Issue
EPL
Volume 80, Number 3, November 2007
Article Number 30005
Number of page(s) 6
Section General
DOI http://dx.doi.org/10.1209/0295-5075/80/30005
Published online 09 October 2007
EPL, 80 (2007) 30005
DOI: 10.1209/0295-5075/80/30005

On a generalised model for time-dependent variance with long-term memory

S. M. Duarte Queirós

Centro Brasileiro de Pesquisas Físicas - Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro-RJ, Brazil

sdqueiro@cbpf.br

received 23 May 2007; accepted in final form 10 September 2007; published November 2007
published online 9 October 2007

Abstract
The ARCH process (ENGLE R. F., Econometrica, 50 (1982) 987) constitutes a paradigmatic generator of stochastic time series with time-dependent variance like it appears on a wide variety of systems besides economics in which ARCH was born. Although the ARCH process captures the so-called "volatility clustering" and the asymptotic power law probability density distribution of the random variable, it is not capable to reproduce further statistical properties of many of these time series such as: the strong persistence of the instantaneous variance characterised by large values of the Hurst exponent (H > 0.8), and asymptotic power law decay of the absolute values self-correlation function. By means of considering an effective return obtained from a correlation of past returns that has a q-exponential form ( $\exp _{q}[ x] \equiv [ 1+(1-q) \,x] ^{\frac{1}{1-q}} $, $(q\in \Re) $, and ${\rm exp}_{1}[x]=e^{x}$) we are able to fix the limitations of the original model. Moreover, this improvement can be obtained through the correct choice of a sole additional parameter, qm. The assessment of its validity and usefulness is made by mimicking daily fluctuations of the SP500 financial index.

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.
89.90.+n - Other topics in areas of applied and interdisciplinary physics.

© Europhysics Letters Association 2007