On the connection between financial processes with stochastic volatility and nonextensive statistical mechanics

The GARCH algorithm is the most renowned generalisation of Engle's original proposal for modelising returns, the ARCH process. Both cases are characterised by presenting a time dependent and correlated variance or volatility. Besides a memory parameter, b, (present in ARCH) and an independent and identically distributed noise, ω, GARCH involves another parameter, c, such that, for c=0, the standard ARCH process is reproduced. In this manuscript we use a generalised noise following a distribution characterised by an index q n , such that q n =1 recovers the Gaussian distribution. Matching low statistical moments of GARCH distribution for returns with a q-Gaussian distribution obtained through maximising the entropy $S_{q}=\frac{1-\sum_{i}p_{i}^{q}}{q-1}$ , basis of nonextensive statistical mechanics, we obtain a sole analytical connection between q and $\left( b,c,q_{n}\right) $ which turns out to be remarkably good when compared with computational simulations. With this result we also derive an analytical approximation for the stationary distribution for the (squared) volatility. Using a generalised Kullback-Leibler relative entropy form based on S q , we also analyse the degree of dependence between successive returns, z t and z t+1 , of GARCH(1,1) processes. This degree of dependence is quantified by an entropic index, q op . Our analysis points the existence of a unique relation between the three entropic indexes q op , q and q n of the problem, independent of the value of (b,c). Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2005