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Asymptotic Behavior of the Stock Price Distribution Density and Implied Volatility in Stochastic Volatility Models

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  • A. Gulisashvili
  • E. M. Stein
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    Abstract

    We study the asymptotic behavior of distribution densities arising in stock price models with stochastic volatility. The main objects of our interest in the present paper are the density of time averages of the squared volatility process and the density of the stock price process in the Stein-Stein and the Heston model. We find explicit formulas for leading terms in asymptotic expansions of these densities and give error estimates. As an application of our results, sharp asymptotic formulas for the implied volatility in the Stein-Stein and the Heston model are obtained.

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    File URL: http://arxiv.org/pdf/0906.0392
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 0906.0392.

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    Date of creation: Jun 2009
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    Handle: RePEc:arx:papers:0906.0392

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    1. Adrian A. Dragulescu & Victor M. Yakovenko, 2002. "Probability distribution of returns in the Heston model with stochastic volatility," Papers cond-mat/0203046, arXiv.org, revised Nov 2002.
    2. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," Review of Financial Studies, Society for Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-52.
    3. Adrian Dragulescu & Victor Yakovenko, 2002. "Probability distribution of returns in the Heston model with stochastic volatility," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 2(6), pages 443-453.
    4. A. Dragulescu & V. M. Yakovenko, 2002. "Probability distribution of returns in the Heston model with stochastic volatility," Computing in Economics and Finance 2002, Society for Computational Economics 127, Society for Computational Economics.
    5. Cox, John C & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1985. "A Theory of the Term Structure of Interest Rates," Econometrica, Econometric Society, Econometric Society, vol. 53(2), pages 385-407, March.
    6. Nielsen, Lars Tyge, 1999. "Pricing and Hedging of Derivative Securities," OUP Catalogue, Oxford University Press, Oxford University Press, number 9780198776192, October.
    7. Leif Andersen & Vladimir Piterbarg, 2007. "Moment explosions in stochastic volatility models," Finance and Stochastics, Springer, Springer, vol. 11(1), pages 29-50, January.
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    Cited by:
    1. P. Friz & S. Gerhold & A. Gulisashvili & S. Sturm, 2010. "On refined volatility smile expansion in the Heston model," Papers 1001.3003, arXiv.org, revised Nov 2010.
    2. Archil Gulisashvili & Josep Vives, 2010. "Two-sided estimates for stock price distribution densities in jump-diffusion models," Papers 1005.1917, arXiv.org.

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