8.6 Conclusion
The BDT model has several positive features:

for positive value of the decay factor (reversion speed) the general shape of the term structure of volatilities which is captured within the BDT model is consistent with the market observed volatility term structure.

due to the lognormal process assumed for the shortterm interest rate, calibration to market prices becomes much simpler. It is possible to fit the model to both the yield curve and to cap volatilities at the same time. Hence the model can simultaneously reproduce the prices of various maturity caps, displaying a declining term structure of volatilities.
However, it also displays several problems:

as with all one factor interest rate models the changes in rates of various maturities are by and large parallel which is not consistent with market observation. Hence the BDT model is not able to capture a tilting effect on the yield curve. This would require a second factor.

no specification is made of the evolution, through time, of the term structure of volatilities.

since the future shortterm interest rate volatilities fully determine the term structure of volatility it is impossible to specify one independently of the other.
This model was developed by practitioners for practitioners and hence allows for easy calibration to observed data and easy pricing of European and American style contingent claims.
Chapter 9: The Black and Karasinski Model
Overview
The discrete time Black, Derman and Toy model [ 6 ], discussed in Chapter 8, makes provision for two timedependent factors: the mean shortterm interest rate and the shortterm 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 timedependent factors allows the future shortterm interest rate volatilities to be specified independently of the initial volatility term structure.
As in the BDT model, the shortterm interest rate is assumed to have a lognormal distribution at any time horizon. The standard assumptions underlying perfect markets are also made.
9.1 The lognormality assumption
Ideally one wants a process for the short term interest rate such that negative interest rates are prevented, but the zero level may be reached and maintained for extended periods of time. None of the processes examined thus far, i.e. normal, lognormal and square root processes, satisfy both these requirements. A lognormal process does not admit a zero interest rate, while the square root process makes the zero level a reflecting barrier .
BK use a lognormal process. A lognormal distribution is fully described by its mean and variance, which are functions of time, so we have a different lognormal distribution of the shortterm interest rate at each future time. When mean reversion is combined with a lognormal model, we have three timedependent factors, an example being the BDT model:
However, here ( t ) is a function of ƒ ( t ). Dropping this functional dependence, and letting ¼ ( t ) be the target interest rate, i.e. the reversion level, the BK model may be written as:
where ( t ) is the speed of the mean reversion and ƒ ( t ) the local volatility, i.e. the volatility of the shortterm interest rate. BK calibrate their model to the initial observed interest rate and volatility term structures as well as the observed cap curve. The cap curve gives the prices of atthemoney caps, which pay the difference between the forward rate (strike) and the realised shortterm interest rate at maturity. BK do not attempt to specify a process which accurately depicts the evolution of the shortterm interest rate, but rather a shortterm interest rate process which can be fitted to observed market prices and hence used to price securities in a consistent manner. The future riskneutral distribution of the shortterm interest rate generated by the model is not the true distribution, but rather a distribution which leads to correct option prices.