Factorial Moments in Complex Systems

TL;DR

This paper explores factorial moments as tools to analyze correlations in complex systems, comparing high energy physics and finance, and predicts gap probabilities in return sequences confirmed by data.

Contribution

It introduces a novel application of factorial moments to finance, linking particle physics methods to financial data analysis and providing empirical validation.

Findings

Correlations cause broadening of multiplicity distributions.

Factorial moments predict exponential suppression of gap probabilities.

Predictions are confirmed with financial data.

Abstract

Factorial moments are convenient tools in particle physics to characterize the multiplicity distributions when phase-space resolution ($\Delta$) becomes small. They include all correlations within the system of particles and represent integral characteristics of any correlation between these particles. In this letter, we show a direct comparison between high energy physics and quantitative finance results. Both for physics and finance, we illustrate that correlations between particles lead to a broadening of the multiplicity distribution and to dynamical fluctuations when the resolution becomes small enough. From the generating function of factorial moments, we make a prediction on the gap probability for sequences of returns of positive or negative signs. The gap is defined as the number of consecutive positive returns after a negative return, thus this is a gap in negative return. Inversely for a gap in positive return. Then, the gap probability is shown to be exponentially suppressed within the gap size. We confirm this prediction with data.