Time evolution of stochastic processes with correlations in the variance: stability in power-law tails of distributions

We model the time series of the S&P500 index by a combined process, the AR+GARCH process, where AR denotes the autoregressive process which we use to account for the short-range correlations in the index changes and GARCH denotes the generalized autoregressive conditional heteroskedastic process which takes into account the long-range correlations in the variance. We study the AR+GARCH process with an initial distribution of truncated L9 evy form. We :nd that this process generates a new probability distribution with a crossover from a L9 evy stable power law to a power law with an exponent outside the L9 evy range, beyond the truncation cuto;. We analyze the sum of n variables of the AR+GARCH process, and :nd that due to the correlations the AR+GARCH process generates a probability distribution which exhibits stable behavior in the tails for a broad range of values n—a feature which is observed in the probability distribution of the S&P500 index. We :nd that this power-law stability depends on the characteristic scale in the correlations. We also :nd that inclusion of short-range correlations through the AR process is needed to obtain convergence to a limiting Gaussian distribution for large n as observed in the data. c © 2001 Elsevier Science B.V. All rights reserved.