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Stock Option Trading

Our objective during the first stage is to build a tree similar to that shown in Figure 23.8 for R*. To do this, we must resolve which of the three branching methods shown in Figure 23.7 will apply at each node. This will determine the overall geometry of the tree. Once this is done, the branching probabilities must also be calculated.

Define (i, j) as the node where t — iSt and R* =j SR. (The variable; is a positive integer andj is a positive or negative integer.) The branching method used at a node must lead to the probabilities on all three branches being positive. Most of the time, the branching in Figure 23.7a is appropriate. When a > 0, it is necessary to switch from the branching in Figure 23.7a to the branching in Figure 23.7c for a sufficiently large / Similarly, it is necessary to switch from the branching in Figure 23.7a to the branching in Figure 23.7b whenj is sufficiently negative. Define y max as the value ofj where we switch from the Figure 23.7a branching to the Figure 23.7c branching and j min as the value ofj where we switch from the Figure 23.7a branching to the Figure 23.7b branching. Hull and White show that probabilities are always positive if we set j max equal to the smallest integer greater than 0. 184/(a St) and j min equal to - j ma x • 13 Define p u , p m , and pd as the probabilities of the highest, middle, and lowest branches emanating from the node. The probabilities are chosen to match the expected change and variance of the change in R * over the next time interval St. The probabilities must also sum to unity. This leads to three equations in the three probabilities.

As already mentioned, the mean change in R * in time St is -aR * St and the variance of the

13 The probabilities are positive for any value of j m .dX between O.184/(a<5() and 0.816/(«5f) and for any value ofjm [ n between — O.\84/(a8t) and -O.816/(ai5t). Changing the branching at the first possible node proves to be computationally most efficient.

SecondStage

The second stage in the tree construction is to convert the tree for R* into a tree for R. This is accomplished by displacing the nodes on the R *-tree so that the initial term structure of interest rates is exactly matched. Define

a(t) =R(t) -R*(t) Because

dR = [6(t)-aR]dt and

dR* = -aR* it follows that

da = [9(t) - aa(t)]dt

If we ignore the distinction between r and R, equation (23.18) shows that the solution to this is

a(t) = F(0, 0 +,- A (l - e~ at f {23.21)

As a tends to zero, this becomes a(t) = F(0, f) + a 2 t 2 /2.

Equation (23.27) can be used to create a tree for R from the corresponding tree for R*. The approach is to set the interest rates on the R-tree at time iSt to be equal to the corresponding interest rates on the R *-tree plus the value of a at time iSt and to keep the probabilities the same.

The tree for R produced using equation (23.27), although satisfactory for most purposes, is not exactly consistent with the initial term structure. An alternative procedure is to calculate the a's iteratively so that the initial term structure is matched exactly. We now explain this approach. It provides a tree-building procedure that can be extended to models where there are no analytic results. Furthermore it is applicable to situations where the initial zero curve is not differentiable everywhere.

Define a, as a(i St), the value of R at time / St on the R-tree minus the corresponding value of R * at time ;' St on the r*-tree. Define Q;j as the present value of a security that pays off $1 if node (i, j) is reached and zero otherwise. The a, and Q,y can be calculated using forward induction in such a way that the initial term structure is matched exactly.