{
  "id": "1711.04536",
  "version": "v1",
  "published": "2017-11-13T11:48:51.000Z",
  "updated": "2017-11-13T11:48:51.000Z",
  "title": "On the Heston Model with Stochastic Volatility: Analytic Solutions and Complete Markets",
  "authors": [
    "Bénédicte Alziary",
    "Peter Takáč"
  ],
  "comment": "61 pages, 4 figures, research finished in September 2017",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the Heston model for pricing European options on stocks with stochastic volatility. This is a Black\\--Scholes\\--type equation whose spatial domain for the logarithmic stock price $x\\in \\RR$ and the variance $v\\in (0,\\infty)$ is the half\\--plane $\\HH = \\RR\\times (0,\\infty)$. The {\\it volatility\\/} is then given by $\\sqrt{v}$. The diffusion equation for the price of the European call option $p = p(x,v,t)$ at time $t\\leq T$ is parabolic and degenerates at the boundary $\\partial \\HH = \\RR\\times \\{0\\}$ as $v\\to 0+$. The goal is to hedge with this option against volatility fluctuations, i.e., the function $v\\mapsto p(x,v,t)\\colon (0,\\infty)\\to \\RR$ and its (local) inverse are of particular interest. We prove that $\\frac{\\partial p}{\\partial v}(x,v,t) \\not= 0$ holds almost everywhere in $\\HH\\times (-\\infty,T)$ by establishing the analyticity of $p$ in both, space $(x,v)$ and time $t$ variables. To this end, we are able to show that the Black\\--Scholes\\--type operator, which appears in the diffusion equation, generates a holomorphic $C^0$-semigroup in a suitable weighted $L^2$-space over $\\HH$. We show that the $C^0$-semigroup solution can be extended to a holomorphic function in a complex domain in $\\CC^2\\times \\CC$, by establishing some new a~priori weighted $L^2$-estimates over certain complex \"shifts\" of $\\HH$ for the unique holomorphic extension. These estimates depend only on the weighted $L^2$-norm of the terminal data over $\\HH$ (at $t=T$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-13T11:48:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B65",
      "91G80",
      "35K65",
      "35K15"
    ],
    "keywords": [
      "heston model",
      "stochastic volatility",
      "complete markets",
      "analytic solutions",
      "diffusion equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 61,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}