Characteristic times in stock market indices

In this study we analyze the Standard and Poorʹs 500 index data of the New York Stock Exchange for more than 32 years. Using a simple random walk model we demonstrate that the proper variable to look at is the logarithmic return. In the statistical analysis we have done fittings to the Lévy distribution using either the index data as such or pre-processing it with ARCH, GARCH or IGARCH methods, which tend to remove the time-dependent variance. For short times the truncated Lévy distribution is found to fit the data quite well. Since this is not a stable distribution, the scaling behavior observed for short times should brake down for longer times. We demonstrate that the characteristic time where this cross-over starts is of the order of one day.