Bridging a paradigmatic financial model and nonextensive entropy

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 qn-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=qn 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