Universality in DAX index returns fluctuations

TL;DR

This paper demonstrates that the fluctuations of DAX index returns, when properly rescaled, follow a universal probability distribution similar to the Bramwell-Holdsworth-Pinton distribution, indicating universality in stock market behavior.

Contribution

The study analytically derives the distribution of DAX index returns and identifies optimal rescaling parameters that reveal a universal distribution across different markets.

Findings

Rescaled positive and negative DAX returns follow the BHP distribution.

Optimal alpha values are approximately 0.50 and 0.48.

Evidence of universality in stock market fluctuations.

Abstract

In terms of the stock exchange returns, we compute the analytic expression of the probability distributions F{DAX,+} and F{DAX,-} of the normalized positive and negative DAX (Germany) index daily returns r(t). Furthermore, we define the alpha re-scaled DAX 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. 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 Bramwell-Holdsworth-Pinton (BHP) probability density function. The optimal parameters that we found are alpha+=0.50 and alpha-=0.48. Since the BHP probability density function appears in several other dissimilar phenomena, our results reveal universality in the stock exchange markets.