Negative volatility for a 2-dimensional square root SDE

TL;DR

This paper investigates the conditions under which negative volatility factors can be avoided in a 2-dimensional square root SDE used in affine term structure models, establishing the necessity of multivariate Feller conditions.

Contribution

It proves the necessity of multivariate Feller conditions for a 2-dimensional square root SDE with one volatility factor using measure transformations and differential equations.

Findings

Multivariate Feller conditions are necessary to prevent negative volatility.

Methodology involves measure transformations and solving linear ODE systems.

Results clarify conditions for positive volatility in affine models.

Abstract

In affine term structure models the short rate is modelled as an affine transformation of a multi-dimensional square root process. Sufficient conditions to avoid negative volatility factors are the multivariate Feller conditions. We will prove their necessity for a 2-dimensional square root SDE with one volatility factor by presenting a methodology based on measure transformations and solving linear systems of ordinary differential equations.