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Expansion formulas for European options in a local volatility model

Author

Listed:
  • Eric Benhamou

    (Pricing Partners - Pricing Partners)

  • Emmanuel Gobet

    () (MATHFI - Mathématiques financières - LJK - Laboratoire Jean Kuntzmann - UPMF - Université Pierre Mendès France - Grenoble 2 - UJF - Université Joseph Fourier - Grenoble 1 - Institut Polytechnique de Grenoble - Grenoble Institute of Technology - CNRS - Centre National de la Recherche Scientifique - UGA - Université Grenoble Alpes)

  • Mohammed Miri

    () (Pricing Partners - Pricing Partners, MATHFI - Mathématiques financières - LJK - Laboratoire Jean Kuntzmann - UPMF - Université Pierre Mendès France - Grenoble 2 - UJF - Université Joseph Fourier - Grenoble 1 - Institut Polytechnique de Grenoble - Grenoble Institute of Technology - CNRS - Centre National de la Recherche Scientifique - UGA - Université Grenoble Alpes)

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.

Suggested Citation

  • Eric Benhamou & Emmanuel Gobet & Mohammed Miri, 2010. "Expansion formulas for European options in a local volatility model," Post-Print hal-00325939, HAL.
  • Handle: RePEc:hal:journl:hal-00325939
    DOI: 10.1142/S0219024910005887
    Note: View the original document on HAL open archive server: https://hal.archives-ouvertes.fr/hal-00325939
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    File URL: https://hal.archives-ouvertes.fr/hal-00325939/document
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    Citations

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    Cited by:

    1. repec:spr:finsto:v:21:y:2017:i:3:d:10.1007_s00780-017-0330-x is not listed on IDEAS
    2. Emmanuel Gobet & Ali Suleiman, 2013. "New approximations in local volatility models," Post-Print hal-00523369, HAL.
    3. Gobet, Emmanuel & Miri, Mohammed, 2014. "Weak approximation of averaged diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 475-504.
    4. repec:wsi:ijtafx:v:20:y:2017:i:05:n:s0219024917500340 is not listed on IDEAS
    5. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "Analytical expansions for parabolic equations," Papers 1312.3314, arXiv.org, revised Nov 2014.
    6. Julien Hok & Philip Ngare & Antonis Papapantoleon, 2018. "Expansion formulas for European quanto options in a local volatility FX-LIBOR model," Papers 1801.01205, arXiv.org, revised Apr 2018.
    7. Alev{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 Jul 2017.
    8. 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.
    9. Pierre Etore & Emmanuel Gobet, 2012. "Stochastic expansion for the pricing of call options with discrete dividends," Post-Print hal-00507787, HAL.
    10. Romain Bompis, 2017. "Weak approximations for arithmetic means of geometric Brownian motions and applications to Basket options," Working Papers hal-01502886, HAL.

    More about this item

    Keywords

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

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