Interest rate modeling Hull-White model

Hull, J., White, A., 1990. Pricing interest-rate derivative securities. Rev. Financ. Stud. 3, 573-592. James, J., Webber, N., 2000. Interest Rate Modelling. Wiley, Chichester. 

The Hull-White (1990) model is an extension of the Vasicek model designed to produce a precise fit with the current term structure of rates. It is also known as the extended Vasicek model, with the interest rate following a process described by Equation (3.48) ... 

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), which is sometimes written with d W or dx in place of dz, is similar to the models first described in Vasicek (1977), Ho and Lee (1986), and Hull and White (1991). It assumes that, on average, the instantaneous change in interest rates is given by the function adt, with random shocks specified by adz. 

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). 

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