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An Asymptotic Expansion with Push-Down of Malliavin Weights

Author

Listed:
  • Akihiko Takahashi

    (Faculty of Ecocnomics, University of Tokyo)

  • Yoshihiro Yajima

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

Abstract

This paper derives asymptotic expansion formulas for option prices and implied volatilities as well as the density of the underlying asset price in multi-dimensional stochastic volatility models. In particular, the integration-byparts formula in Malliavin calculus and the push-down of Malliavin weights are effectively applied. We provide an expansion formula for generalized Wiener functionals and closed-form approximation formulas in stochastic volatility environment. In addition, we present applications of the general formula to expansions of option prices for the shifted log-normal model with stochastic volatility. Moreover, with some results of Malliavin calculus in jump-type models, we derive an approximation formula for the jump-diffusion model in stochastic volatility environment. Some numerical examples are also shown.

Suggested Citation

  • Akihiko Takahashi & Yoshihiro Yajima, 2011. "An Asymptotic Expansion with Push-Down of Malliavin Weights," CIRJE F-Series CIRJE-F-824, CIRJE, Faculty of Economics, University of Tokyo.
    • Handle: RePEc:tky:fseres:2011cf824
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    File URL: http://www.cirje.e.u-tokyo.ac.jp/research/dp/2011/2011cf824.pdf
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    References listed on IDEAS

    as
    1. Akihiko Takahashi & Nakahiro Yoshida, 2005. "Monte Carlo Simulation with Asymptotic Method (Published in "Journal of Japan Statistical Society", Vol.35-2, 171-203, 2005. )," CARF F-Series CARF-F-030, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    2. 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.
    3. Akihiko Takahashi & Nakahiro Yoshida, 2005. "Monte Carlo Simulation with Asymptotic Method," CIRJE F-Series CIRJE-F-335, CIRJE, Faculty of Economics, University of Tokyo.
    4. 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.
    5. 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.
    6. Maria Siopacha & Josef Teichmann, 2010. "Weak and strong Taylor methods for numerical solutions of stochastic differential equations," Quantitative Finance, Taylor & Francis Journals, vol. 11(4), pages 517-528.
    7. Fabio Antonelli & Sergio Scarlatti, 2009. "Pricing options under stochastic volatility: a power series approach," Finance and Stochastics, Springer, vol. 13(2), pages 269-303, April.
    8. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
    9. 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.
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