{
  "id": "math/0509503",
  "version": "v1",
  "published": "2005-09-22T00:10:21.000Z",
  "updated": "2005-09-22T00:10:21.000Z",
  "title": "A filtering approach to tracking volatility from prices observed at random times",
  "authors": [
    "Jaksa Cvitanic",
    "Robert Liptser",
    "Boris Rozovskii"
  ],
  "categories": [
    "math.PR",
    "q-fin.ST"
  ],
  "abstract": "This paper is concerned with nonlinear filtering of the coefficients in asset price models with stochastic volatility. More specifically, we assume that the asset price process $ S=(S_{t})_{t\\geq0} $ is given by \\[ dS_{t}=r(\\theta_{t})S_{t}dt+v(\\theta_{t})S_{t}dB_{t}, \\] where $B=(B_{t})_{t\\geq0}$ is a Brownian motion, $v$ is a positive function, and $\\theta=(\\theta_{t})_{t\\geq0}$ is a c\\'{a}dl\\'{a}g strong Markov process. The random process $\\theta$ is unobservable. We assume also that the asset price $S_{t}$ is observed only at random times $0<\\tau_{1}<\\tau_{2}<....$ This is an appropriate assumption when modelling high frequency financial data (e.g., tick-by-tick stock prices). In the above setting the problem of estimation of $\\theta$ can be approached as a special nonlinear filtering problem with measurements generated by a multivariate point process $(\\tau_{k},\\log S_{\\tau_{k}})$. While quite natural, this problem does not fit into the standard diffusion or simple point process filtering frameworks and requires more technical tools. We derive a closed form optimal recursive Bayesian filter for $\\theta_{t}$, based on the observations of $(\\tau_{k},\\log S_{\\tau_{k}})_{k\\geq1}$. It turns out that the filter is given by a recursive system that involves only deterministic Kolmogorov-type equations, which should make the numerical implementation relatively easy.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-09-22T00:10:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random times",
      "high frequency financial data",
      "filtering approach",
      "optimal recursive bayesian filter",
      "form optimal recursive bayesian"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......9503C"
    }
  }
}