From short to fat tails in financial markets: A unified description

TL;DR

This paper provides a unified framework to describe the transition from Gaussian to fat-tailed distributions in financial market returns across different time scales, using empirical data and stochastic modeling.

Contribution

It introduces a novel approach employing Kramers-Moyal coefficients to derive a unified evolution equation for return distributions, linking tail behavior to underlying dynamics.

Findings

Identifies a dynamical transition from Gaussian to fat-tailed statistics.

Derives relationships between tail exponents and Kramers-Moyal parameters.

Provides asymptotic solutions explaining the crossover in return distributions.

Abstract

In complex systems such as turbulent flows and financial markets, the dynamics in long and short time-lags, signaled by Gaussian and fat-tailed statistics, respectively, calls for a unified description. To address this issue we analyze a real dataset, namely, price fluctuations, in a wide range of temporal scales to embrace both regimes. By means of Kramers-Moyal (KM) coefficients evaluated from empirical time series, we obtain the evolution equation for the probability density function (PDF) of price returns. We also present consistent asymptotic solutions for the timescale dependent equation that emerges from the empirical analysis. From these solutions, new relationships connecting PDF characteristics, such as tail exponents, to parameters of KM coefficients arise. The results reveal a dynamical path that leads from Gaussian to fat-tailed statistics, furnishing insights on other complex systems where akin crossover is observed.