On the connection between financial processes with stochastic volatility and nonextensive statistical mechanics

TL;DR

This paper links GARCH models with nonextensive statistical mechanics by deriving analytical relations between model parameters and entropic indices, providing new insights into the distribution of returns and volatility dependence.

Contribution

It introduces a novel analytical connection between GARCH parameters and nonextensive entropy indices, enhancing understanding of financial return distributions.

Findings

Derived a relation between GARCH parameters and q-Gaussian distribution.

Provided an analytical approximation for the stationary volatility distribution.

Established a link between dependence measures and entropic indices.

Abstract

The $GARCH$ algorithm is the most renowned generalisation of Engle's original proposal for modelising {\it returns}, the $ARCH$ process. Both cases are characterised by presenting a time dependent and correlated variance or {\it volatility}. Besides a memory parameter, $b$, (present in $ARCH$) and an independent and identically distributed noise, $\omega $, $GARCH$ involves another parameter, $c$, such that, for $c=0$, the standard $ARCH$ process is reproduced. In this manuscript we use a generalised noise following a distribution characterised by an index $q_{n}$, such that $q_{n}=1$ recovers the Gaussian distribution. Matching low statistical moments of $GARCH$ distribution for returns with a $q$-Gaussian distribution obtained through maximising the entropy $S_{q}=\frac{1-\sum_{i}p_{i}^{q}}{q-1}$, basis of nonextensive statistical mechanics, we obtain a sole analytical connection between $q$ and $(b,c,q_{n}) $ which turns out to be remarkably good when compared with computational simulations. With this result we also derive an analytical approximation for the stationary distribution for the (squared) volatility. Using a generalised Kullback-Leibler relative entropy form based on $S_{q}$, we also analyse the degree of dependence between successive returns, $z_{t}$ and $z_{t+1}$, of GARCH(1,1) processes. This degree of dependence is quantified by an entropic index, $q^{op}$. Our analysis points the existence of a unique relation between the three entropic indexes $q^{op}$, $q$ and $q_{n}$ of the problem, independent of the value of $(b,c)$.