The discrete time Black, Derman and Toy model [ 6 ], discussed in Chapter 8, makes provision for two time-dependent factors: the mean short-term interest rate and the short-term interest rate volatility. The continuous time equivalent of the model clearly shows that the rate of mean reversion is a function of the volatility. This is equivalent to future short- term interest rate volatilities being fully determined by the observed volatility term structure. This dependence makes it impossible to specify these two factors independently.
Black and Karasinski (BK) [ 7 ] develop a model, within a discrete time framework, where the target rate, mean reversion rate and local volatility are deterministic functions of time. The specification of three time-dependent factors allows the future short-term interest rate volatilities to be specified independently of the initial volatility term structure.
As in the BDT model, the short-term interest rate is assumed to have a lognormal distribution at any time horizon. The standard assumptions underlying perfect markets are also made.