The Black Derman Toy model Excel implementation guide. A one factor interest rate model and some simplifying assumptions made in its construction.
In order to value instruments whose cash flows are contingent on interest rates we would require a term structure of interest rates which define the evolution of spot rates over time. These term structures are in turn derived using term structure models which define the probabilistic nature of all rates. There are two main categories of term structure models; Equilibrium models and No-Arbitrage models.
Equilibrium models use assumptions about various economic variables and derive a process for determining the short term risk free rate. The initial term structure of rates is an output of such a model and may not automatically be consistent with today’s term structure. On the other hand, No-Arbitrage models are exactly consistent with today’s term structure as the initial term structure is taken as an input in the model. The model’s output ( future short rates in the case of the BDT model), is calibrated to this initial term structure using a risk-neutral pricing process of a binomial tree lattice of valuing bonds over single time periods.
The Black-Derman-Toy term structure model was developed by Fischer Black, Emmanuel Derman and William Toy in 1990. It is an example of a No-Arbitrage model. It is also a one factor model, i.e., it assumes that all security prices and rates depend on only one single factor- the short rate.
The inputs to a BDT model are the existing term structure of zero coupon rates and their estimated volatilities. The result or output from the models is a binomial lattice of possible future short rates. These short rates can be used to value interest-rate-sensitive instruments such as options, swaps, etc.
Like all models, the BDT model is based on certain simplifying assumptions that need to be kept in mind when using the model to value instruments in the real world. These include:
- The changes in all bond yields are assumed to be perfectly correlated. As the BDT model is a one factor model the output from this model is the short rate. In other words, the uncertainty in the short rates is the only thing assumed to affect the rates of different maturities. In reality rates with different maturities tend not to be perfectly correlated.
Also given that it is a one factor model it cannot be extended to include the impacts of another factor (e.g. slope) even if such factors are relevant to the term structure being considered. Further, the volatilities of yields of different maturities cannot be expressed independently of the future volatility of the short rate.
- Expected returns on all securities over one period are equal.
- Short rates are log-normally distributed. The advantage of this is that the model results for the interest rates can never be negative. However, under certain specifications of the volatility function the short rates derived by the BDT may not be mean reverting which is one of the fundamental ideas of a short rate process.
- There are no taxes or transaction costs.
We have reviewed some of the basic features and assumptions of the Black-Derman-Toy (BDT) one factor interest rate model. In our next post, we will look at the step-by-step methodology for building such a model in EXCEL.