical results for American style of perpetual put options through transformation into nonlinear stationary Black-Scholes equations," Papers 1707.00356,
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    9. Brennan, Michael J & Schwartz, Eduardo S, 1977. "The Valuation of American Put Options," Journal of Finance, American Finance Association, vol. 32(2), pages 449-462, May.
    10. Rüdiger Frey & Alexander Stremme, 1997. "Market Volatility and Feedback Effects from Dynamic Hedging," Mathematical Finance, Wiley Blackwell, vol. 7(4), pages 351-374, October.
    11. RØdiger Frey, 1998. "Perfect option hedging for a large trader," Finance and Stochastics, Springer, vol. 2(2), pages 115-141.
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