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Probability distribution of returns in the Heston model with stochastic volatility

Listed author(s):
  • A. Dragulescu
  • V. M. Yakovenko

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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Paper provided by Society for Computational Economics in its series Computing in Economics and Finance 2002 with number 127.

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Date of creation: 01 Jul 2002
Handle: RePEc:sce:scecf2:127
Contact details of provider: Web page: http://www.cepremap.cnrs.fr/sce2002.html/

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