Bridging the ARCH model for finance and nonextensive entropy

Abstract

Engle's ARCH algorithm is a generator of stochastic time series for financial returns (and similar quantities) characterized by a time-dependent variance. It involves a memory parameter $b$ ($b=0$ corresponds to {\it no memory}), and the noise is currently chosen to be Gaussian. We assume here a generalized noise, namely $q_n$-Gaussian, characterized by an index $q_{n} \in {\cal R}$ ($q_{n}=1$ recovers the Gaussian case, and $q_n>1$ corresponds to tailed distributions). We then match the second and fourth momenta of the ARCH return distribution with those associated with the $q$-Gaussian distribution obtained through optimization of the entropy $S_{q}=\frac{% 1-\sum_{i} {p_i}^q}{q-1}$, basis of nonextensive statistical mechanics. The outcome is an {\it analytic} distribution for the returns, where an unique $q\ge q_n$ corresponds to each pair $(b,q_n)$ ($q=q_n$ 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.