Eur. Phys. J. B 48, 139-148 (2005)
https://doi.org/10.1140/epjb/e2005-00366-1

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

¹ Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro-RJ, Brazil
² Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA

Corresponding authors: ^a sdqueiro@cbpf.br - ^b tsallis@santafe.edu

Received: 5 February 2005
Revised: 12 July 2005
Published online: 9 December 2005

Abstract

The GARCH algorithm is the most renowned generalisation of Engle's original proposal for modelising returns, the ARCH process. Both cases are characterised by presenting a time dependent and correlated variance or volatility. Besides a memory parameter, b, (present in ARCH) and an independent and identically distributed noise, , 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 recovers the Gaussian distribution. Matching low statistical moments of GARCH distribution for returns with a q-Gaussian distribution obtained through maximising the entropy , basis of nonextensive statistical mechanics, we obtain a sole analytical connection between q and 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 , of GARCH(1,1) processes. This degree of dependence is quantified by an entropic index, . Our analysis points the existence of a unique relation between the three entropic indexes , q and q_n of the problem, independent of the value of (b,c).