Analysis of market weights under volatility-stabilized market models

TL;DR

This paper derives the joint density of market weights in volatility-stabilized market models, linking them to the Wright-Fisher diffusion, and provides a new proof of its transition density.

Contribution

It offers a novel derivation of the joint density of market weights and connects market models to population genetics models, providing a new proof of the Wright-Fisher transition density.

Findings

Derived joint density of market weights at fixed and stopping times.

Established equivalence between market weights and Wright-Fisher diffusion.

Provided a new proof for the Wright-Fisher transition density.

Abstract

We derive the joint density of market weights, at fixed times and suitable stopping times, of the volatility-stabilized market models introduced by Fernholz and Karatzas in [Ann. Finan. 1 (2005) 149-177]. The argument rests on computing the exit density of a collection of independent Bessel-square processes of possibly different dimensions from the unit simplex. We show that the law of the market weights is the same as that of the multi-allele Wright-Fisher diffusion model, well known in population genetics. Thus, as a side result, we furnish a novel proof of the transition density function of the Wright-Fisher model which was originally derived by Griffiths by bi-orthogonal series expansion.