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

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  • Akihiko Takahashi

    (Faculty of Economics, The University of Tokyo)

  • Toshihiro Yamada

    (Graduate School of Economics, Hitotsubashi University)

Abstract

This paper proposes a new analytical 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 for the forward SDEs combined with the Picard iteration scheme for the BSDEs. We also show numerical examples for pricing option with counterparty risk under local and stochastic volatility models, where the credit value adjustment (CVA) is taken into account.

Suggested Citation

  • Akihiko Takahashi & Toshihiro Yamada, 2016. "An Asymptotic Expansion for Forward-Backward SDEs: A Malliavin Calculus Approach," CIRJE F-Series CIRJE-F-1009, CIRJE, Faculty of Economics, University of Tokyo.
    • Handle: RePEc:tky:fseres:2016cf1009
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    File URL: http://www.cirje.e.u-tokyo.ac.jp/research/dp/2016/2016cf1009.pdf
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    References listed on IDEAS

    as
    1. Kenichiro Shiraya & Akihiko Takahashi & Toshihiro Yamada, 2012. "Pricing Discrete Barrier Options Under Stochastic Volatility," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 19(3), pages 205-232, September.
    2. Akihiko Takahashi & Kohta Takehara, 2007. "An Asymptotic Expansion Approach to Currency Options with a Market Model of Interest Rates under Stochastic Volatility Processes of Spot Exchange Rates," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 14(1), pages 69-121, March.
    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 1-25.
    4. Akihiko Takahashi & Kohta Takehara, 2007. "An Asymptotic Expansion Approach to Currency Options with a Market Model of Interest Rates under Stochastic Volatility Processes of Spot Exchange Rates," CIRJE F-Series CIRJE-F-474, CIRJE, Faculty of Economics, University of Tokyo.
    5. Masaaki Fujii & Akihiko Takahashi, 2012. "Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method," CARF F-Series CARF-F-278, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Jan 2015.
    6. 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.
    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. Masaaki Fujii & Akihiko Takahashi, 2012. "Collateralized CDS and Default Dependence -Implications for the Central Clearing," CARF F-Series CARF-F-281, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    9. Chenxu Li, 2014. "Closed-Form Expansion, Conditional Expectation, and Option Valuation," Mathematics of Operations Research, INFORMS, vol. 39(2), pages 487-516, May.
    10. Masaaki Fujii & Akihiko Takahashi, 2015. "Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 22(3), pages 283-304, September.
    11. Masaaki Fujii & Akihiko Takahshi, 2015. "Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method," CIRJE F-Series CIRJE-F-954, CIRJE, Faculty of Economics, University of Tokyo.
    12. Akihiko Takahashi & Toshihiro Yamada, 2015. "On Error Estimates for Asymptotic Expansions with Malliavin Weights: Application to Stochastic Volatility Model," Mathematics of Operations Research, INFORMS, vol. 40(3), pages 513-541, March.
    13. Kyo Yamamoto & Akihiko Takahashi, 2009. "A Remark on a Singular Perturbation Method for Option Pricing Under a Stochastic Volatility Model," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 16(4), pages 333-345, December.
    14. Masaaki Fujii & Seisho Sato & Akihiko Takahashi, 2015. "An FBSDE Approach to American Option Pricing with an Interacting Particle Method," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 22(3), pages 239-260, September.
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