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Valuation of power options under Heston's stochastic volatility model

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  • Kim, Jerim
  • Kim, Bara
  • Moon, Kyoung-Sook
  • Wee, In-Suk
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    Abstract

    We derive semi-analytic solutions for power option prices under the Heston model; specifically, the pricing formula is shown to be valid whenever the power of the underlying asset price has a finite moment. Unlike the majority of stochastic volatility models, there remains a significant problem to check the existence of moments of assets prices of order higher than one. Fortunately, the moment explosion property under the Heston model is examined systematically in Andersen and Piterbarg (2000). Incorporating with their results, we present explicit formulas for moment generating function of log price and for power option prices under the circumstances when the corresponding moments are finite. In case that the corresponding moment explodes, we provide two numerical methods to derive prices of power put and capped power call options. In spite of a simple idea, numerical examples show that the approximations are extremely accurate and efficient.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Economic Dynamics and Control.

    Volume (Year): 36 (2012)
    Issue (Month): 11 ()
    Pages: 1796-1813

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    Handle: RePEc:eee:dyncon:v:36:y:2012:i:11:p:1796-1813

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    Web page: http://www.elsevier.com/locate/jedc

    Related research

    Keywords: Power option; Stochastic volatility; Heston model; Change of numeraire; Fourier transform;

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    1. Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    2. Richard T. Baillie & Claudio Morana, 2007. "Modeling Long Memory and Structural Breaks in Conditional Variances: an Adaptive FIGARCH Approach," ICER Working Papers - Applied Mathematics Series 11-2007, ICER - International Centre for Economic Research.
    3. Suh, Sangwon & Zapatero, Fernando, 2008. "A class of quadratic options for exchange rate stabilization," Journal of Economic Dynamics and Control, Elsevier, vol. 32(11), pages 3478-3501, November.
    4. 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.
    5. Scott, Louis O., 1987. "Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(04), pages 419-438, December.
    6. Angelo Melino & Stuart M. Turnbull, 1991. "The Pricing of Foreign Currency Options," Canadian Journal of Economics, Canadian Economics Association, vol. 24(2), pages 251-81, May.
    7. Holger Kraft, 2005. "Optimal portfolios and Heston's stochastic volatility model: an explicit solution for power utility," Quantitative Finance, Taylor & Francis Journals, vol. 5(3), pages 303-313.
    8. Charles Quanwei Cao & Gurdip S. Bakshi & Zhiwu Chen, 1997. "Empirical Performance of Alternative Option Pricing Models," Yale School of Management Working Papers ysm54, Yale School of Management.
    9. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-52.
    10. Melino, Angelo & Turnbull, Stuart M., 1990. "Pricing foreign currency options with stochastic volatility," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 239-265.
    11. Charles Quanwei Cao & Gurdip S. Bakshi & Zhiwu Chen, 1997. "Empirical Performance of Alternative Option Pricing Models," Yale School of Management Working Papers ysm65, Yale School of Management.
    12. Clark, Todd E. & Davig, Troy, 2011. "Decomposing the declining volatility of long-term inflation expectations," Journal of Economic Dynamics and Control, Elsevier, vol. 35(7), pages 981-999, July.
    13. Bakshi, Gurdip & Cao, Charles & Chen, Zhiwu, 1997. " Empirical Performance of Alternative Option Pricing Models," Journal of Finance, American Finance Association, vol. 52(5), pages 2003-49, December.
    14. Wiggins, James B., 1987. "Option values under stochastic volatility: Theory and empirical estimates," Journal of Financial Economics, Elsevier, vol. 19(2), pages 351-372, December.
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