IDEAS home Printed from https://ideas.repec.org/p/tky/fseres/2008cf538.html
   My bibliography  Save this paper

Fourier Transform Method with an Asymptotic Expansion Approach: an Application to Currency Options

Author

Listed:
  • Akihiko Takahashi

    (Faculty of Economics, University of Tokyo)

  • Kohta Takehara

    (Graduate School of Economics, University of Tokyo)

Abstract

This paper develops a Fourier transform method with an asymptotic expansion approach for option pricing. The method is applied to European currency options with a libor market model of interest rates and jump-diffusion stochastic volatility models of spot exchange rates. In particular, we derive closed-form approximation formulas of the characteristic functions of log-prices of the underlying assets and the prices of currency options based on a third order asymptotic expansion scheme; we use a jump- diffusion model with a mean-reverting stochastic variance process such as in Heston [1993] / Bates [1996] and log-normal market models for domestic and foreign interest rates. Finally, the validity of our method is confirmed through numerical examples.

Suggested Citation

  • 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.
    • Handle: RePEc:tky:fseres:2008cf538
    as

    Download full text from publisher

    File URL: http://www.cirje.e.u-tokyo.ac.jp/research/dp/2008/2008cf538.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
    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. 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.
    5. 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.
    6. Akihiko Takahashi & Kohta Takehara, 2007. "An Asymptotic Expansion Approach to Currency Options with a Market Model of Interest Rates under Stochastic Volatility Processes of Spot Exchange Rates," CIRJE F-Series CIRJE-F-474, CIRJE, Faculty of Economics, University of Tokyo.
    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. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    9. 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.
    10. 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.
    11. 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..
    12. 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.
    13. Yoshifumi Muroi, 2005. "Pricing contingent claims with credit risk: Asymptotic expansion approach," Finance and Stochastics, Springer, vol. 9(3), pages 415-427, July.
    14. 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.
    15. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. Akihiko Takahashi & Masashi Toda, 2013. "Note on an Extension of an Asymptotic Expansion Scheme," CARF F-Series CARF-F-312, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    3. Chenxu Li, 2014. "Closed-Form Expansion, Conditional Expectation, and Option Valuation," Mathematics of Operations Research, INFORMS, vol. 39(2), pages 487-516, May.
    4. Akihiko Takahashi & Kohta Takehara, 2010. "A Hybrid Asymptotic Expansion Scheme: an Application to Long-term Currency Options," CIRJE F-Series CIRJE-F-734, CIRJE, Faculty of Economics, University of Tokyo.
    5. Yoshifumi Muroi & E. Takino, 2011. "Pricing Derivatives using the Asymptotic Expansion Approach: Credit Migration Models with Stochastic Credit Spreads," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 18(4), pages 345-372, November.
    6. 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.
    7. Lech A. Grzelak & Cornelis W. Oosterlee, 2012. "On Cross-Currency Models with Stochastic Volatility and Correlated Interest Rates," Applied Mathematical Finance, Taylor & Francis Journals, vol. 19(1), pages 1-35, February.
    8. Akihiko Takahashi & Kohta Takehara, 2007. "An Asymptotic Expansion Approach to Currency Options with a Market Model of Interest Rates under Stochastic Volatility Processes of Spot Exchange Rates (Revised in August 2007 and January 2009; subseq," CARF F-Series CARF-F-092, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    9. Akihiko Takahashi & Masashi Toda, 2012. "Note on an Extension of an Asymptotic Expansion Scheme," CARF F-Series CARF-F-286, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Dec 2012.
    10. Akihiko Takahashi & Yukihiro Tsuzuki, 2013. "A New Improvement Scheme for Approximation Methods of Probability Density Functions," CARF F-Series CARF-F-305, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    11. 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.
