| Issue |
EPL
Volume 80, Number 3, November 2007
|
|
|---|---|---|
| Article Number | 30005 | |
| Number of page(s) | 6 | |
| Section | General | |
| DOI | https://doi.org/10.1209/0295-5075/80/30005 | |
| Published online | 09 October 2007 | |
On a generalised model for time-dependent variance with long-term memory
Centro Brasileiro de Pesquisas Físicas - Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro-RJ, Brazil
Corresponding author: sdqueiro@cbpf.br
Received:
23
May
2007
Accepted:
10
September
2007
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 (
), 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}} $](/articles/epl/abs/2007/21/epl10536/epl10536_tex_eq2.png)
, 
, and ![${\rm exp}_{1}[x]=e^{x}$](/articles/epl/abs/2007/21/epl10536/epl10536_tex_eq4.png)
) 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 
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
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