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A Call-Put Duality for Perpetual American Options

Author

Listed:
  • Aur'elien Alfonsi

    (CERMICS)

  • Benjamin Jourdain

    (CERMICS)

Abstract

It is well known that in models with time-homogeneous local volatility functions and constant interest and dividend rates, the European Put prices are transformed into European Call prices by the simultaneous exchanges of the interest and dividend rates and of the strike and spot price of the underlying. This paper investigates such a Call Put duality for perpetual American options. It turns out that the perpetual American Put price is equal to the perpetual American Call price in a model where, in addition to the previous exchanges between the spot price and the strike and between the interest and dividend rates, the local volatility function is modified. We prove that equality of the dual volatility functions only holds in the standard Black-Scholes model with constant volatility. Thanks to these duality results, we design a theoretical calibration procedure of the local volatility function from the perpetual Call and Put prices for a fixed spot price $x_0$. The knowledge of the Put (resp. Call) prices for all strikes enables to recover the local volatility function on the interval $(0,x_0)$ (resp. $(x_0,+\infty)$).

Suggested Citation

  • Aur'elien Alfonsi & Benjamin Jourdain, 2006. "A Call-Put Duality for Perpetual American Options," Papers math/0612648, arXiv.org.
    • Handle: RePEc:arx:papers:math/0612648
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    File URL: https://arxiv.org/pdf/math/0612648
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    References listed on IDEAS

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    1. Fajardo, J. & Mordecki, E., 2003. "Put-Call Duality and Symmetry," Finance Lab Working Papers flwp_54, Finance Lab, Insper Instituto de Ensino e Pesquisa.
    2. repec:cup:astinb:v:24:y:1994:i:02:p:195-220_00 is not listed on IDEAS
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    Cited by:

    1. B. Jourdain, 2007. "Stochastic flow approach to Dupire’s formula," Finance and Stochastics, Springer, vol. 11(4), pages 521-535, October.
    2. Hyong-chol O & Song-San Jo, 2019. "Variational inequality for perpetual American option price and convergence to the solution of the difference equation," Papers 1903.05189, arXiv.org.
    3. Aurélien Alfonsi & Benjamin Jourdain, 2008. "General Duality For Perpetual American Options," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 11(06), pages 545-566.
    4. Aur'elien Alfonsi & Benjamin Jourdain, 2006. "General Duality for Perpetual American Options," Papers math/0612649, arXiv.org.

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    1. José Fajardo & Ernesto Mordecki, 2005. "Duality and Derivative Pricing with Time-Changed Lévy Processes," IBMEC RJ Economics Discussion Papers 2005-12, Economics Research Group, IBMEC Business School - Rio de Janeiro.
    2. B. Jourdain, 2007. "Stochastic flow approach to Dupire’s formula," Finance and Stochastics, Springer, vol. 11(4), pages 521-535, October.

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