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Smart expansion and fast calibration for jump diffusion

  • Eric Benhamou

    (Pricing Partners - Pricing Partners)

  • Emmanuel Gobet


    (LJK - Laboratoire Jean Kuntzmann - Institut Polytechnique de Grenoble - Grenoble Institute of Technology - Grenoble 2 UPMF - Université Pierre Mendès France - UJF - Université Joseph Fourier - CNRS)

  • Mohammed Miri


    (MATHFI - LJK - Laboratoire Jean Kuntzmann - Institut Polytechnique de Grenoble - Grenoble Institute of Technology - Grenoble 2 UPMF - Université Pierre Mendès France - UJF - Université Joseph Fourier - CNRS - Pricing Partners - Pricing Partners)

Registered author(s):

    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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    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 in Finance and Stochastics, Springer Verlag (Germany), 2009, 13 (4), pp.563-589. <10.1007/s00780-009-0102-3>
    Handle: RePEc:hal:journl:hal-00200395
    DOI: 10.1007/s00780-009-0102-3
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    1. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
    2. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    3. Rubinstein, Mark, 1994. " Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
    4. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
    5. Bouchard, Bruno & Elie, Romuald, 2008. "Discrete-time approximation of decoupled Forward-Backward SDE with jumps," Stochastic Processes and their Applications, Elsevier, vol. 118(1), pages 53-75, January.
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