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

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  • S. M.D. Queirós
  • C. Tsallis
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 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

Suggested Citation

  • S. M.D. Queirós & C. Tsallis, 2005. "On the connection between financial processes with stochastic volatility and nonextensive statistical mechanics," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 48(1), pages 139-148, November.
    • Handle: RePEc:spr:eurphb:v:48:y:2005:i:1:p:139-148
    DOI: 10.1140/epjb/e2005-00366-1
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    Cited by:

    1. Potirakis, Stelios M. & Zitis, Pavlos I. & Eftaxias, Konstantinos, 2013. "Dynamical analogy between economical crisis and earthquake dynamics within the nonextensive statistical mechanics framework," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(13), pages 2940-2954.
    2. Xu, Dan & Beck, Christian, 2016. "Transition from lognormal to χ2-superstatistics for financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 453(C), pages 173-183.
    3. Rashidi Ranjbar, Hedieh & Seifi, Abbas, 2015. "A path-independent method for barrier option pricing in hidden Markov models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 440(C), pages 1-8.
    4. Geoffrey Ducournau, 2021. "Bayesian inference and superstatistics to describe long memory processes of financial time series," Papers 2105.04171, arXiv.org.

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