We now move to multi-step binomial trees. The key is to realize that a multi-step tree is nothing more than a sequence of one-step trees. Therefore, any argument we made for each little tree in Chapter 9 holds here too.
Consider extending the binomial tree in Table 9.2 in Chapter 9 to one more period. Recall that the interest rates on the tree are continuously compounded (see discussion in Section 9.1.1 in Chapter 9). The binomial tree in Table 10.1 is recombining, which means that an “up and down” movement in interest rates leads to the same level as a “down and up” movement. This, of course, need not be the case in general. But using recombining trees becomes particularly helpful when we move to very long trees, with hundreds of steps. Non-recombining trees require massive computing power to be solved.
Finally, we assume that the probability p of an up movement is constant and equal to 1/2 along the tree. This assumption is not necessary, and it is made here only for convenience. This interest rate tree was computed on January 8, 2002, in a way that its implied forecasts of future interest rates are reasonable given the information available at that time. In particular, since the top node uu is reached with probability p × p = 1/4, the bottom one dd is reached with probability (1 – p) × (1 – p) = 1/4, and the middle with probability 2 × p × (1 – p) = 1/2, we have the predicted rate in six and twelve months