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A Hybrid Asymptotic Expansion Scheme: an Application to Long-term Currency Options

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  • Akihiko Takahashi

    (Faculty of Economics, University of Tokyo)

  • Kohta Takehara

    (Graduate School of Economics, University of Tokyo)

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    Abstract

    This paper develops a general approximation scheme, henceforth called a hybrid asymptotic expansion scheme for 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. An asymptotic expansion approach provides a closed-form approximation formula for their values, which can be calculated in a moment and thus can be used for calibration or for an explicit approximation of Greeks of options. Moreover, this scheme develops Fourier transform method with an asymptotic expansion as well as with closed-form characteristic functions obtainable in parts of a model, extending the method proposed by Takehara and Takahashi[2008] to be applicable to a general class of models. It 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 the closed-form characteristic functions. Finally, a series of numerical examples shows the effectiveness of our scheme.

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    File URL: http://www.cirje.e.u-tokyo.ac.jp/research/dp/2010/2010cf734.pdf
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    Bibliographic Info

    Paper provided by CIRJE, Faculty of Economics, University of Tokyo in its series CIRJE F-Series with number CIRJE-F-734.

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    Length: 39pages
    Date of creation: Apr 2010
    Date of revision:
    Handle: RePEc:tky:fseres:2010cf734

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    References

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    1. Takao Kobayashi & Akihiko Takahashi & Norio Tokioka, 2004. "Dynamic Optimality of Yield Curve Strategies," CIRJE F-Series CIRJE-F-299, CIRJE, Faculty of Economics, University of Tokyo.
    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, 07.
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    Cited by:
    1. Akihiko Takahashi & Masashi Toda, 2012. "Note on an Extension of an Asymptotic Expansion Scheme," CIRJE F-Series CIRJE-F-860, CIRJE, Faculty of Economics, University of Tokyo.
    2. 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.

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