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

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 (, , and exp₁[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, q_m. The assessment of its validity and usefulness is made by mimicking daily fluctuations of the SP500 financial index.