Bridging a paradigmatic financial model and nonextensive entropy

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, NM 87501, USA

Corresponding authors: sdqueiro@cbpf.br tsallis@cbpf.br

Received: 2 September 2004
Accepted: 17 January 2005

Abstract

Engle's ARCH algorithm is a generator of stochastic time series for financial returns (and similar quantities) characterised by a time-dependent variance. It involves a memory parameter b ( corresponds to no memory), and a noise currently chosen to be Gaussian. We assume here a generalised noise, namely q_n-Gaussian, characterised by an index ( recovers the Gaussian case, and corresponds to tailed distributions). Supported by the recently introduced concept of superstatistics, we match the second and fourth moments of ARCH return distribution with those associated with the q-Gaussian distribution obtained through optimisation of the entropy , basis of nonextensive statistical mechanics. The outcome is an analytic distribution for returns, where a unique corresponds to each pair (q=q_n if ). This distribution is compared with numerical results and appears to be remarkably precise. This system constitutes a simple, low-dimensional, dynamical mechanism which accommodates well within the current nonextensive framework.