Both the Vasicek and CIR models, as discussed in Chapters 1 and 2, incorporate a deterministically mean reverting process. In these models, the mean reversion is incorporated without additional assumptions about the future behaviour of the short-term interest rate volatility. This is a highly desirable feature. By allowing time-dependent parameters and therefore the matching of any arbitrary initial yield curve, HW manage to overcome one of the major drawbacks of the Vasicek and CIR models. The HW-extended Vasicek model is usually implemented with constant absolute volatility and reversion speed. However, in some yield curve environments, such as rising term structure of rates and declining term structure of volatilities, this version of the model provides a rather poor fit to observed cap prices [ 45 ]. On the other hand, allowing time-dependent absolute volatility and reversion speed results in unsuitable behaviour of short-term interest rate volatilities in the future. If these shortcomings are recognised and accounted for during the calibration process, the extended Vasicek model can be of great value, since it allows for closed-form solutions of discount bond and discount bond option prices.
Chapter 8: The Black, Derman and Toy One-Factor Interest Rate Model
Black, Derman and Toy (BDT) [ 6 ] make use of a binomial tree approach to model interest rates in a discrete time framework. The model has one fundamental factor, the short-term interest rate, which is used to determine all rates and security prices. The current term structure of interest rates and related volatilities are used to construct a binomial tree of possible short-term interest rates in the future. Since an interest rate sensitive security is characterised by its payoff at expiry, the constructed tree of possible interest rates is used to determine the current price of a security by means of an iterative procedure.
8.1 Model characteristics
The fundamental variable which drives security prices within the model is the short-term interest rate, which is defined as the annualised one period rate of interest.
The model inputs are a set of long-term interest rates of various maturities and their corresponding volatilities. Hence a yield curve and a volatility curve are required to calibrate the model.
These inputs are used to determine mean values and volatilities of future realisation of the short-term interest rate. As the input yield and volatility curves change, so do the means and volatilities of future short-term interest rates. Changes in future volatility have an impact on the degree of mean reversion.
As with most models, the assumption of perfect markets is made, hence:
changes in the yields of all zero coupon bonds are perfectly correlated,
the expected one period returns are the same for all securities,
short-term interest rates are lognormally distributed and
the market is free of taxes and transaction costs.
The lognormality feature holds several advantages for calibration of the model [ 45 ]. Negative interest rates are prevented and the volatility input may be specified in percentage terms, i.e. the volatility refers to relative price moves. This is the market convention for quoting volatilities, so calibration to market- observed volatilities is simplified.
Figure 8.1: Tree with one time step
8.2 Pricing contingent claims
The short-term interest rate at each node in the tree is found such that the term structure produced by the model matches the current observed term structure. European-style contingent claims may then be priced. The value at a node is the discounted expected value one time period in the future. Since the binomial tree is calibrated to the market- observed risk-free rate, the contingent claim is priced in a risk-neutral environment, where the probabilities of an up and down move are equal. Hence our expectation of the price of the contingent claim after one period is:
where S u and S d are the prices of the contingent claim after an up and down move respectively. Discounting by the current one period interest rate r ,the current price of the contingent claim S ,is:
This method may be used to determine the price at any node in the tree, from the prices one step in the future. Iterative application of (8.1) allows valuation of contingent claims of any duration, as long as the tree of short-term interest rates extends sufficiently far into the future.
To value bond options, the tree must first be used to determine the bond value at every node according to the interest rate associated with that node.
Then, making use of the known option value at expiry  , and working backwards through time, the bond option value may be found at every node prior to expiry.
Figure 8.2: Valuation of a 2-year zero coupon bond
 This is the option payoff at expiry, hence for a call option the payoff is spot less the strike, while for a put option it is the strike less the spot.