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An Asymptotic Expansion with Malliavin Weights: An Application to Pricing Discrete Barrier Options

Author

Listed:
  • Akihiko Takahashi

    (Faculty of Economics, University of Tokyo)

  • Toshihiro Yamada

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

Abstract

This paper proposes a new approximation method for pricing barrier options with discrete monitoring under stochastic volatility environment. In particular, the integration-by-parts formula in Malliavin calculus is effectively applied in an asymptotic expansion approach. First, the paper derives an expansion formula for generalized Wiener functionals. After it is applied to pricing path-dependent derivatives with discrete monitoring, the paper presents an analytic (approximation) formula for valuation of discrete barrier options under stochastic volatility environment. To our knowledge, this paper is the first one that shows an analytical formula for pricing discrete barrier options with stochastic volatility models.

Suggested Citation

  • Akihiko Takahashi & Toshihiro Yamada, 2009. "An Asymptotic Expansion with Malliavin Weights: An Application to Pricing Discrete Barrier Options," CARF F-Series CARF-F-193, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    • Handle: RePEc:cfi:fseres:cf193
    as

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    File URL: https://www.carf.e.u-tokyo.ac.jp/old/pdf/workingpaper/fseries/200.pdf
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    References listed on IDEAS

    as
    1. Maria Siopacha & Josef Teichmann, 2007. "Weak and Strong Taylor methods for numerical solutions of stochastic differential equations," Papers 0704.0745, arXiv.org.
    2. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux, 2001. "Applications of Malliavin calculus to Monte-Carlo methods in finance. II," Finance and Stochastics, Springer, vol. 5(2), pages 201-236.
    3. 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.
    4. Kenichiro Shiraya & Akihiko Takahashi & Masashi Toda, 2009. "Pricing Barrier and Average Options under Stochastic Volatility Environment," CIRJE F-Series CIRJE-F-682, CIRJE, Faculty of Economics, University of Tokyo.
    5. Gianluca Fusai & I. Abrahams & Carlo Sgarra, 2006. "An exact analytical solution for discrete barrier options," Finance and Stochastics, Springer, vol. 10(1), pages 1-26, January.
    6. Kenichiro Shiraya & Akihiko Takahashi & Masashi Toda, 2009. "Pricing Average Options under Stochastic Volatility Models," CARF J-Series CARF-J-059, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    7. Akihiko Takahashi & Toshihiro Yamada, 2009. "An Asymptotic Expansion with Push-Down of Malliavin Weights," CIRJE F-Series CIRJE-F-695, CIRJE, Faculty of Economics, University of Tokyo.
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