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An Asymptotic Expansion with Push-Down of Malliavin Weights

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  • Akihiko Takahashi

    (Faculty of Economics, University of Tokyo)

  • Toshihiro Yamada

    (Mitsubishi UFJ Trust Investment Technology Institute Co.,Ltd. (MTEC))

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    Abstract

    This paper derives asymptotic expansion formulas for option prices and implied volatilities as well as the density of the underlying asset price in a stochastic volatility model. In particular, the integration-by-parts formula in Malliavin calculus and the push-down of Malliavin weights are effectively applied. It provides an expansion formula for generalized Wiener functionals and closed-form approximation formulas in stochastic volatility environment. In addition, it presents applications of the general formula to a local volatility expansion as well as to expansions of option prices for the shifted log-normal model with stochastic volatility. Moreover, with some result of Malliavin calculus in jump-type models, this paper derives an approximation formula for the jump-diffusion model in stochastic volatility environment. Some numerical examples are also shown.

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    File URL: http://www.cirje.e.u-tokyo.ac.jp/research/dp/2009/2009cf695.pdf
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    Bibliographic Info

    Paper provided by CIRJE, Faculty of Economics, University of Tokyo in its series CIRJE F-Series with number CIRJE-F-695.

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    Length: 31pages
    Date of creation: Dec 2009
    Date of revision:
    Handle: RePEc:tky:fseres:2009cf695

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    1. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux & Nizar Touzi, 1999. "Applications of Malliavin calculus to Monte Carlo methods in finance," Finance and Stochastics, Springer, vol. 3(4), pages 391-412.
    2. Akihiko Takahashi & Nakahiro Yoshida, 2004. "An Asymptotic Expansion Scheme for Optimal Investment Problems," Statistical Inference for Stochastic Processes, Springer, vol. 7(2), pages 153-188, May.
    3. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, July.
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    Cited by:
    1. Kenichiro Shiraya & Akihiko Takahashi & Toshihiro Yamada, 2010. "On Pricing Barrier Options with Discrete Monitoring," CIRJE F-Series CIRJE-F-725, CIRJE, Faculty of Economics, University of Tokyo.
    2. Akihiko Takahashi & Toshihiro Yamada, 2009. "An Asymptotic Expansion with Malliavin Weights: An Application to Pricing Discrete Barrier Options," CIRJE F-Series CIRJE-F-696, CIRJE, Faculty of Economics, University of Tokyo.

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