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Pricing Options under Stochastic Interest Rates: A New Approach

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  • Yong-Jin Kim
  • Naoto Kunitomo
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    Abstract

    We will generalize the Black-Scholes option pricing formula by incorporating stochastic interest rates. Although the existing literature has obtained some formulae for stock options under stochastic interest rates, the closed-form solutions have been known only under the Gaussian (Merton type) interest rate processes. We will show that an explicit solution, which is an extended Black-Scholes formula under stochastic interest rates in certain asymptotic sense, can be obtained by extending the asymptotic expansion approach when the interest rate volatility is small. This method, called the small-disturbance asymptotics for Itô processes, has recently been developed by Kunitomo and Takahashi (1995, 1998) and Takahashi (1997). We found that the extended Black-Scholes formula is decomposed into the original Black-Scholes formula under the deterministic interest rates and the adjustment term driven by the volatility of interest rates. We will illustrate the numerical accuracy of our new formula by using the Cox–Ingersoll–Ross model for the interest rates. Copyright Kluwer Academic Publishers 1999

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    File URL: http://hdl.handle.net/10.1023/A:1010006525552
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    Bibliographic Info

    Article provided by Springer in its journal Asia-Pacific Financial Markets.

    Volume (Year): 6 (1999)
    Issue (Month): 1 (January)
    Pages: 49-70

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    Handle: RePEc:kap:apfinm:v:6:y:1999:i:1:p:49-70

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    Web page: http://springerlink.metapress.com/link.asp?id=102851

    Related research

    Keywords: asymptotic expansion approach; Black-Scholes economy; Cox–Ingersoll–Ross model; stochastic interest rates;

    References

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    1. Turnbull, Stuart M & Milne, Frank, 1991. "A Simple Approach to Interest-Rate Option Pricing," Review of Financial Studies, Society for Financial Studies, vol. 4(1), pages 87-120.
    2. Kaushik I. Amin & Robert A. Jarrow, 1992. "Pricing Options On Risky Assets In A Stochastic Interest Rate Economy," Mathematical Finance, Wiley Blackwell, vol. 2(4), pages 217-237.
    3. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
    4. Duffie, Darrell, 1988. "An extension of the Black-Scholes model of security valuation," Journal of Economic Theory, Elsevier, vol. 46(1), pages 194-204, October.
    5. 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.
    6. Cheng, Susan T., 1991. "On the feasibility of arbitrage-based option pricing when stochastic bond price processes are involved," Journal of Economic Theory, Elsevier, vol. 53(1), pages 185-198, February.
    7. Cox, John C & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1985. "A Theory of the Term Structure of Interest Rates," Econometrica, Econometric Society, vol. 53(2), pages 385-407, March.
    8. Heath, David & Jarrow, Robert & Morton, Andrew, 1992. "Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation," Econometrica, Econometric Society, vol. 60(1), pages 77-105, January.
    9. Amin, Kaushik I & Ng, Victor K, 1993. " Option Valuation with Systematic Stochastic Volatility," Journal of Finance, American Finance Association, vol. 48(3), pages 881-910, July.
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    Cited by:
    1. Yoshida, Nakahiro, 2003. "Conditional expansions and their applications," Stochastic Processes and their Applications, Elsevier, vol. 107(1), pages 53-81, September.
    2. Naoto Kunitomo & Akihiko Takahashi, 2003. "Applications of the Asymptotic Expansion Approach based on Malliavin-Watanabe Calculus in Financial Problems," CIRJE F-Series CIRJE-F-245, CIRJE, Faculty of Economics, University of Tokyo.
    3. Nikolai Dokuchaev, 2011. "On martingale measures and pricing for continuous bond-stock market with stochastic bond," Papers 1108.0719, arXiv.org, revised Apr 2014.
    4. Naoto Kunitomo & Yong-Jin Kim, 2001. "Effects of Stochastic Interest Rates and Volatility on Contingent Claims (Revised Version)," CIRJE F-Series CIRJE-F-129, CIRJE, Faculty of Economics, University of Tokyo.
    5. Cocozza, Rosa & De Simone, Antonio, 2011. "One numerical procedure for two risk factors modeling," MPRA Paper 30859, University Library of Munich, Germany.

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