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.

click to expand
Figure 8.1: Tree with one time step



Interest Rate Modelling
Interest Rate Modelling (Finance and Capital Markets Series)
ISBN: 1403934703
EAN: 2147483647
Year: 2004
Pages: 132

flylib.com © 2008-2017.
If you may any questions please contact us: flylib@qtcs.net