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On Error Estimates for Asymptotic Expansions with Malliavin Weights ï¼ Application to Stochastic Volatility Model ï¼

  • Akihiko Takahashi

    (Faculty of Economics, The University of Tokyo)

  • Toshihiro Yamada

    (Graduate School of Economics, The University of Tokyo and Mitsubishi UFJ Trust Investment Technology Institute Co.,Ltd. (MTEC))

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       This paper proposes a unified method for precise estimates of the error bounds in asymptotic expansions of an option price and its Greeks (sensitivities) under a stochastic volatility model. More generally, we also derive an error estimate for an asymptotic expansion around a general partially elliptic diffusion and a more general Wiener functional, which is applicable to various important valuation and risk management tasks in the financial business such as the ones for multi-dimensional diffusion and non-diffusion models. In particular, we take the Malliavin calculus approach, and estimate the error bounds for the Malliavin weights of both the coefficient and the residual terms in the expansions by effectively applying the properties of Kusuoka-Stroock functions. Moreover, a numerical experiment under the Heston-type model confirms the effectiveness of our method.

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    File URL: http://www.cirje.e.u-tokyo.ac.jp/research/dp/2013/2013cf897.pdf
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    Paper provided by CIRJE, Faculty of Economics, University of Tokyo in its series CIRJE F-Series with number CIRJE-F-897.

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    Length: 21 pages
    Date of creation: Jul 2013
    Date of revision:
    Handle: RePEc:tky:fseres:2013cf897
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    1. Akihiko Takahashi & Kohta Takehara, 2010. "A Hybrid Asymptotic Expansion Scheme: An Application To Long-Term Currency Options," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 13(08), pages 1179-1221.
    2. Kenichiro Shiraya & Akihiko Takahashi, 2012. "Pricing Multi-Asset Cross Currency Options," CARF F-Series CARF-F-276, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    3. Akihiko Takahashi & Kohta Takehara & Masashi Toda, 2012. "A General Computation Scheme For A High-Order Asymptotic Expansion Method," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(06), pages 1250044-1-1.
    4. Akihiko Takahashi & Kohta Takehara & Masashi Toda, 2012. "A General Computation Scheme for a High-Order Asymptotic Expansion Method," CARF F-Series CARF-F-272, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    5. Takashi Kato & Akihiko Takahashi & Toshihiro Yamada, 2013. "An Asymptotic Expansion Formula for Up-and-Out Barrier Option Price under Stochastic Volatility Model," Papers 1302.3306, arXiv.org.
    6. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-43.
    7. Kenichiro Shiraya & Akihiko Takahashi, 2012. "Pricing Multi-Asset Cross Currency Options," CARF F-Series CARF-F-290, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
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