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Stochastic expansion for the pricing of call options with discrete dividends

Author

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  • Pierre Etoré

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

  • Emmanuel Gobet

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

Abstract

In the context of an asset paying affine-type discrete dividends, we present closed analytical approximations for the pricing of European vanilla options in the Black-Scholes model with time-dependent parameters. They are obtained using a stochastic Taylor expansion around a shifted lognormal proxy model. The final formulae are respectively first, second and third order approximations w.r.t. the fixed part of the dividends. Using Cameron-Martin transformations, we provide explicit representations of the correction terms as Greeks in the Black-Scholes model. The use of Malliavin calculus enables us to provide tight error estimates for our approximations. Numerical experiments show that the current approach yields very accurate results, in particular compared to known approximations of [BGS03,VW09], and quicker than the iterated integration procedure of [HHL03] or than the binomial tree method of [VN06].

Suggested Citation

  • Pierre Etoré & Emmanuel Gobet, 2012. "Stochastic expansion for the pricing of call options with discrete dividends," Post-Print hal-00507787, HAL.
    • Handle: RePEc:hal:journl:hal-00507787
    DOI: 10.1080/1350486X.2011.620397
    Note: View the original document on HAL open archive server: https://hal.science/hal-00507787
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    References listed on IDEAS

    as
    1. 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.
    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. 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.
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    Cited by:

    1. Gobet, Emmanuel & Miri, Mohammed, 2014. "Weak approximation of averaged diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 475-504.
    2. Fabien Le Floc'h, 2021. "More stochastic expansions for the pricing of vanilla options with cash dividends," Papers 2106.12051, arXiv.org.

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