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Computation in an Asymptotic Expansion Method

Author

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  • Akihiko Takahashi

    (Faculty of Economics, University of Tokyo)

  • Kohta Takehara

    (Graduate School of Economics, University of Tokyo)

  • Masashi Toda

    (Graduate School of Economics, University of Tokyo)

Abstract

An asymptotic expansion scheme in finance initiated by Kunitomo and Takahashi [15] and Yoshida[68] is a widely applicable methodology for analytic approximation of the expectation of a certain functional of diffusion processes. [46], [47] and [53] provide explicit formulas of conditional expectations necessary for the asymptotic expansion up to the third order. In general, the crucial step in practical applications of the expansion is calculation of conditional expectations for a certain kind of Wiener functionals. This paper presents two methods for computing the conditional expectations that are powerful especially for high order expansions: The first one, an extension of the method introduced by the preceding papers presents a general scheme for computation of the conditional expectations and show the formulas useful for expansions up to the fourth order explicitly. The second one develops a new calculation algorithm for computing the coefficients of the expansion through solving a system of ordinary differential equations that is equivalent to computing the conditional expectations. To demonstrate their effectiveness, the paper gives numerical examples of the approximation for ă-SABR model up to the fifth order and a cross-currency Libor market model with a general stochastic volatility model of the spot foreign exchange rate up to the fourth order.

Suggested Citation

  • 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.
  • Handle: RePEc:tky:fseres:2009cf621
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    File URL: http://www.cirje.e.u-tokyo.ac.jp/research/dp/2009/2009cf621.pdf
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    References listed on IDEAS

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    1. Yoshida, Nakahiro, 1996. "Asymptotic Expansions for Perturbed Systems on Wiener Space: Maximum Likelihood Estimators," Journal of Multivariate Analysis, Elsevier, vol. 57(1), pages 1-36, April.
    2. David Heath & Robert Jarrow & Andrew Morton, 2008. "Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation," World Scientific Book Chapters,in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 13, pages 277-305 World Scientific Publishing Co. Pte. Ltd..
    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. Yoshifumi Muroi, 2005. "Pricing contingent claims with credit risk: Asymptotic expansion approach," Finance and Stochastics, Springer, vol. 9(3), pages 415-427, July.
    5. Sakamoto, Yuji & Yoshida, Nakahiro, 1996. "Expansion of Perturbed Random Variables Based on Generalized Wiener Functionals," Journal of Multivariate Analysis, Elsevier, vol. 59(1), pages 34-59, October.
    6. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    7. Atsushi Kawai, 2003. "A new approximate swaption formula in the LIBOR market model: an asymptotic expansion approach," Applied Mathematical Finance, Taylor & Francis Journals, vol. 10(1), pages 49-74.
    8. Yoshida, Nakahiro, 2003. "Conditional expansions and their applications," Stochastic Processes and their Applications, Elsevier, vol. 107(1), pages 53-81, September.
    9. Masuda, Hiroki & Yoshida, Nakahiro, 2004. "An application of the double Edgeworth expansion to a filtering model with Gaussian limit," Statistics & Probability Letters, Elsevier, vol. 70(1), pages 37-48, October.
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    Citations

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    Cited by:

    1. Kenichiro Shiraya & Akihiko Takahashi, 2015. "An Asymptotic Expansion for Local-Stochastic Volatility with Jump Models (Forthcoming in Stochastics)," CARF F-Series CARF-F-377, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    2. Kenichiro Shiraya & Akihiko Takahashi, 2015. "An Asymptotic Expansion for Local-Stochastic Volatility with Jump Models," CIRJE F-Series CIRJE-F-998, CIRJE, Faculty of Economics, University of Tokyo.
    3. repec:wsi:ijtafx:v:16:y:2013:i:05:n:s0219024913500313 is not listed on IDEAS
    4. Akihiko Takahashi & Toshihiro Yamada, 2015. "A weak approximation with asymptotic expansion and multidimensional Malliavin weights," CARF F-Series CARF-F-358, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Mar 2015.
    5. Fernández, J.L. & Ferreiro, A.M. & García-Rodríguez, J.A. & Leitao, A. & López-Salas, J.G. & Vázquez, C., 2013. "Static and dynamic SABR stochastic volatility models: Calibration and option pricing using GPUs," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 94(C), pages 55-75.

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