Abstract.
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 qn, such that qn=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 Sq, we also analyse the degree of dependence between successive returns, zt and zt+1, of GARCH(1,1) processes. This degree of dependence is quantified by an entropic index, qop. Our analysis points the existence of a unique relation between the three entropic indexes qop, q and qn of the problem, independent of the value of (b,c).
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Queirós, S., Tsallis, C. On the connection between financial processes with stochastic volatility and nonextensive statistical mechanics. Eur. Phys. J. B 48, 139–148 (2005). https://doi.org/10.1140/epjb/e2005-00366-1
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DOI: https://doi.org/10.1140/epjb/e2005-00366-1