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A weak approximation with asymptotic expansion and multidimensional Malliavin weights (Revised version of CARF-F-335; Forthcoming in Annals of Applied Probability")"

Author

Listed:
  • Akihiko Takahashi

    (The University of Tokyo)

  • Toshihiro Yamada

    (The University of Tokyo and MTEC)

Abstract

This paper develops a new efficient scheme for approximations of expectations of the solutions to stochastic differential equations (SDEs). In particular, we present a method for connecting approximate operators based on an asymptotic expansion with multidimensional Malliavin weights to compute a target expectation value precisely. The mathematical validity is given based on Watanabe and Kusuoka theories in Malliavin calculus. Moreover, numerical experiments for option pricing under local and stochastic volatility models confirm the effectiveness of our scheme. Especially, our weak approximation substantially improve the accuracy at deep Out-of-The-Moneys (OTMs).

Suggested Citation

  • Akihiko Takahashi & Toshihiro Yamada, 2015. "A weak approximation with asymptotic expansion and multidimensional Malliavin weights (Revised version of CARF-F-335; Forthcoming in Annals of Applied Probability")"," CARF F-Series CARF-F-358, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Apr 2016.
    • Handle: RePEc:cfi:fseres:cf358
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    References listed on IDEAS

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    1. Hideyuki Tanaka & Toshihiro Yamada, 2014. "Strong Convergence For Euler–Maruyama And Milstein Schemes With Asymptotic Method," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 17(02), pages 1-22.
    2. 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.
    3. Akihiko Takahashi & Toshihiro Yamada, 2013. "On Error Estimates for Asymptotic Expansions with Malliavin Weights -Application to Stochastic Volatility Model-," CARF F-Series CARF-F-324, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Mar 2014.
    4. Akihiko Takahashi & Masashi Toda, 2013. "Note On An Extension Of An Asymptotic Expansion Scheme," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(05), pages 1-23.
    5. Akihiko Takahashi & Toshihiro Yamada, 2013. "On an Asymptotic Expansion of Forward-Backward SDEs with a Perturbed Driver," CARF F-Series CARF-F-326, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Oct 2013.
    6. Akihiko Takahashi & Toshihiro Yamada, 2013. "On an Asymptotic Expansion of Forward-Backward SDEs with a Perturbed Driver," CIRJE F-Series CIRJE-F-902, CIRJE, Faculty of Economics, University of Tokyo.
    7. Masaaki Fujii, 2012. "Momentum-Space Approach to Asymptotic Expansion for Stochastic Filtering," Papers 1209.1893, arXiv.org, revised Mar 2013.
    8. Masaaki Fujii, 2013. "Momentum-Space Approach to Asymptotic Expansion for Stochastic Filtering," CIRJE F-Series CIRJE-F-883, CIRJE, Faculty of Economics, University of Tokyo.
    9. 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.
    10. 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.
    11. 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.
    12. Akihiko Takahashi & Masashi Toda, 2013. "Note on an Extension of an Asymptotic Expansion Scheme," CARF F-Series CARF-F-312, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    13. Masaaki Fujii, 2012. "Momentum-Space Approach to Asymptotic Expansion for Stochastic Filtering," CARF F-Series CARF-F-311, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Mar 2013.
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    Cited by:

    1. Yamada, Toshihiro, 2015. "A formula of small time expansion for Young SDE driven by fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 101(C), pages 64-72.
    2. Masaaki Fujii & Akihiko Takahashi, 2018. "Asymptotic Expansion for Forward-Backward SDEs with JumpsAsymptotic Expansion for Forward-Backward SDEs with Jumps (Forthcoming in Stochastics) (Revised version of F-372)," CARF F-Series CARF-F-445, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.

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