Transition from lognormal to chi-square superstatistics for financial time series

TL;DR

This paper investigates the appropriate superstatistics models for financial time series, revealing a transition from lognormal to chi-square superstatistics as the time scale increases, and proposing a unified interpolating model.

Contribution

It identifies the time-scale-dependent transition between lognormal and chi-square superstatistics in financial data and introduces a general interpolating model for better data fitting.

Findings

Chi-square superstatistics fits daily data well.

Lognormal superstatistics describes minute-scale data better.

Correlation functions show exponential decay at large scales and long-range correlations at small scales.

Abstract

Share price returns on different time scales can be well modelled by a superstatistical dynamics. Here we provide an investigation which type of superstatistics is most suitable to properly describe share price dynamics on various time scales. It is shown that while chi-square superstatistics works well on a time scale of days, on a much smaller time scale of minutes the price changes are better described by lognormal superstatistics. The system dynamics thus exhibits a transition from lognormal to chi-square superstatistics as a function of time scale. We discuss a more general model interpolating between both statistics which fits the observed data very well. We also present results on correlation functions of the extracted superstatistical volatility parameter, which exhibits exponential decay for returns on large time scales, whereas for returns on small time scales there are long-range correlations and power-law decay.