Interest Rate Dynamics
The goal is to derive a model that is consistent with the observed interest rate dynamics and matches the term structure. We will look at the Ho-Lee and the Black-Derman-Toy (BDT) models. Ho and Lee was one of the early models that matched the term structure. I illustrate how this is accomplished in the chapter spreadsheet, using Excel's Solver.
Ho and Lee propose a linear short-rate model 
, where b is a constant volatility parameter, specifically b = 2σ, where σ is the annual volatility of the short rate. The annual drift rate is given by 
which, following Luenberger (Chapter 14) is treated as a variable to be solved in such a way that the short-rate lattice constructed is matched to the current term structure. The continuous time counterpart to the Ho and Lee model is given by 
, where dz is a standard Wiener process. Therefore, the parameter 
is a drift component, while (b × s) is the volatility factor. In our lattice, s is the state variable that measures the number of up-movements in rates (similar to the binomial lattice). It therefore represents the volatility given by the standard extrapolation measure used in the options lattice of Chapters 16 and 17, that is, 
. The results for the August 1, 2008, term structure are given in Figure 20.7.
Figure 20.7 Ho-Lee Model
This lattice is tied to the term structure by solving for the parameters 
so that the differences between the observed spot rates and those computed off the estimated bond prices are minimized (see Solver on the spreadsheet for details). One problem with Ho and Lee, however, is that negative short rates are not ruled out. The BDT model resolves this problem using the form 
. Again, we assume that the volatility parameter is constant across the term structure. The continuous time version of BDT is:
where, again, dz is a Wiener process. This is a lognormal model similar to the one derived in Chapter 17. Note that the drift component is lognormal and proportional to the short rate. The BDT lattice is given in the chapter spreadsheet for comparison purposes. Notice that the bond prices are identical to those from the Ho-Lee model.