Quantitative Finance, Volume 0, Issue 0, Page 1-13, Ahead of Print.

Lisa Borland

Abstract

We study properties of the cross-sectional distribution of returns. A significant anti-correlation between dispersion and cross-sectional kurtosis is found such that dispersion is high but kurtosis is low in panic times, and the opposite in normal times. The co-movement of stock returns also increases in panic times. We define a simple statistic s, the normalized sum of signs of returns on a given day, to capture the degree of correlation in the system. s can be seen as the order parameter of the system because if s?=?0 there is no correlation (a disordered state), whereas for s? ??0 there is correlation among stocks (an ordered state). We make an analogy to non-equilibrium phase transitions and hypothesize that financial markets undergo self-organization when the external volatility perception rises above some critical value. Indeed, the distribution of s is unimodal in normal times, shifting to bimodal in times of panic. This is consistent with a second-order phase transition. Simulations of a joint stochastic process for stocks use a multi-timescale process in the temporal direction and an equation for the order parameter s for the dynamics of the cross-sectional correlation. Numerical results show good qualitative agreement with the stylized facts of real data, in both normal and panic times.