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Building HW Trinomial Trees with a Constant Timestep

The following brief description of the lattice construction process is based on Hull and White (1994), Hull and White (1996), Hull (2000), and Brigo and Mercurio (2001). Although the method was first applied to the HW model, it can also be used to built lattices that are consistent with other models.

The HW approach is based on a two-step process. The first step is to build a tree that is consistent with the modified process of:

 

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where the initial value of the short rate is set to zero. In the first instance, assume that the tree uses a constant "state" step and a constant time interval. That is, the spacing between the rates at any two nodes is set equal to:

 

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At each time step, i Dt, the trinomial process for r is symmetrical around the initial value of zero. That means that within 'normal' branches of the tree the short rate at any node is just equal to j Dr, where the node index j runs from -i, -(i + 1), -(i +2), ..., 0, ..., (i - 1), i. For any given node (i, j) the rate can move from r i, j to r i + 1, j + 1, r i + 1, j, or r i + 1, j - 1. Note that the state index j increments by 1 in the trinomial tree rather than by 2 as in the binomial tree.

However at some nodes in the tree, the branching needs to change to account for the impact of mean reversion. At nodes towards the top of the tree (representing 'high' interest rates), the branching alters to reflect the fact that rates are more likely to decline during the next period. Likewise, when rates are at 'low' levels the branching must change to reflect the increasing likelihood that rates will increase. The various branching alternatives are depicted below.

Alternative Branching for HW Trinomial Trees

 

 

 

 

 

 

(a) Normal

 

(b) Top

 

(c) Bottom

 

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Trinomial Tree for Mean Reverting Models (Stage 1)

With the space step and timestep both set equal to a constant, the procedure now involves determining the probabilities for each of the 3 potential paths that emanate from each node. The probabilities are chosen to ensure that the expected value and volatility for the short rate reflected on the tree are consistent with the statistics implied by dr = -ar dt + s dz (above). That is, the process in this equation implies that the variable ri+1 -ri is normally distributed with an expected value of -ariDt and a variance equal to s2 Dt. By adding the sensible restriction that the probabilities must sum to 1, the solutions for the normal branching probabilities can be determined from the following three equations (note that the second equation below is derived using the condition that Var(X) = E(X2) - [E(X)]2):

 

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HW show that the solutions to these equations are:

 

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where they have used the relation Dr2 = 3s2Dt, and where we define -aDt M. In a similar manner, it can be shown that the probabilities for the top branching process shown above are given by:

 

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Finally, if the branching is bottom branching like that shown in (c) above, then the probabilities are computed as:

 

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The only remaining thing to do to complete the first stage of the tree construction is to determine at what point each of the alternative branching methods should be used. Recall that the value of the short rate at any node, ri,j, is computed as jDr. Thus, when j is sufficiently high, the branching will have to switch from the scheme in (a) to that depicted as branch (b). HW have shown that the precise value of j at which the switch should occur is given by the next integer greater than -0.184/M. Similarly, at values of j lower than 0.184/M, the tree branching should switch from the standard scheme to the upward sloping scheme shown in (c).

The second stage in the tree construction is similar to the approach used in the BDT tree building process. The aim is to find the step-dependent shift in the tree, ai, so that the values of zero-coupon bonds implied by the tree are consistent with the observed zero curve. State prices are again fundamental to the calculations. Recall that the state price at the initial node (Q0,0) is equal to 1, and that the original tree is built assuming a starting value for the short rate that is equal to 0. The value of a0 is therefore the rate that will give a value for the Dt -maturity bond equal to that observed in the actual zero curve. This is simply equal to the Dt -maturity interest rate.

We now move to the calculation of a1. This is based on the state prices at the first branch in the tree (Q1,j) and the known value of a zero-coupon bond that matures at time 2.Dt. The state prices are defined as:

 

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where,

Pj = the probability of moving to the up, middle, or down node.

DF(0,1) = the discount factor at time 0 for maturity at time 1.

The prices of the 2.Dt -maturity bond seen from time Dt can be written as:

 

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The value of the bond implied by the tree at time 0 can therefore be computed (assuming continuously compounded interest rates) as:

 

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All of the variables in the above equation other than a1 are known, and the solution is given by:

 

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Calculation of a2 proceeds in the same manner, this time based on the updated state prices for all nodes at the time 2.Dt branch in the tree, together with the observed 3.Dt -maturity zero-coupon bond price. The state prices are updated in a similar fashion to that used in the construction of BDT binomial trees, while the general solution for ai is given by:

 

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