WIAS Preprint No. 1796, (2013)

On the probability density function of baskets



Authors

  • Bayer, Christian
    ORCID: 0000-0002-9116-0039
  • Friz, Peter
    ORCID: 0000-0003-2571-8388
  • Laurence, Peter

2010 Mathematics Subject Classification

  • 91G20 60F10 65C50

Keywords

  • Sums of lognormals, Focality, Pricing of butterfly spreads on baskets

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.

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