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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)

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.

Suggested Citation

  • Jérôme Detemple & Weidong Tian, 2002. "The Valuation of American Options for a Class of Diffusion Processes," Management Science, INFORMS, vol. 48(7), pages 917-937, July.
  • Handle: RePEc:inm:ormnsc:v:48:y:2002:i:7:p:917-937
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    File URL: http://dx.doi.org/10.1287/mnsc.48.7.917.2815
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. 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.
    2. 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.
    3. repec:eee:dyncon:v:80:y:2017:i:c:p:75-100 is not listed on IDEAS
    4. Antonio Cosma & Stefano Galluccio & Paola Pederzoli & O. Scaillet, "undated". "Valuing American Options Using Fast Recursive Projections," Swiss Finance Institute Research Paper Series 12-26, Swiss Finance Institute.
    5. Jérôme Detemple, 2014. "Optimal Exercise for Derivative Securities," Annual Review of Financial Economics, Annual Reviews, vol. 6(1), pages 459-487, December.
    6. 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.
    7. Vicky Henderson & Kamil Klad'ivko & Michael Monoyios, 2017. "Executive stock option exercise with full and partial information on a drift change point," Papers 1709.10141, arXiv.org.
    8. Jonathan Ziveyi, 2011. "The Evaluation of Early Exercise Exotic Options," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 12.
    9. 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.
    10. Antonio Cosma & Stefano Galluccio & Paola Pederzoli & Olivier Scaillet, 2016. "Early exercise decision in American options with dividends, stochastic volatility and jumps," Papers 1612.03031, arXiv.org.
    11. Jing Zhao & Hoi Ying Wong, 2012. "A closed-form solution to American options under general diffusion processes," Quantitative Finance, Taylor & Francis Journals, vol. 12(5), pages 725-737, July.
    12. repec:spr:fininn:v:2:y:2016:i:1:d:10.1186_s40854-016-0042-9 is not listed on IDEAS
    13. Minqiang Li, 2010. "A quasi-analytical interpolation method for pricing American options under general multi-dimensional diffusion processes," Review of Derivatives Research, Springer, vol. 13(2), pages 177-217, July.
    14. 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.
    15. 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.
    16. 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.
    17. 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.
    18. 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.
    19. Dulluri, Sandeep & Raghavan, N.R. Srinivasa, 2008. "Collaboration in tool development and capacity investments in high technology manufacturing networks," European Journal of Operational Research, Elsevier, vol. 187(3), pages 962-977, June.
    20. Mark Broadie & Jerome B. Detemple, 2004. "ANNIVERSARY ARTICLE: Option Pricing: Valuation Models and Applications," Management Science, INFORMS, vol. 50(9), pages 1145-1177, September.

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