On a generalised model for time-dependent variance with long-term memory

TL;DR

This paper introduces a generalized model for time-dependent variance that incorporates long-term memory effects, improving upon the ARCH process by capturing persistent volatility and power-law decay in financial time series.

Contribution

The paper proposes a new model that extends ARCH by including a q-exponential correlation structure, effectively modeling long-term memory with only one additional parameter.

Findings

Successfully mimics daily fluctuations of SP500 index

Captures high Hurst exponent and power-law decay in autocorrelations

Provides a simple extension to improve volatility modeling

Abstract

The ARCH process (R. F. Engle, 1982) constitutes a paradigmatic generator of stochastic time series with time-dependent variance like it appears on a wide broad of systems besides economics in which ARCH was born. Although the ARCH process captures the so-called "volatility clustering" and the asymptotic power-law probability density distribution of the random variable, it is not capable to reproduce further statistical properties of many of these time series such as: the strong persistence of the instantaneous variance characterised by large values of the Hurst exponent (H > 0.8), and asymptotic power-law decay of the absolute values self-correlation function. By means of considering an effective return obtained from a correlation of past returns that has a q-exponential form we are able to fix the limitations of the original model. Moreover, this improvement can be obtained through the correct choice of a sole additional parameter, $q_{m}$. The assessment of its validity and usefulness is made by mimicking daily fluctuations of SP500 financial index.