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Pricing Options under Stochastic Interest Rates: A New Approach

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  • Yong-Jin Kim
  • Naoto Kunitomo

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

Suggested Citation

  • Yong-Jin Kim & Naoto Kunitomo, 1999. "Pricing Options under Stochastic Interest Rates: A New Approach," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 6(1), pages 49-70, January.
    • Handle: RePEc:kap:apfinm:v:6:y:1999:i:1:p:49-70
    DOI: 10.1023/A:1010006525552
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    Cited by:

    1. Cocozza, Rosa & De Simone, Antonio, 2011. "One numerical procedure for two risk factors modeling," MPRA Paper 30859, University Library of Munich, Germany.
    2. Naoto Kunitomo & Yong-Jin Kim, 2000. "Effects of Stochastic Interest Rates and Volatility on Contingent Claims," CIRJE F-Series CIRJE-F-67, CIRJE, Faculty of Economics, University of Tokyo.
    3. 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.
    4. Nikolai Dokuchaev, 2011. "On martingale measures and pricing for continuous bond-stock market with stochastic bond," Papers 1108.0719, arXiv.org, revised Sep 2014.
    5. 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.
    6. Wang, Xiaoyu & Xie, Dejun & Jiang, Jingjing & Wu, Xiaoxia & He, Jia, 2017. "Value-at-Risk estimation with stochastic interest rate models for option-bond portfolios," Finance Research Letters, Elsevier, vol. 21(C), pages 10-20.
    7. 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.
    8. Benjamin Cheng & Christina Nikitopoulos-Sklibosios & Erik Schlogl, 2016. "Hedging Futures Options with Stochastic Interest Rates," Research Paper Series 375, Quantitative Finance Research Centre, University of Technology, Sydney.
    9. Yoshida, Nakahiro, 2003. "Conditional expansions and their applications," Stochastic Processes and their Applications, Elsevier, vol. 107(1), pages 53-81, September.

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