{
  "id": "1306.2793",
  "version": "v2",
  "published": "2013-06-12T11:44:56.000Z",
  "updated": "2016-04-05T09:10:24.000Z",
  "title": "On the probability density function of baskets",
  "authors": [
    "Christian Bayer",
    "Peter Friz",
    "Peter Laurence"
  ],
  "comment": "Appeared in: Large Deviations and Asymptotic Methods in Finance, Springer proceedings in Mathematics & Statistics, Editors: Friz, P.K., Gatheral, J., Gulisashvili, A., Jacquier, A., Teichmann, J., 2015, with minor typos removed",
  "doi": "10.1007/978-3-319-11605-1_16",
  "categories": [
    "math.PR",
    "q-fin.PR"
  ],
  "abstract": "The state price density of a basket, even under uncorrelated Black-Scholes dynamics, does not allow for a closed from density. (This may be rephrased as statement on the sum of lognormals and is especially annoying for such are used most frequently in Financial and Actuarial Mathematics.) In this note we discuss short time and small volatility expansions, respectively. The method works for general multi-factor models with correlations and leads to the analysis of a system of ordinary (Hamiltonian) differential equations. Surprisingly perhaps, even in two asset Black-Scholes situation (with its flat geometry), the expansion can degenerate at a critical (basket) strike level; a phenomena which seems to have gone unnoticed in the literature to date. Explicit computations relate this to a phase transition from a unique to more than one \"most-likely\" paths (along which the diffusion, if suitably conditioned, concentrates in the afore-mentioned regimes). This also provides a (quantifiable) understanding of how precisely a presently out-of-money basket option may still end up in-the-money.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-06-12T11:44:56.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2016-04-05T09:10:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "probability density function",
      "presently out-of-money basket option",
      "general multi-factor models",
      "explicit computations relate",
      "asset black-scholes situation"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.2793B"
    }
  }
}