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An Asymptotic Expansion for Forward-Backward SDEs: A Malliavin Calculus Approach

Author

Listed:
  • Akihiko Takahashi

    (Graduate School of Economics, The University of Tokyo, Tokyo)

  • Toshihiro Yamada

    (Graduate School of Economics, The University of Tokyo, Tokyo & MTEC)

Abstract

This paper proposes a new closed-form approximation scheme for the representation of the forward-backward stochastic differential equations (FBSDEs) of Ma and Zhang (2002). In particular, we obtain an error estimate for the scheme applying Malliavin calculus method of Kunitomo and Takahashi (2001, 2003), Kusuoka (2003), Takahashi and Yamada (2012) for the forward SDEs combined with the Picard iteration scheme for the BSDEs. We also show numerical examples for pricing options with counterparty risk under the local and stochastic volatility models, where the credit value adjustment (CVA) is taken into account.

Suggested Citation

  • Akihiko Takahashi & Toshihiro Yamada, 2012. "An Asymptotic Expansion for Forward-Backward SDEs: A Malliavin Calculus Approach," CARF F-Series CARF-F-296, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Sep 2013.
    • Handle: RePEc:cfi:fseres:cf296
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    References listed on IDEAS

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    1. Masaaki Fujii & Akihiko Takahashi, 2011. "Collateralized CDS and Default Dependence -Implications for the Central Clearing-," CIRJE F-Series CIRJE-F-799, CIRJE, Faculty of Economics, University of Tokyo.
    2. Hisashi Nakamura & Wataru Nozawa & Akihiko Takahashi, 2009. "Macroeconomic Implications of Term Structures of Interest Rates Under Stochastic Differential Utility with Non-Unitary EIS," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 16(3), pages 231-263, September.
    3. Bender, Christian & Denk, Robert, 2007. "A forward scheme for backward SDEs," Stochastic Processes and their Applications, Elsevier, vol. 117(12), pages 1793-1812, December.
    4. Masaaki Fujii & Akihiko Takahashi, 2010. "Asymmetric and Imperfect Collateralization, Derivative Pricing, and CVA," CIRJE F-Series CIRJE-F-781, CIRJE, Faculty of Economics, University of Tokyo.
    5. Masaaki Fujii & Akihiko Takahashi, 2011. "Collateralized CDS and Default Dependence," Papers 1104.1855, arXiv.org.
    6. Bouchard, Bruno & Touzi, Nizar, 2004. "Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 175-206, June.
    7. Naoto Kunitomo & Akihiko Takahashi, 2001. "The Asymptotic Expansion Approach to the Valuation of Interest Rate Contingent Claims," Mathematical Finance, Wiley Blackwell, vol. 11(1), pages 117-151, January.
    8. Duffie, Darrell & Huang, Ming, 1996. "Swap Rates and Credit Quality," Journal of Finance, American Finance Association, vol. 51(3), pages 921-949, July.
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    Cited by:

    1. Akihiko Takahashi & Toshihiro Yamada, 2015. "An Asymptotic Expansion of Forward-Backward SDEs with a Perturbed Driver," CARF F-Series CARF-F-363, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    2. Kenichiro Shiraya & Akihiko Takahashi, 2015. "An approximation formula for basket option prices under local stochastic volatility with jumps: an application to commodity markets," CARF F-Series CARF-F-361, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Aug 2015.
    3. Akihiko Takahashi & Toshiaki Watanabe, 2015. "An Asymptotic Expansion of Forward-Backward SDEs with a Perturbed Driver ," CIRJE F-Series CIRJE-F-976, CIRJE, Faculty of Economics, University of Tokyo.
    4. Akihiko Takahashi & Toshihiro Yamada, 2015. "An asymptotic expansion of forward-backward SDEs with a perturbed driver," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 2(02), pages 1-29.

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