The Hull-White Model under Volatility Uncertainty

TL;DR

This paper extends the Hull-White interest rate model to account for volatility uncertainty using G-Brownian motion, proposing an arbitrage-free, affine term structure adjustment compatible with traditional models.

Contribution

It introduces a novel approach to incorporate volatility uncertainty into the Hull-White model, adjusting the term structure to remain arbitrage-free under a set of beliefs.

Findings

The adjusted model remains affine and consistent with traditional Hull-White after yield curve fitting.

The approach generalizes to a multifactor setting with multiple risk factors.

Classical martingale methods are inadequate under volatility uncertainty.

Abstract

We study the Hull-White model for the term structure of interest rates in the presence of volatility uncertainty. The uncertainty about the volatility is represented by a set of beliefs, which naturally leads to a sublinear expectation and a G-Brownian motion. The main question in this setting is how to find an arbitrage-free term structure. This question is crucial, since we can show that the classical approach, martingale modeling, does not work in the presence of volatility uncertainty. Therefore, we need to adjust the model in order to find an arbitrage-free term structure. The resulting term structure is affine with respect to the short rate and the adjustment factor. Although the adjustment changes the structure of the model, it is still consistent with the traditional Hull-White model after fitting the yield curve. In addition, we extend the model and the results to a multifactor version, driven by multiple risk factors.