Universality in the stock exchange market

Abstract

We consider the alpha re-scaled Standard & Poor's 100 (SP100) daily index positive returns r(t)(alpha) and negative returns (-r(t))(alpha) that we call, after normalization, the alpha positive fluctuations and alpha negative fluctuations, respectively. We use the Kolmogorov-Smirnov statistical test as a method to find the values of alpha that optimize the data collapse of the histogram of the alpha fluctuations with the truncated Bramwell-Holdsworth-Pinton (BHP) probability density function (pdf) and the truncated generalized log-normal pdf f(LN) that best approximates the truncated BHP pdf. The optimal parameters we found are alpha(+)(BHP) = 0.52, alpha(-)(BHP) = 0.48, alpha(+)(LN) = 0.52 and alpha(-)(LN) = 0.50. Using the optimal alpha's, we compute analytical approximations of the probability distributions of the normalized positive and negative SP100 index daily returns r(t). Since the BHP pdf appears in several other dissimilar phenomena, our result reveals a universal feature of the stock exchange markets.