A Review of the Vasicek & Hull-White Models & the Likelihood of Negative Rates

Abstract

In this paper we review the Vasicek and Hull-White 1 factor (HW1F) models. For each model we summarize the model stochastic process, solution and Gaussian or normal dynamics. For pricing purposes we might opt to use more advanced models, however for risk management and complex calculations such as XVA or Financial Review of the Trading Book (FRTB) such models remain competitive. Models with Gaussian or normal distributions allow for the easy derivation of analytical formulas. Furthermore such models are efficient in that they simplify the implementation and performance of numerical procedures. They also cater for the pricing and risking a rich variety of derivative securities and are readily extendible to include a higher number factors if required. However the possibility of negative rates makes such models undesirable. Whilst in several markets we do indeed observe negative rates, the dynamics of normally distributed models can imply rates are negative more often than they ought to be. We review these models and perform a brief case study into the likelihood of negative rates in current interest rate markets, highlighting that the approach can applied not only to evaluate the probability of spot rates being negative, but also Libor rates at any future point in time. This implies that we could utilize the same approach to evaluate the likelihood of derivatives such as caps and floors being in the money say. We conclude that the Hull-White 1 factor model exhibits reasonable behaviour with negligible probability of rates being negative for currencies in positive territory such as AUD & USD and offer a reasonable reflection of the current market regime for rates in or very close to negative territory such as CHF, EUR and JPY. These results suggest that the Hull-White 1 factor model is a reasonable risk management tool with respect to the handling of negative rates.