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On Pricing Barrier Options with Discrete Monitoring

Author

Listed:
  • Kenichiro Shiraya

    (Mizuho-DL Financial Technology Co., Ltd.)

  • 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 and the duality formula in Malliavin calculus are effectively applied in an asymptotic expansion approach. First, the paper derives an asymptotic expansion 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 approximation for pricing discrete barrier options with stochastic volatility models. Finally, it provides numerical examples for pricing double barrier call options with discrete monitoring under the Heston model.

Suggested Citation

  • 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.
    • Handle: RePEc:tky:fseres:2010cf725
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    References listed on IDEAS

    as
    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 & Kohta Takehara & Masashi Toda, 2009. "Computation in an Asymptotic Expansion Method," CIRJE F-Series CIRJE-F-621, CIRJE, Faculty of Economics, University of Tokyo.
    3. Maria Siopacha & Josef Teichmann, 2007. "Weak and Strong Taylor methods for numerical solutions of stochastic differential equations," Papers 0704.0745, arXiv.org.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. Akihiko Takahashi & Kohta Takehara & Masashi Toda, 2009. "Computation in an Asymptotic Expansion Method," CARF F-Series CARF-F-149, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    9. 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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