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The Valuation of American Options for a Class of Diffusion Processes

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  • Jérôme Detemple

    ()
    (Boston University School of Management, 595 Commonwealth Avenue, Boston, Massachusetts 02215, and CIRANO, 2020 University Street, 25th Floor, Montreal, Quebec, Canada H3A 2A5, and Wholesale Group Quantitative Research, Reliant Resource, Louisiana #1111, Houston, Texas 77210-4567)

  • Weidong Tian

    ()
    (Boston University School of Management, 595 Commonwealth Avenue, Boston, Massachusetts 02215, and CIRANO, 2020 University Street, 25th Floor, Montreal, Quebec, Canada H3A 2A5, and Wholesale Group Quantitative Research, Reliant Resource, Louisiana #1111, Houston, Texas 77210-4567)

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    Abstract

    We present an integral equation approach for the valuation of American-style derivatives when the underlying asset price follows a general diffusion process and the interest rate is stochastic. Our contribution is fourfold. First, we show that the exercise region is determined by a single exercise boundary under very general conditions on the interest rate and the dividend yield. Second, based on this result, we derive a recursive integral equation for the exercise boundary and provide a parametric representation of the American option price. Third, we apply the results to models with stochastic volatility or stochastic interest rate, and to American bond options in one-factor models. For the cases studied, explicit parametric valuation formulas are obtained. Finally, we extend results on American capped options to general diffusion prices. Numerical schemes based on approximations of the optimal stopping time (such as approximations based on a lower bound, or on a combination of lower and upper bounds) are shown to be valid in this context.

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    File URL: http://dx.doi.org/10.1287/mnsc.48.7.917.2815
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    Bibliographic Info

    Article provided by INFORMS in its journal Management Science.

    Volume (Year): 48 (2002)
    Issue (Month): 7 (July)
    Pages: 917-937

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    Handle: RePEc:inm:ormnsc:v:48:y:2002:i:7:p:917-937

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    Related research

    Keywords: american options; valuation; optimal exercise; diffusion; stochastic interest rate; stochastic volatility; integral equation; capped options; bounds and approximations;

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    Cited by:
    1. Gao, Jianwei, 2009. "Optimal portfolios for DC pension plans under a CEV model," Insurance: Mathematics and Economics, Elsevier, vol. 44(3), pages 479-490, June.
    2. Nagae, Takeshi & Akamatsu, Takashi, 2008. "A generalized complementarity approach to solving real option problems," Journal of Economic Dynamics and Control, Elsevier, vol. 32(6), pages 1754-1779, June.
    3. Li, Minqiang, 2009. "A Quasi-analytical Interpolation Method for Pricing American Options under General Multi-dimensional Diffusion Processes," MPRA Paper 17348, University Library of Munich, Germany.
    4. Paola Zerilli, 2005. "Option pricing and spikes in volatility: theoretical and empirical analysis," Money Macro and Finance (MMF) Research Group Conference 2005 76, Money Macro and Finance Research Group.
    5. Zhao, Hui & Rong, Ximin, 2012. "Portfolio selection problem with multiple risky assets under the constant elasticity of variance model," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 179-190.
    6. Ruas, João Pedro & Dias, José Carlos & Vidal Nunes, João Pedro, 2013. "Pricing and static hedging of American-style options under the jump to default extended CEV model," Journal of Banking & Finance, Elsevier, vol. 37(11), pages 4059-4072.
    7. Gu, Mengdi & Yang, Yipeng & Li, Shoude & Zhang, Jingyi, 2010. "Constant elasticity of variance model for proportional reinsurance and investment strategies," Insurance: Mathematics and Economics, Elsevier, vol. 46(3), pages 580-587, June.
    8. Medvedev, Alexey & Scaillet, Olivier, 2010. "Pricing American options under stochastic volatility and stochastic interest rates," Journal of Financial Economics, Elsevier, vol. 98(1), pages 145-159, October.
    9. Gao, Jianwei, 2009. "Optimal investment strategy for annuity contracts under the constant elasticity of variance (CEV) model," Insurance: Mathematics and Economics, Elsevier, vol. 45(1), pages 9-18, August.
    10. João Nunes, 2011. "American options and callable bonds under stochastic interest rates and endogenous bankruptcy," Review of Derivatives Research, Springer, vol. 14(3), pages 283-332, October.

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