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Asymptotic Expansion Approaches in Finance: Applications to Currency Options

Author

Listed:
  • Akihiko Takahashi

    (Faculty of Economics, University of Tokyo)

  • Kohta Takehara

    (Graduate School of Economics, University of Tokyo)

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.

Suggested Citation

  • Akihiko Takahashi & Kohta Takehara, 2009. "Asymptotic Expansion Approaches in Finance: Applications to Currency Options," CARF F-Series CARF-F-165, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    • Handle: RePEc:cfi:fseres:cf165
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    References listed on IDEAS

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    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. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    4. Akihiko Takahashi & Nakahiro Yoshida, 2005. "Monte Carlo Simulation with Asymptotic Method," CIRJE F-Series CIRJE-F-335, CIRJE, Faculty of Economics, University of Tokyo.
    5. 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.
    6. Akihiko Takahashi & Kohta Takehara & Masashi Toda, 2009. "Computation in an Asymptotic Expansion Method," CIRJE F-Series CIRJE-F-621, CIRJE, Faculty of Economics, University of Tokyo.
    7. Maria Siopacha & Josef Teichmann, 2007. "Weak and Strong Taylor methods for numerical solutions of stochastic differential equations," Papers 0704.0745, arXiv.org.
    8. 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.
    9. 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.
    10. Amin, Kaushik I & Bodurtha, James N, Jr, 1995. "Discrete-Time Valuation of American Options with Stochastic Interest Rates," The Review of Financial Studies, Society for Financial Studies, vol. 8(1), pages 193-234.
    11. 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..
    12. Orlin J. Grabbe, "undated". "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.
    13. Mikkelsen, Peter, 2001. "Cross-Currency LIBOR Market Models," Finance Working Papers 01-6, University of Aarhus, Aarhus School of Business, Department of Business Studies.
    14. 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(2), pages 139-151, June.
    15. 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.
    16. 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.
    17. Orlin J. Grabbe, "undated". "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.
    18. 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.
    19. Miltersen, Kristian R & Sandmann, Klaus & Sondermann, Dieter, 1997. "Closed Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates," Journal of Finance, American Finance Association, vol. 52(1), pages 409-430, March.
    20. Kaushik I. Amin & Robert A. Jarrow, 2008. "Pricing foreign currency options under stochastic interest rates," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 14, pages 307-326, World Scientific Publishing Co. Pte. Ltd..
    21. 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..
    22. Yuji Sakamoto & Nakahiro Yoshida, 1998. "Asymptotic Expansion of M ‐Estimator Over Wiener Space," Statistical Inference for Stochastic Processes, Springer, vol. 1(1), pages 85-103, January.
    23. Akihiko Takahashi & , Kota Takehara & Akira Yamazaki, 2006. "Pricing Currency Options with a Market Model of Interest Rates under Jump-Diffusion Stochastic Volatility Processes of Spot Exchange Rates," CIRJE F-Series CIRJE-F-451, CIRJE, Faculty of Economics, University of Tokyo.
    24. Akihiko Takahashi & Kohta Takehara & Masashi Toda, 2009. "Computation in an Asymptotic Expansion Method," CARF F-Series CARF-F-149, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    25. Yoshifumi Muroi, 2005. "Pricing contingent claims with credit risk: Asymptotic expansion approach," Finance and Stochastics, Springer, vol. 9(3), pages 415-427, July.
    26. 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, January.
    27. 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.
    28. 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 637-654, May-June.
    29. Akihiko Takahashi & Kota Takehara & Akira Yamazaki, 2006. "Pricing Currency Options with a Market Model of Interest Rates under Jump-Diffusion Stochastic Volatility Processes of Spot Exchange Rates," CARF F-Series CARF-F-082, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
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