Hull-White model

The well-known model described in Hull and White (1993) uses Vasicek s model to obtain a theoretical yield curve and fit it to the observed market curve. It is therefore sometimes referred to as the extended Vasicek 

Given this description of the short-rate process, the price at time r of a zero-coupon bond with maturity Tmay be expressed as (4.10). 

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The Hull-White model can be fitted to an initial term structure, and also a volatility term structure. A comprehensive analysis is given in Pelsser (1996) as well as James and Webber (2000). 

The forward rate function/at time t is not static and is a function of the short-rate r at time f, the time t and the time to maturity T. The Hull-White model can be calibrated in terms of the forward rate /. That is, at time t the information (parameters) required to implement this are the short-rate r(t), the standard deviation a of the short-rate, the forward rate / and the standard deviations Bj t, T)a t) of the forward rates at time t. If the forward rates are known in a form that allows their first differential to be calculated with respect to t, using the other information, it is possible to calculate the function Bj, the derivative of this function and thereby the value for a f), using the relationship in Equation (3.56) ... 

To recap on the issues involved in fitting the extended Vasicek model or Hull-White model this describes the short-rate process as following the form... 

The Vasicek, Cox-Ingersoll-Ross, Hull-White and other models incorporate mean reversion. As the time to maturity increases and as it approaches infinity, the forward rates converge to a point at the long-run mean reversion level of the current short-rate. This is the limiting level of the forward rate and is a function of the volatility of the current short-rate. As the time to maturity approaches zero, the short-term forward rate converges to the same level as the instantaneous short-rate. In the Merton and Vasicek models, the mean of the short-rate over the maturity period T is assumed to be constant. The same constant for the mean, or the drift of the interest rate, is described in the Ho-Lee model, but not the extended Vasicek or Hull-White model. 

Cap prices can also be valued analytically using the Hull-White model. The cap prices calculated using the implied volatilities of interest rate caps and the Black-Scholes model serve as the calibrating instruments. After the Hull-White model has been calibrated, the parameters a and o that minimize a goodness-of-fit measure can be used to solve for the convexity bias. 

Equation (4.3) describes a stochastic short-rate process modified to include the direction of change. To be more realistic, it should also include a term describing the tendency of interest rates to drift back to their long-run average level. This process is known as mean reversion and is perhaps best captured in the Hull-White model. Adding a general specification of mean reversion to (4.3) results in (4.4). 

The academic literature and market participants have proposed a large number of alternatives to the Vasicek term-structure model and models, such as the Hull-White model, that are based on it. Like those they seek to replace, each of the alternatives has advantages and disadvant es. 

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