Minding impacting events in a model of stochastic variance

TL;DR

This paper generalizes the ARCH process by incorporating impact events that influence volatility regimes, resulting in a model that captures fat tails and persistent variance in complex systems.

Contribution

It introduces a novel two-regime ARCH-based model that accounts for impact events affecting volatility, enhancing the modeling of long-term correlations and non-Gaussian distributions.

Findings

Model produces fat-tailed probability density functions.

Achieves high persistence in variance with Hurst exponent > 0.8.

Effectively captures impact events influencing volatility regimes.

Abstract

We introduce a generalisation of the well-known ARCH process, widely used for generating uncorrelated stochastic time series with long-term non-Gaussian distributions and long-lasting correlations in the (instantaneous) standard deviation exhibiting a clustering profile. Specifically, inspired by the fact that in a variety of systems impacting events are hardly forgot, we split the process into two different regimes: a first one for regular periods where the average volatility of the fluctuations within a certain period of time is below a certain threshold and another one when the local standard deviation outnumbers it. In the former situation we use standard rules for heteroscedastic processes whereas in the latter case the system starts recalling past values that surpassed the threshold. Our results show that for appropriate parameter values the model is able to provide fat tailed probability density functions and strong persistence of the instantaneous variance characterised by large values of the Hurst exponent is greater than 0.8, which are ubiquitous features in complex systems.