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Expansion formulas for European options in a local volatility model

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Author Info

  • Eric Benhamou

    (Pricing Partners - Pricing Partners)

  • Emmanuel Gobet

    ()
    (LJK - Laboratoire Jean Kuntzmann - CNRS : UMR5224 - Université Joseph Fourier - Grenoble I - Université Pierre Mendès-France - Grenoble II - Institut Polytechnique de Grenoble - Grenoble Institute of Technology)

  • Mohammed Miri

    (Pricing Partners - Pricing Partners, LJK - Laboratoire Jean Kuntzmann - CNRS : UMR5224 - Université Joseph Fourier - Grenoble I - Université Pierre Mendès-France - Grenoble II - Institut Polytechnique de Grenoble - Grenoble Institute of Technology)

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    Abstract

    Because of its very general formulation, the local volatility model does not have an analytical solution for European options. In this article, we present a new methodology to derive closed form solutions for the price of any European options. The formula results from an asymptotic expansion, terms of which are Black-Scholes price and related Greeks. The accuracy of the formula depends on the payoff smoothness and it converges with very few terms.

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    File URL: http://hal.archives-ouvertes.fr/docs/00/32/59/39/PDF/BenhamouGobetMiri_dupireSept08.pdf
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    Bibliographic Info

    Paper provided by HAL in its series Post-Print with number hal-00325939.

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    Date of creation: Jun 2010
    Date of revision:
    Publication status: Published, International Journal of Theoretical and Applied Finance, 2010, 13, 4, 603-634
    Handle: RePEc:hal:journl:hal-00325939

    Note: View the original document on HAL open archive server: http://hal.archives-ouvertes.fr/hal-00325939
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    Web page: http://hal.archives-ouvertes.fr/

    Related research

    Keywords: Local volatility model; European options; asymptotic expansion; Malliavin calculus; small diffusion process; CEV model;

    This paper has been announced in the following NEP Reports:

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    Cited by:
    1. Elisa Alòs, 2012. "A decomposition formula for option prices in the Heston model and applications to option pricing approximation," Finance and Stochastics, Springer, vol. 16(3), pages 403-422, July.
    2. Pierre Etore & Emmanuel Gobet, 2012. "Stochastic expansion for the pricing of call options with discrete dividends," Post-Print hal-00507787, HAL.
    3. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "Analytical expansions for parabolic equations," Papers 1312.3314, arXiv.org.
    4. Gobet, Emmanuel & Miri, Mohammed, 2014. "Weak approximation of averaged diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 475-504.
    5. Emmanuel Gobet & Ali Suleiman, 2013. "New approximations in local volatility models," Post-Print hal-00523369, HAL.
    6. Ale\v{s} \v{C}ern\'y & Stephan Denkl & Jan Kallsen, 2013. "Hedging in L\'evy models and the time step equivalent of jumps," Papers 1309.7833, arXiv.org, revised Nov 2013.

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