Emergent invariance and scaling properties in the collective return dynamics of a stock market

A key metric to determine the performance of a stock in a market is its return over different investment horizons ( τ ). Several works have observed heavy-tailed behavior in the distributions of returns in different markets, which are observable indicators of underlying complex dynamics. Such prior works study return distributions that are marginalized across the individual stocks in the market, and do not track statistics about the joint distributions of returns conditioned on different stocks, which would be useful for optimizing inter-stock asset allocation strategies. As a step towards this goal, we study emergent phenomena in the distributions of returns as captured by their pairwise correlations. In particular, we consider the pairwise (between stocks i , j ) partial correlations of returns with respect to the market mode, c i , j ( τ ), (thus, correcting for the baseline return behavior of the market), over different time horizons ( τ ), and discover two novel emergent phenomena: (i) the standardized distributions of the c i , j ( τ )’s are observed to be invariant of τ ranging from from 1000min (2.5 days) to 30000min (2.5 months); (ii) the scaling of the standard deviation of c i , j ( τ )’s with τ admits good fits to simple model classes such as a power-law τ −λ or stretched exponential function e - τ β (λ, β > 0). Moreover, the parameters governing these fits provide a summary view of market health: for instance, in years marked by unprecedented financial crises—for example 2008 and 2020—values of λ (scaling exponent) are substantially lower. Finally, we demonstrate that the observed emergent behavior cannot be adequately supported by existing generative frameworks such as single- and multi-factor models. We introduce a promising agent-based Vicsek model that closes this gap.