Probability distribution of returns for a model with stochastic volatility

  • 4 March 2002
Abstract
We study a model where stock price dynamics is governed by a geometrical (multiplicative) Brownian motion with stochastic variance. We solve the corresponding Fokker-Planck equation exactly, using either the method of characteristics or path integrals. After integrating out the variance, we 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 follow the scaling function for seven orders of magnitude.

This publication has 0 references indexed in Scilit: