## What Is the Hull-White Model?

The Hull-White model is a single-factor interest model used to price interest rate derivatives. The Hull-White model assumes that short rates have a normal distribution and that the short rates are subject to mean reversion. Volatility is thus likely to be low when short rates are near zero, which is reflected in a larger mean reversion in the model.

The Hull-White model extends the Vasicek Model and Cox-Ingersoll-Ross (CIR) model.

### Key Takeaways

• The Hull-White model is an interest rate derivatives pricing model.
• This model makes the assumption that very short-term rates are normally distributed and revert to the mean.
• The Hull-White model calculates the price of a derivative security as a function of the entire yield curve rather than a single rate.

## Understanding the Hull-White Model

An interest rate derivative is a financial instrument with a value that is linked to the movements of an interest rate or rates. Interest rate derivatives are often used as hedges by institutional investors, banks, companies, and individuals to protect themselves against changes in market interest rates, but they can also be used to increase or refine the holder's risk profile or to speculate on rate moves. These may include interest rate caps and floors.

Investments whose values are dependent upon interest rates, such as bond options and mortgage-backed securities (MBS), have grown in popularity as financial systems have become more sophisticated. Determining the value of these investments often entailed using different models, with each model having its own set of assumptions. This made it difficult to match the volatility parameters of one model with another model, and also made it difficult to understand risk across a portfolio of different investments.

## Special Considerations

Like the Ho-Lee model, the Hull-White model treats interest rates as normally distributed. This creates a scenario in which interest rates are negative, though there is a low probability of this occurring as a model output.

The Hull-White model also prices the derivative as a function of the entire yield curve, rather than at a single point. Because the yield curve estimates future interest rates rather than observable market rates, analysts will hedge against different scenarios that economic conditions might create.

Unlike the Hull-White model, which uses the instantaneous short rate, or the Heath-Jarrow-Morton (HJM) model, which uses the instantaneous forward rate, the Brace Gatarek Musiela Model (BGM) model only uses rates that are observable; i.e., forward LIBOR rates.

## Who Are Hull and White?

John C. Hull and Alan D. White are finance professors at the Rotman School of Management at the University of Toronto. Together they developed the model in 1990. Professor Hull is the author of Risk Management and Financial Institutions and Fundamentals of Futures and Options Markets. Professor White, also recognized internationally as an authority on financial engineering, is the Associate Editor of the Journal of Financial and Quantitative Analysis and the Journal of Derivatives.