Autoregressive processes with anomalous scaling behavior: applications to high-frequency variations of a stock market index.

We employ autoregressive conditional heteroskedasticity processes to model the probability distribution function (PDF) of high-frequency relative variations of the Standard & Poors 500 market index data, obtained at the time horizon of 1 min. The model reproduces quantitatively the shape of the PDF, characterized by a Lévy-type power-law decay around its center, followed by a crossover to a faster decay at the tails. Furthermore, it is able to reproduce accurately the anomalous decay of the central part of the PDF at larger time horizons and, by the introduction of a short-range memory, also the crossover behavior of the corresponding standard deviations and the time scale of the exponentially decaying autocorrelation function of returns displayed by the empirical data.