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A Hybrid Asymptotic Expansion Scheme: an Application to Long-term Currency Options ( Revised in April 2008, January 2009 and April 2010; forthcoming in "International Journal of Theoretical and Applied Finance". )

Author

Listed:
  • Akihiko Takahashi

    (Faculty of Economics, University of Tokyo)

  • Kohta Takehara

    (Graduate School of Economics, University of Tokyo)

Abstract

This paper develops a general approximation scheme, henceforth called a hybrid asymptotic expansion scheme for the valuation of multi-factor European path-independent derivatives. Specifically, we apply it to pricing long-term currency options under a market model of interest rates and a general diffusion stochastic volatility model with jumps of spot exchange rates. Our scheme is very effective for a type of models in which there exist correlations among all the factors whose dynamics are not necessarily affine nor even Markovian so long as the randomness is generated by Brownian motions. It can also handle models that include jump components under an assumption of their independence of the other random variables when the characteristic functions for the jump parts can be analytically obtained. Moreover, the hybrid scheme develops Fourier transform method with an asymptotic expansion to utilize closed-form characteristic functions obtainable in parts of a model. Our scheme also introduces a characteristic-function-based Monte Carlo simulation method with the asymptotic expansion as a control variable in order to make full use of analytical approximations by the asymptotic expansion and of closed-form characteristic functions. Finally, a series of numerical examples shows the validity of our scheme.

Suggested Citation

  • Akihiko Takahashi & Kohta Takehara, 2008. "A Hybrid Asymptotic Expansion Scheme: an Application to Long-term Currency Options ( Revised in April 2008, January 2009 and April 2010; forthcoming in "International Journal of Theoretical and A," CARF F-Series CARF-F-116, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    • Handle: RePEc:cfi:fseres:cf116
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    References listed on IDEAS

    as
    1. Pierre Henry-Labordere, 2006. "Unifying the BGM and SABR Models: A short Ride in Hyperbolic Geometry," Papers physics/0602102, arXiv.org.
    2. Takao Kobayashi & Akihiko Takahashi & Norio Tokioka, 2003. "Dynamic Optimality of Yield Curve Strategies," International Review of Finance, International Review of Finance Ltd., vol. 4(1‐2), pages 49-78, March.
    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. Maria Siopacha & Josef Teichmann, 2007. "Weak and Strong Taylor methods for numerical solutions of stochastic differential equations," Papers 0704.0745, arXiv.org.
    5. Akihiko Takahashi & Nakahiro Yoshida, 2004. "An Asymptotic Expansion Scheme for Optimal Investment Problems," Statistical Inference for Stochastic Processes, Springer, vol. 7(2), pages 153-188, May.
    6. 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.
    7. 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..
    8. Akihiko Takahashi & Kohta Takehara, 2008. "Fourier Transform Method with an Asymptotic Expansion Approach: an Application to Currency Options," CIRJE F-Series CIRJE-F-538, CIRJE, Faculty of Economics, University of Tokyo.
    9. Akihiko Takahashi & Kohta Takehara, 2008. "Fourier Transform Method With An Asymptotic Expansion Approach: An Application To Currency Options," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 11(04), pages 381-401.
    10. Naoto Kunitomo & Yong‐Jin Kim, 2007. "Effects Of Stochastic Interest Rates And Volatility On Contingent Claims," The Japanese Economic Review, Japanese Economic Association, vol. 58(1), pages 71-106, March.
    11. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    12. 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.
    13. Naoto Kunitomo & Akihiko Takahashi, 2004. "Applications of the Asymptotic Expansion Approach based on Malliavin-Watanabe Calculus in Financial Problems," World Scientific Book Chapters, in: Jiro Akahori & Shigeyoshi Ogawa & Shinzo Watanabe (ed.), Stochastic Processes And Applications To Mathematical Finance, chapter 10, pages 195-232, World Scientific Publishing Co. Pte. Ltd..
    14. Yoshifumi Muroi, 2005. "Pricing contingent claims with credit risk: Asymptotic expansion approach," Finance and Stochastics, Springer, vol. 9(3), pages 415-427, July.
    15. Jiro Akahori & Shigeyoshi Ogawa & Shinzo Watanabe (ed.), 2004. "Stochastic Processes and Applications to Mathematical Finance," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 5487, August.
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