Chapter 7: The Hull and White Model

The Vasicek [ 50 ] and CIR [ 18 ] models, studied in Chapters 1 and 2 respectively, allow all interest rate contingent claims to be valued in a consistent manner, but involve unobservable parameters and do not provide a perfect fit for the current interest rate term structure.

However, the process describing the evolution of the short-term interest rate may be deduced from the observed term structure of interest rates and interest rate volatilities. Hence the Vasicek and CIR models may be extended so as to be consistent with the current term structure of interest rates and the current spot interest rate volatilities or current forward rate volatilities.

7.1 General model formulation

The Vasicek and CIR models are special cases of a general mean reverting process of the form:

where ² = 0 for the Vasicek model and ² = ½ for the CIR model.

Since market expectations of interest rate movements can be time-dependent, the drift and volatility parameters should be functions of time:

where ( t ) is the drift rate imposed on the interest rate which otherwise reverts to a constant level b . Rewriting (7.2) in the form:

gives a mean reverting model where the reversion level is a function of time.

Hull and White (HW) [ 28 ] make assumptions about the market price of interest rate risk and fit the Vasicek and CIR special cases of the above mean reverting model to the current term structure of interest rates and spot (or forward) interest rate volatilities.