{
  "id": "1508.06797",
  "version": "v1",
  "published": "2015-08-27T10:50:55.000Z",
  "updated": "2015-08-27T10:50:55.000Z",
  "title": "Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility",
  "authors": [
    "A. Paliathanasis",
    "K. Krishnakumar",
    "K. M. Tamizhmani",
    "P. G. L. Leach"
  ],
  "comment": "14pages, 4 figures",
  "categories": [
    "math.AP",
    "q-fin.PR"
  ],
  "abstract": "We perform a classification of the Lie point symmetries for the Black-Scholes-Merton model for European options with stochastic volatility $% \\sigma$, in which the last is defined by a stochastic differential equation with the Orstein-Uhlenbeck term. In this model the value of the option is given by a linear (1+2) evolution partial differential equation, in which the price of the option depends on two independent variables, the value of the underlying asset $S$ and a new variable, $y$, which follow from the Orstein-Uhlenbeck process. We find that for arbitrary functional form of the volatility, $\\sigma(y)$, the (1+2) evolution equation admits always two Lie symmetries, plus the linear symmetry and the infinity number of solution symmetries. However when $\\sigma(y)=\\sigma_{0}$ and since the price of the option depends on the second Brownian motion in which the volatility is defined, the (1+2) evolution is not reduced to the Black-Scholes-Merton equation, the model admits five Lie symmetries, plus the linear symmetry and the infinity number of solution symmetries. Furthermore we apply the zero-order invariants of the Lie symmetries and we reduce the (1+2) evolution equation to a linear second-order ordinary differential equation. Finally we study two models of special interest, the Heston model and the Stein-Stein model.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-27T10:50:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E60",
      "35Q91"
    ],
    "keywords": [
      "lie symmetry analysis",
      "black-scholes-merton model",
      "european options",
      "stochastic volatility",
      "linear second-order ordinary differential equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}