Bridging a paradigmatic financial model and nonextensive entropy

S. M. Duarte Queirós; C. Tsallis

doi:10.1209/epl/i2004-10436-6

All issues

Volume 69 / No 6 (March 2005)

Europhys. Lett., 69 6 (2005) 893-899

Abstract

Issue		Europhys. Lett. Volume 69, Number 6, March 2005


Page(s)		893 - 899
Section		General
DOI		https://doi.org/10.1209/epl/i2004-10436-6
Published online		18 February 2005

Europhys. Lett., 69 (6), pp. 893-899 (2005)

Bridging a paradigmatic financial model and nonextensive entropy

S. M. Duarte Queirós¹ and C. Tsallis¹^,2

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 ( $b=0$ 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 $q_{n}\in{\Re}$ ( $q_{n}=1$ recovers the Gaussian case, and $q_n>1$ 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 $S_{q}=\frac{1-\sum_{i}{p_i}^q}{q-1}$ , basis of nonextensive statistical mechanics. The outcome is an analytic distribution for returns, where a unique $q\ge q_n$ corresponds to each pair $(b,q_n)$ (q=q_n if $b=0$ ). 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.

PACS: 05.90.+m – Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems / 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion / 89.65.Gh – Economics; econophysics, financial markets, business and management

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