TERM STRUCTURE RECOVERY USING FORWARD INDUCTION
In the last section we constructed a tree for r*. This however is not the tree that we want for pricing our option. In this section we will discuss the methodology used to convert the tree for r* to the tree for r. From equation 4.13 we see that r* is just r but with an offset α defined by equation 4.30:
The goal is to determine α at each node such that the tree, for r, produces expected prices that match the yield curve. An analytical solution to α exists—it was developed by Kijima and Nagayama—but using it in this scenario will result in a loss of accuracy since the tree is a discrete presentation of a continuous stochastic differential equation. Instead Hull and White developed a step-by-step method using forward induction to adjust the r* tree such that the final tree exactly matches the yield curve. The reason for the forward induction will become clear once we discuss the methodology in detail, but for now it would be helpful to state that the expected rate at time step i is determined in part by the yield at time i +1. Therefore if the yield curve, for example, is valid up to time step 10, then we can only price an option up to time step 9.
Let us first define a few terms. As we have mentioned before, let i be the index for the time step away from the origin and j be the branch increment away from the mean. The combination of ...