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Probability distribution of returns in the Heston model with stochastic volatility

  • Adrian A. Dragulescu
  • Victor M. Yakovenko
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    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 the 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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    File URL: http://arxiv.org/pdf/cond-mat/0203046
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    Paper provided by arXiv.org in its series Papers with number cond-mat/0203046.

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    Date of creation: Mar 2002
    Date of revision: Nov 2002
    Publication status: Published in Quantitative Finance 2, 443 (2002)
    Handle: RePEc:arx:papers:cond-mat/0203046
    Contact details of provider: Web page: http://arxiv.org/

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