Exponential distribution of financial returns at mesoscopic time lags: a new stylized fact

TL;DR

This paper reveals that stock return distributions at mesoscopic time lags follow an exponential law, transitioning to Gaussian at longer times, with the Heston model accurately describing this evolution.

Contribution

It identifies a new stylized fact that the bulk of stock returns at mesoscopic times follow an exponential distribution, bridging microscopic power-law tails and macroscopic Gaussian behavior.

Findings

Returns follow exponential distribution at mesoscopic times

Variance of returns increases proportionally with time lag

Exponential-to-Gaussian crossover is modeled by the Heston model

Abstract

We study the probability distribution of stock returns at mesoscopic time lags (return horizons) ranging from about an hour to about a month. While at shorter microscopic time lags the distribution has power-law tails, for mesoscopic times the bulk of the distribution (more than 99% of the probability) follows an exponential law. The slope of the exponential function is determined by the variance of returns, which increases proportionally to the time lag. At longer times, the exponential law continuously evolves into Gaussian distribution. The exponential-to-Gaussian crossover is well described by the analytical solution of the Heston model with stochastic volatility.