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

Abstract

The $GARCH$ algorithm is the most renowned generalisation of Engle's original proposal for modelising {\it returns}, the $ARCH$ process. Both cases are characterised by presenting a time dependent and correlated variance or {\it volatility}. Besides a memory parameter, $b$, (present in $ARCH$) and an independent and identically distributed noise, $\omega$, $GARCH$ 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 $(b,c,q_n)$ 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 $(b,c)$.

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