On the probability density function of baskets

TL;DR

This paper investigates the complex probability density function of baskets in financial models, revealing phase transition phenomena and providing expansions for short time and small volatility regimes, with implications for basket option pricing.

Contribution

It introduces a method using differential equations to analyze basket densities, uncovering critical strike levels and phase transitions in multi-factor models.

Findings

Density expansion can degenerate at critical strike levels.

Phase transition from unique to multiple most-likely paths.

Out-of-money basket options can become in-the-money unexpectedly.

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.