Building HW Trinomial Trees with Changing Timestep

 Trinomial trees that are constructed with a constant timestep often do not contain a branch in the tree that coincides with all of the cashflow dates of the underlying instrument. For example, the tree might be built to value a callable bond that pays a semi-annual coupon, with payments that could potentially continue for a ten year period. The tree construction process would be based on splitting the entire period into a large enough number of timesteps so that the resulting bond value is 'sufficiently' accurate. Even if a very large number of steps is chosen, it is unlikely that each coupon payment date will fall exactly on the dates represented by the branches in the tree. There are a number of alternative ways to deal with the problem: Discount the cashflow back to the next earliest node. To demonstrate the procedure, suppose that a coupon payment occurs at a date denoted by r, which falls between two nodal dates, ti and ti + 1. The cashflow could be discounted back to time ti at the appropriate r-ti rates prevailing at all of the nodes at time ti. Unfortunately these rates are not directly included in the tree, and must be estimated separately. Apportion the cashflow between the adjacent nodes. This approach assumes that a proportion of the cashflow occurs at time ti with the remainder paid at time ti + 1. Change the length of the timestep so that a tree branch always coincides exactly with the coupon payment dates. While the last method does provide the most complete solution, it also requires that the tree building procedure described in Building HW Trinomial Trees with a Constant Timestep is slightly amended. A brief outline of the new procedure is given below. Consider a time period between time 0 and time T that is divided into a set of finite times denoted as t0, t1, t2, ......, tn, where t0 = 0 and tn = T. Also define Dti = ti + 1 - ti as the length of the period beginning at time ti. Now when the initial tree is constructed, the state step also becomes time dependent and is calculated as: The tree branching also differs from the constant time-step case. Rather than determine which of the three alternative branching methods are used as a function of the current position in the tree, we now choose the position of the central node as that which is closest to the expected value of the short rate at the end of the period. The branching at all positions in the tree can therefore be depicted as: Note that the state subscript k is now a parameter to be chosen. Other than that, the tree construction follows the same general steps that are detailed in the previous section. At each node (i, j): 1. determine the value of k such that the central node emanating from the initial node corresponds to a rate equal to the expected value of ri + 1: E(ri + 1) = ri - a.ri.Dti. This is computed as the closest integer to E(ri + 1)/Dri + 1. 2. Compute the probabilities for the top, middle, and bottom branches in the tree. These are determined so that the constructed tree is consistent with the conditional mean and variance of the interest rate process. By also restricting the probabilities to sum to 1, Brigo and Mercurio (2001) show that the probabilities are calculated as: where,  3. Apply stage two of the tree building process in exactly the same way as for the constant timestep model. 