  The ARCH process (R. F. Engle, 1982) constitutes a paradigmatic generator of
stochastic time series with time-dependent variance like it appears on a wide
broad 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 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 SP500 financial
index.
