Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2002-03-03
Quantitative Finance 2, 443 (2002)
Physics
Condensed Matter
Statistical Mechanics
11 pages, 7 figures, RevTeX 4. V.2: substantial revision - new figures, sections, and references; V.3: accepted to Quantitativ
Scientific paper
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.
Dragulescu Adrian A.
Yakovenko Victor M.
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