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Smart expansion and fast calibration for jump diffusion

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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

    Using Malliavin calculus techniques, we derive an analytical formula for the price of European options, for any model including local volatility and Poisson jump process. We show that the accuracy of the formula depends on the smoothness of the payoff function. Our approach relies on an asymptotic expansion related to small diffusion and small jump frequency/size. Our formula has excellent accuracy (the error on implied Black-Scholes volatilities for call option is smaller than 2 bp for various strikes and maturities). Additionally, model calibration becomes very rapid.

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    File URL: http://hal.archives-ouvertes.fr/docs/00/32/59/30/PDF/smart_expansion_J-16Sep08.pdf
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    Bibliographic Info

    Paper provided by HAL in its series Post-Print with number hal-00200395.

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    Date of creation: Sep 2009
    Date of revision:
    Publication status: Published, Finance and Stochastics, 2009, 13, 4, 563-589
    Handle: RePEc:hal:journl:hal-00200395

    Note: View the original document on HAL open archive server: http://hal.archives-ouvertes.fr/hal-00200395
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    Web page: http://hal.archives-ouvertes.fr/

    Related research

    Keywords: asymptotic expansion; Malliavin calculus; volatility skew and smile; small diffusion process; small jump frequency/size;

    References

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    1. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, Elsevier, vol. 3(1-2), pages 125-144.
    2. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers, University of California at Berkeley RPF-232, University of California at Berkeley.
    3. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, Finance Press, number ovsv.
    4. Rubinstein, Mark, 1994. " Implied Binomial Trees," Journal of Finance, American Finance Association, American Finance Association, vol. 49(3), pages 771-818, July.
    5. Bouchard, Bruno & Elie, Romuald, 2008. "Discrete-time approximation of decoupled Forward-Backward SDE with jumps," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 118(1), pages 53-75, January.
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    Cited by:
    1. Emmanuel Gobet & Ali Suleiman, 2013. "New approximations in local volatility models," Post-Print, HAL hal-00523369, HAL.
    2. Gobet, Emmanuel & Miri, Mohammed, 2014. "Weak approximation of averaged diffusion processes," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 124(1), pages 475-504.
    3. Elisa Alòs, 2012. "A decomposition formula for option prices in the Heston model and applications to option pricing approximation," Finance and Stochastics, Springer, Springer, vol. 16(3), pages 403-422, July.

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