{
  "id": "1001.3003",
  "version": "v2",
  "published": "2010-01-18T10:13:06.000Z",
  "updated": "2010-11-12T14:07:23.000Z",
  "title": "On refined volatility smile expansion in the Heston model",
  "authors": [
    "P. Friz",
    "S. Gerhold",
    "A. Gulisashvili",
    "S. Sturm"
  ],
  "categories": [
    "q-fin.PR",
    "math.PR"
  ],
  "abstract": "It is known that Heston's stochastic volatility model exhibits moment explosion, and that the critical moment $s_+$ can be obtained by solving (numerically) a simple equation. This yields a leading order expansion for the implied volatility at large strikes: $\\sigma_{BS}( k,T)^{2}T\\sim \\Psi (s_+-1) \\times k$ (Roger Lee's moment formula). Motivated by recent \"tail-wing\" refinements of this moment formula, we first derive a novel tail expansion for the Heston density, sharpening previous work of Dragulescu and Yakovenko [Quant. Finance 2, 6 (2002), 443--453], and then show the validity of a refined expansion of the type $\\sigma_{BS}( k,T) ^{2}T=( \\beta_{1}k^{1/2}+\\beta_{2}+...)^{2}$, where all constants are explicitly known as functions of $s_+$, the Heston model parameters, spot vol and maturity $T$. In the case of the \"zero-correlation\" Heston model such an expansion was derived by Gulisashvili and Stein [Appl. Math. Optim. 61, 3 (2010), 287--315]. Our methods and results may prove useful beyond the Heston model: the entire quantitative analysis is based on affine principles: at no point do we need knowledge of the (explicit, but cumbersome) closed form expression of the Fourier transform of $\\log S_{T}$\\ (equivalently: Mellin transform of $S_{T}$ ); what matters is that these transforms satisfy ordinary differential equations of Riccati type. Secondly, our analysis reveals a new parameter (\"critical slope\"), defined in a model free manner, which drives the second and higher order terms in tail- and implied volatility expansions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-11-12T14:07:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E99",
      "91B70"
    ],
    "keywords": [
      "heston model",
      "refined volatility smile expansion",
      "transforms satisfy ordinary differential equations",
      "hestons stochastic volatility model",
      "roger lees moment formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1001.3003F"
    }
  }
}