{
  "id": "cond-mat/0502151",
  "version": "v1",
  "published": "2005-02-05T17:18:13.000Z",
  "updated": "2005-02-05T17:18:13.000Z",
  "title": "On the connection between financial processes with stochastic volatility and nonextensive statistical mechanics",
  "authors": [
    "Silvio M. Duarte Queiros",
    "Constantino Tsallis"
  ],
  "journal": "Eur. Phys. J. B 48, 139--148 (2005)",
  "doi": "10.1140/epjb/e2005-00366-1",
  "categories": [
    "cond-mat.stat-mech",
    "q-fin.ST"
  ],
  "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 $(b,c)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-02-05T17:18:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05.90.+m",
      "05.40.-a",
      "89.65.Gh"
    ],
    "keywords": [
      "nonextensive statistical mechanics",
      "stochastic volatility",
      "financial processes",
      "connection",
      "gaussian distribution"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "European Physical Journal B",
      "year": 2005,
      "month": "Nov",
      "volume": 48,
      "number": 1,
      "pages": 139
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005EPJB...48..139Q"
    }
  }
}