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We analyze daily log-returns data for a set of 1200 stocks, taken from US stock markets, over a period of 2481 trading days (January 1996-November 2005. We estimate the degree of non-stationarity in daily market volatility employing a polynomial fit, used as a detrending function. We find that the autocorrelation function of absolute detrended log-returns departs strongly from the corresponding original data autocorrelation function, while the observed leverage effect depends only weakly on trends. Such effect is shown to occur when both skewness and long-time memory are simultaneously present. A fractional derivative random walk model is discussed yielding a quantitative agreement with the empirical results.

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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.