Scaling Laws in Stock Markets. An Analysis of Prices and Volumes

The scaling behaviour of both log-price and volume is analyzed for three stock indexes. The traditional approach, mainly consisting of the evaluation of particular moments such as variance or higher absolute moments, is replaced by a new technique which allows the estimation of the self-similarity parameter on the whole empirical distribution designed by any time horizon. In this way, the method we propose attaches its own scaling parameter to any two given time lags, so defining a scaling surface whose properties give information about the nature of the analyzed process. We conclude that, for the log-price process, self-similarity is rejected with a frequency much larger than that assumed by the confidence interval and, when not rejected, the scaling parameter heavily changes with the considered pair of time horizons. Opposite evidence is provided for the volumes, characterized by (generally low) self-similarity parameters which are somewhat uniform with respect to the pairs of time horizons.