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Asymptotic Expansion Approaches in Finance: Applications to 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 chapter presents a basic of the methodology so-called an asymptotic expansion approach, and applies this method to approximation of prices of currency options with a libor market model of interest rates and stochastic volatility models of spot exchange rates. The scheme enables us to derive closed-form approximation formulas for pricing currency options even with high flexibility of the underlying model; we do not model a foreign exchange rate's variance such as in Heston [27], but its volatility that follows a general time-inhomogeneous Markovian process. Further, the correlations among all the factors such as domestic and foreign interest rates, a spot foreign exchange rate and its volatility, are allowed. At the end of this chapter some numerical examples are provided and the pricing formula is applied to the calibration of volatility surfaces in the JPY/USD option market.

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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-654.

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    Length: 51 pages
    Date of creation: Aug 2009
    Date of revision:
    Handle: RePEc:tky:fseres:2009cf654

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    1. Akihiko Takahashi & Nakahiro Yoshida, 2005. "Monte Carlo Simulation with Asymptotic Method," CIRJE F-Series CIRJE-F-335, CIRJE, Faculty of Economics, University of Tokyo.
    2. Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring.
    3. Hilliard, Jimmy E. & Madura, Jeff & Tucker, Alan L., 1991. "Currency Option Pricing with Stochastic Domestic and Foreign Interest Rates," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 26(02), pages 139-151, June.
    4. Amin, Kaushik I & Bodurtha, James N, Jr, 1995. "Discrete-Time Valuation of American Options with Stochastic Interest Rates," Review of Financial Studies, Society for Financial Studies, vol. 8(1), pages 193-234.
    5. Miltersen, K. & K. Sandmann & D. Sondermann, 1994. "Closed Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates," Discussion Paper Serie B 308, University of Bonn, Germany.
    6. Orlin J. Grabbe, . "The Pricing of Call and Put Options on Foreign Exchange," Rodney L. White Center for Financial Research Working Papers 06-83, Wharton School Rodney L. White Center for Financial Research.
    7. Amin, Kaushik I. & Jarrow, Robert A., 1991. "Pricing foreign currency options under stochastic interest rates," Journal of International Money and Finance, Elsevier, vol. 10(3), pages 310-329, September.
    8. 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.
    9. Orlin J. Grabbe, . "The Pricing of Call and Put Options on Foreign Exchange," Rodney L. White Center for Financial Research Working Papers 6-83, Wharton School Rodney L. White Center for Financial Research.
    10. 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.
    11. Yoshifumi Muroi, 2005. "Pricing contingent claims with credit risk: Asymptotic expansion approach," Finance and Stochastics, Springer, vol. 9(3), pages 415-427, 07.
    12. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
    13. Orlin Grabbe, J., 1983. "The pricing of call and put options on foreign exchange," Journal of International Money and Finance, Elsevier, vol. 2(3), pages 239-253, December.
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