Alternative branching methods in a trinomial tree node. At the final nodes, the value of the derivative equals the payoff. For example; at node E, the value is 100 x (0.14 0.11) = 3. At earlier nodes, the value of the derivative is calculated using the rollback procedure explained in Chapters 10 and 18. At node B, the one-year interest rate is 12%. This is used for discounting to obtain the value of the derivative at node B from its values at nodes E, F, and G as (0.25 x 3 + 0.5 x 1 + 0.25 x 0)e~ 012xl = 1.11 At node C, the one-year interest rate is 10%. This is used for discounting to obtain the value of the derivative at node C as (0.25 x 1 + 0.5 x 0 + 0.25 x 0)e" 0lxl = 0.23 At the initial node, A, the interest rate is also 10% and the value of the derivative is (0.25 x 1.11+0.5x0.23 +0.25 x 0)e" 01xl =0.35 Nonstandard Branching It sometimes proves convenient to modify the standard branching pattern, which is used at all nodes in Figure 23.6. Three alternative branching possibilities are shown in Figure 23.7. The usual branching is shown in Figure 23.7a. It is "up one/straight along/down one". One alternative to this is "up two/up one/straight along", as shown in Figure 23.7b. This proves useful for incorporating mean reversion when interest rates are very low. A third branching pattern, shown in Figure 23.7c, is "straight along/one down/two down". This is useful for incorporating mean reversion when interest rates are very high. We illustrate the use of different branching patterns in the following section. A GENERAL TREE-BUILDING PROCEDURE Hull and White have proposed a robust two-stage procedure for constructing trinomial trees to represent a wide range of one-factor models. 12 This section first explains how the procedure can be See J. Hull and A. White, "Numerical Procedures for Implementing Term Structure Models I: Single-Factor Models,"'Journal ofDerivatives, 2, no. 1 (1994), 7-16; J. Hull and A. White, "Using Hull-White Interest Rate Trees," Journal ofDerivatives, Spring 1996, 26-36. used for the Hull-White model in Section 23.9 and then shows how it can be extended to represent other models. First Stage The Hull-White model for the instantaneous short rate r is dr = [0(t) - ar] dt + adz For the purposes of our initial discussion, we suppose that the time step on the tree is constant and equal to St. We assume that the St rate, R, follows the same process as r. dR = [0(t) - aR]dt + adz Clearly, this is reasonable in the limit as St tends to zero. The first stage in building a tree for this model is to construct a tree for a variable R* that is initially zero and follows the process dR* = aR*dt + adz This process is symmetrical about R * = 0. The variable R *(t + St) R *(t) is normally distributed. If terms of order higher than St are ignored, the expected value of R *(t + St) - R *(t) is -aR *(t) St and the variance of R*(t + St) - R*(t) is a 2 St. We define SR as the spacing between interest rates on the tree and set This proves to be a good choice of SR from the viewpoint of error minimization. stock option trading | forex trader | making money |