Europhys. Lett.
Volume 69, Number 6, March 2005
Page(s) 893 - 899
Section General
Published online 18 February 2005
Europhys. Lett., 69 (6), pp. 893-899 (2005)
DOI: 10.1209/epl/i2004-10436-6

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

received 2 September 2004; accepted in final form 17 January 2005
published online 18 February 2005

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 qn-Gaussian, characterised by an index $q_{n}\in{\Re}$ (qn=1 recovers the Gaussian case, and qn>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,qn) (q=qn 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.

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.

© EDP Sciences 2005