    12. Akihiko Takahashi & Kohta Takehara, 2007. "An Asymptotic Expansion Approach to Currency Options with a Market Model of Interest Rates under Stochastic Volatility Processes of Spot Exchange Rates," CIRJE F-Series CIRJE-F-474, CIRJE, Faculty of Economics, University of Tokyo.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Akihiko Takahashi & Kohta Takehara, 2007. "Fourier Transform Method with an Asymptotic Expansion Approach: an Application to Currency Options ( Revised in December 2008; subsequently published in "International Journal of Theoretical and ," CARF F-Series CARF-F-097, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    2. 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.
    3. Akihiko Takahashi & Kohta Takehara, 2009. "Asymptotic Expansion Approaches in Finance: Applications to Currency Options," CIRJE F-Series CIRJE-F-654, CIRJE, Faculty of Economics, University of Tokyo.
    4. Akihiko Takahashi & Kohta Takehara, 2007. "An Asymptotic Expansion Approach to Currency Options with a Market Model of Interest Rates under Stochastic Volatility Processes of Spot Exchange Rates (Revised in August 2007 and January 2009; subseq," CARF F-Series CARF-F-092, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    5. Akihiko Takahashi & Kohta Takehara, 2007. "An Asymptotic Expansion Approach to Currency Options with a Market Model of Interest Rates under Stochastic Volatility Processes of Spot Exchange Rates," CIRJE F-Series CIRJE-F-474, CIRJE, Faculty of Economics, University of Tokyo.
    6. 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.
    7. Akihiko Takahashi & Kohta Takehara, 2010. "A Hybrid Asymptotic Expansion Scheme: an Application to Long-term Currency Options," CIRJE F-Series CIRJE-F-734, CIRJE, Faculty of Economics, University of Tokyo.
    8. Akihiko Takahashi & Kohta Takehara, 2007. "An Asymptotic Expansion Approach to Currency Options with a Market Model of Interest Rates under Stochastic Volatility Processes of Spot Exchange Rates," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 14(1), pages 69-121, March.
    9. 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.
    10. Akihiko Takahashi, 2015. "Asymptotic Expansion Approach in Finance," CARF F-Series CARF-F-356, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Aug 2015.
    11. 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.
    12. Anatoliy Swishchuk & Maksym Tertychnyi & Robert Elliott, 2014. "Pricing Currency Derivatives with Markov-modulated Levy Dynamics," Papers 1402.1953, arXiv.org.
    13. Masahiro Nishiba, 2013. "Pricing Exotic Options and American Options: A Multidimensional Asymptotic Expansion Approach," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 20(2), pages 147-182, May.
    14. Akihiko Takahashi & Kohta Takehara & Masashi Toda, 2012. "A General Computation Scheme for a High-Order Asymptotic Expansion Method," CARF F-Series CARF-F-272, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    15. Akihiko Takahashi & Kohta Takehara & Masashi Toda, 2011. "A General Computation Scheme for a High-Order Asymptotic Expansion Method," CARF F-Series CARF-F-242, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Jul 2011.
    16. Akihiko Takahashi & Toshihiro Yamada, 2013. "On Error Estimates for Asymptotic Expansions with Malliavin Weights -- Application to Stochastic Volatility Model --," CIRJE F-Series CIRJE-F-897, CIRJE, Faculty of Economics, University of Tokyo.
    17. Akihiko Takahashi & Toshihiro Yamada, 2014. "On Error Estimates for Asymptotic Expansions with Malliavin Weights -Application to Stochastic Volatility Model- (Revised version of CARF-F-324; Forthcoming in "Mathematics of Operations Research," CARF F-Series CARF-F-347, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Sep 2014.
    18. Akihiko Takahashi & Kohta Takehara & Masashi Toda, 2011. "A General Computation Scheme for a High-Order Asymptotic Expansion Method," CIRJE F-Series CIRJE-F-787, CIRJE, Faculty of Economics, University of Tokyo.
    19. Kohta Takehara & Masashi Toda & Akihiko Takahashi, 2010. "Application Of A High-Order Asymptotic Expansion Scheme To Long-Term Currency Options," CARF F-Series CARF-F-225, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    20. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:tky:fseres:2008cf538. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: CIRJE administrative office (email available below). General contact details of provider: https://edirc.repec.org/data/ritokjp.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.