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Probability distribution of returns in the Heston model with stochastic volatility

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  • Adrian Dragulescu
  • Victor Yakovenko
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    Abstract

    We study the Heston model, where the stock price dynamics is governed by a geometrical (multiplicative) Brownian motion with stochastic variance. We solve the corresponding Fokker-Planck equation exactly and, after integrating out the variance, find an analytic formula for the time-dependent probability distribution of stock price changes (returns). The formula is in excellent agreement with the Dow-Jones index for time lags from 1 to 250 trading days. For large returns, the distribution is exponential in log-returns with a time-dependent exponent, whereas for small returns it is Gaussian. For time lags longer than the relaxation time of variance, the probability distribution can be expressed in a scaling form using a Bessel function. The Dow-Jones data for 1982-2001 follow the scaling function for seven orders of magnitude.

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    Bibliographic Info

    Article provided by Taylor & Francis Journals in its journal Quantitative Finance.

    Volume (Year): 2 (2002)
    Issue (Month): 6 ()
    Pages: 443-453

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    Handle: RePEc:taf:quantf:v:2:y:2002:i:6:p:443-453

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    Cited by:
    1. Bernardo Spagnolo & Davide Valenti, 2008. "Volatility Effects on the Escape Time in Financial Market Models," Papers 0810.1625, arXiv.org.
    2. Nakamura, Tomomichi & Small, Michael, 2007. "Tests of the random walk hypothesis for financial data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 377(2), pages 599-615.
    3. Andreas Behr & Ulrich Pötter, 2009. "Alternatives to the normal model of stock returns: Gaussian mixture, generalised logF and generalised hyperbolic models," Annals of Finance, Springer, vol. 5(1), pages 49-68, January.
    4. López Martín, María del Mar & García, Catalina García & García Pérez, José, 2012. "Treatment of kurtosis in financial markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(5), pages 2032-2045.
    5. Buchbinder, G.L. & Chistilin, K.M., 2007. "Multiple time scales and the empirical models for stochastic volatility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 379(1), pages 168-178.
    6. Oguzhan Cepni & Ahmet Goncu & Mehmet Oguz Karahan & Tolga Umut Kuzubas, 2013. "Goodness-of-fit of the Heston, Variance-Gamma and Normal-Inverse Gaussian Models," Working Papers 2013/16, Bogazici University, Department of Economics.
    7. A. Monteiro & R. Tütüncü & L. Vicente, 2011. "Estimation of risk-neutral density surfaces," Computational Management Science, Springer, vol. 8(4), pages 387-414, November.
    8. repec:sfi:sfiwpa:500029 is not listed on IDEAS
    9. A. Gulisashvili & E. M. Stein, 2009. "Asymptotic Behavior of the Stock Price Distribution Density and Implied Volatility in Stochastic Volatility Models," Papers 0906.0392, arXiv.org.
    10. Anca Gheorghiu & Ion Spanulescu, 2012. "An Econophysics Model for the Migration Phenomena," Papers 1202.0996, arXiv.org.
    11. del Baño Rollin, Sebastian & Ferreiro-Castilla, Albert & Utzet, Frederic, 2010. "On the density of log-spot in the Heston volatility model," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 2037-2063, September.
    12. Agnieszka Janek & Tino Kluge & Rafal Weron & Uwe Wystup, 2010. "FX Smile in the Heston Model," HSC Research Reports HSC/10/02, Hugo Steinhaus Center, Wroclaw University of Technology.
    13. repec:sfi:sfiwpa:500059 is not listed on IDEAS
    14. Andrew Papanicolaou, 2014. "Stochastic Analysis Seminar on Filtering Theory," Papers 1406.1936, arXiv.org.
    15. Ballestra, Luca Vincenzo & Pacelli, Graziella & Zirilli, Francesco, 2007. "A numerical method to price exotic path-dependent options on an underlying described by the Heston stochastic volatility model," Journal of Banking & Finance, Elsevier, vol. 31(11), pages 3420-3437, November.
    16. Chiang, Thomas C. & Yu, Hai-Chin & Wu, Ming-Chya, 2009. "Statistical properties, dynamic conditional correlation and scaling analysis: Evidence from Dow Jones and Nasdaq high-frequency data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(8), pages 1555-1570.
    17. Giacomo Bormetti & Valentina Cazzola & Danilo Delpini, 2009. "Option pricing under Ornstein-Uhlenbeck stochastic volatility: a linear model," Papers 0905.1882, arXiv.org, revised May 2010.
    18. 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.
    19. Anca Gheorghiu & Ion Spanulescu, 2009. "Macrostate Parameter, an Econophysics Approach for the Risk Analysis of the Stock Exchange Market Transactions," Papers 0907.5600, arXiv.org.

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