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Application Of A High-Order Asymptotic Expansion Scheme To Long-Term Currency Options

Author

Listed:
  • Kohta Takehara
  • Masashi Toda
  • Akihiko Takahashi

Abstract

Recently academic researchers and practitioners have use the asymptotic expansion method to examine a variety of financial issues under high-dimensional stochastic environments. This methodology is mathematically justified by Watanabe theory (Watanabe, 1987), and Malliavin calculus (Yoshida, 1992a,b) and essentially based on the framework initiated by Kunitomo and Takahashi (2003) and Takahashi (1995, 1999) in a financial context. In practical applications, it is desirable to investigate the accuracy and stability of the method especially with expansion to higher orders in situations where the underlying processes are highly volatile. After Takahashi (1995,1999) and Takahashi and Takehara (2007) provided explicit formulas for the expansion to the third order, Takahashi, Takehara and Toda (2009) develop general computation schemes and formulas for an arbitrary-order expansion under general diffusion-type stochastic environments. In this paper, we describe these techniques in a simple setting to illustrate thier key ideas. To demonstrate their effectiveness the techniques are applied to pricing long-term currency options.

Suggested Citation

  • Kohta Takehara & Masashi Toda & Akihiko Takahashi, 2011. "Application Of A High-Order Asymptotic Expansion Scheme To Long-Term Currency Options," The International Journal of Business and Finance Research, The Institute for Business and Finance Research, vol. 5(3), pages 87-99.
    • Handle: RePEc:ibf:ijbfre:v:5:y:2011:i:3:p:87-99
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    Citations

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    Cited by:

    1. Akihiko Takahashi & Yukihiro Tsuzuki, 2013. "A New Improvement Scheme for Approximation Methods of Probability Density Functions," CIRJE F-Series CIRJE-F-874, CIRJE, Faculty of Economics, University of Tokyo.
    2. Akihiko Takahashi & Yukihiro Tsuzuki, 2013. "A New Improvement Scheme for Approximation Methods of Probability Density Functions," CARF F-Series CARF-F-305, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    3. Akihiko Takahashi & Kohta Takehara & Masashi Toda, 2011. "A General Computation Scheme for a High-Order Asymptotic Expansion Method," CIRJE F-Series CIRJE-F-787, CIRJE, Faculty of Economics, University of Tokyo.
    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. Akihiko Takahashi & Kohta Takehara & Masashi Toda, 2011. "A General Computation Scheme for a High-Order Asymptotic Expansion Method," CARF F-Series CARF-F-242, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Jul 2011.
    6. Akihiko Takahashi & Toshihiro Yamada, 2014. "On Error Estimates for Asymptotic Expansions with Malliavin Weights -Application to Stochastic Volatility Model- (Revised version of CARF-F-324; Forthcoming in "Mathematics of Operations Research," CARF F-Series CARF-F-347, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Sep 2014.
    7. Akihiko Takahashi, 2015. "Asymptotic Expansion Approach in Finance," CARF F-Series CARF-F-356, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Aug 2015.
    8. 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.

    More about this item

    Keywords

    Asymptotic Expansion; Malliavin Calculus; Stochastic Volatility; Libor Market Model; Currency Options;
    All these keywords.

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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