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New approximations in local volatility models

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  • Emmanuel Gobet

    (CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique - X - École polytechnique - CNRS - Centre National de la Recherche Scientifique)

  • Ali Suleiman

    (ENSIMAG - École nationale supérieure d'informatique et de mathématiques appliquées - Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology)

Abstract

For general time-dependent local volatility models, we propose new approximation formulas for the price of call options. This extends previous results of [BGM10b] where stochastic expansions combined with Malliavin calculus were performed to obtain approximation formulas based on the local volatility At The Money. Here, we derive alternative expansions involving the local volatility at strike. Averaging both expansions give even more accurate results. Approximations of the implied volatility are provided as well.

Suggested Citation

  • Emmanuel Gobet & Ali Suleiman, 2013. "New approximations in local volatility models," Post-Print hal-00523369, HAL.
    • Handle: RePEc:hal:journl:hal-00523369
    Note: View the original document on HAL open archive server: https://hal.science/hal-00523369
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    References listed on IDEAS

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    1. Patrick Hagan & Diana Woodward, 1999. "Equivalent Black volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 6(3), pages 147-157.
    2. E. Benhamou & E. Gobet & M. Miri, 2012. "Analytical formulas for a local volatility model with stochastic rates," Quantitative Finance, Taylor & Francis Journals, vol. 12(2), pages 185-198, September.
    3. Roger W. Lee, 2004. "The Moment Formula For Implied Volatility At Extreme Strikes," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 469-480, July.
    4. Vladimir Piterbarg, 2005. "Stochastic Volatility Model with Time-dependent Skew," Applied Mathematical Finance, Taylor & Francis Journals, vol. 12(2), pages 147-185.
    5. E. Benhamou & E. Gobet & M. Miri, 2009. "Smart expansion and fast calibration for jump diffusions," Finance and Stochastics, Springer, vol. 13(4), pages 563-589, September.
    6. E. Benhamou & E. Gobet & M. Miri, 2010. "Expansion Formulas For European Options In A Local Volatility Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 13(04), pages 603-634.
    7. Schroder, Mark Douglas, 1989. " Computing the Constant Elasticity of Variance Option Pricing Formula," Journal of Finance, American Finance Association, vol. 44(1), pages 211-219, March.
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