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Application of a High-Order Asymptotic Expansion Scheme to Long-Term Currency Options

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  • Kohta Takehara

    (Graduate School of Economics, University of Tokyo)

  • Masashi Toda

    (Graduate School of Economics, University of Tokyo)

  • Akihiko Takahashi

    (Faculty of Economics, University of Tokyo)

Abstract

Recently, not only academic researchers but also many practitioners have used the methodology so-called "an asymptotic expansion method" in their proposed techniques for a variety of financial issues. e.g. pricing or hedging complex derivatives under high-dimensional stochastic environments. This methodology is mathematically justified by Watanabe theory(Watanabe [1987], Yoshida [1992a,b]) in Malliavin calculus and essentially based on the framework initiated by Kunitomo and Takahashi [2003], 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 up to high orders in situations where the underlying processes are highly volatile as seen in recent financial markets. After Takahashi [1995,1999] and Takahashi and Takehara [2007] had provided explicit formulas for the expansion up to the third order, Takahashi, Takehara and Toda [2009] develops general computation schemes and formulas for an arbitrary-order expansion under general diffusion-type stochastic environments. In this paper, we describe them in a simple setting to illustrate thier key idea, and to demonstrate their effectiveness apply them to pricing long-term currency options under a cross-currency Libor market model and a general stochastic volatility of a spot exchange rate with maturities up to twenty years.

Suggested Citation

  • Kohta Takehara & Masashi Toda & Akihiko Takahashi, 2010. "Application of a High-Order Asymptotic Expansion Scheme to Long-Term Currency Options," CIRJE F-Series CIRJE-F-753, CIRJE, Faculty of Economics, University of Tokyo.
  • Handle: RePEc:tky:fseres:2010cf753
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    File URL: http://www.cirje.e.u-tokyo.ac.jp/research/dp/2010/2010cf753.pdf
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    References listed on IDEAS

    as
    1. 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.
    2. Akihiko Takahashi & Nakahiro Yoshida, 2005. "Monte Carlo Simulation with Asymptotic Method," CIRJE F-Series CIRJE-F-335, CIRJE, Faculty of Economics, University of Tokyo.
    3. Yoshifumi Muroi, 2005. "Pricing contingent claims with credit risk: Asymptotic expansion approach," Finance and Stochastics, Springer, vol. 9(3), pages 415-427, July.
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    Cited by:

    1. 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.

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