Pricing bond options with the Black Scholes model, or its binomial approximation, as done in chapter 17, does not always get it right. For example, it ignores the fact that at the maturity of the bond, the bond volatility is zero. The bond volatility decreases as one gets closer to the bond maturity. This behaviour is not captured by the assumptions underlying the Black Scholes assumption. We therefore look at more complicated term structure models, the unifying theme of which is that they are built by building trees of the interest rate.
The Rendleman and Bartter approach to valuation of interest rate contingent claims (see Rendleman (1979) and Rendleman and Bartter (1980)) is a particular simple one. Essentially, it is to apply the same binomial approach that is used to approximate options in the Black Scholes world, but the random variable is now the interest rate. This has implications for multiperiod discounting: Taking the present value is now a matter of choosing the correct sequence of spot rates, and it may be necessary to keep track of the whole ``tree'' of interest rates.
The general idea is to construct a tree as shown in figure 20.1.
Code 20.1 shows how one can construct such an interest rate tree.
Code 20.2 implements the original algorithm for a call option on a (long maturity) zero coupon bond.
General references include Sundaresan (2001).
Rendleman (1979) and Rendleman and Bartter (1980) are the original references for building standard binomial interest rate trees.