In Chapter 11 we covered the Black, Derman, and Toy (BDT) model. In particular, in Section 11.3.2 we covered the notion of forward volatility implicit in caps and floors. To build the BDT tree, at each step i we had to match both a zero coupon bond with maturity i + 1 and the cap with maturity i + 1 by searching for two quantities, the interest rate ri,1 and the volatility σi, A more widespread practice is to use directly the forward volatilities σi, computed from caps and floors from the Black formula, as in Exercise 2. Using this methodology, the BDT tree results are simpler to build, as at each step i it is necessary to search only for one variable, ri.1 that matches the term structure of interest rates.
(a) Using the data in Table 20.6 compare the two methodologies.
(b) Does the tree obtained using the methodology illustrated here (i.e. first cr, from the Black formula) price the caps correctly? Comment.