Bridging a paradigmatic financial model and nonextensive entropy

S. M. Duarte Queirós; C. Tsallis

doi:doi:10.1209/epl/i2004-10436-6

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. Tsallis^{1, 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

sdqueiro@cbpf.br
tsallis@cbpf.br

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