The underlying stochastic differential equation for the HW model is: 



where, 
dr = the change in the instantaneous short rate, r q(t) = a time dependent mean reversion level a = constant mean reversion rate s = volatility of the instantaneous short rate dz = an increment in a standard Weiner process 
When compared to the equivalent stochastic processes for other models, the HW model can be described as the HoLee model with mean reversion, or as the Vasicek model with a timedependent mean reversion level. If we assume that a = 0, then the model above reduces to the HL model and the procedure explained below for constructing the HW model is also relevant for implementing that model. 

Because the HW model assumes that the changes in the short rate are normally distributed, there are closedform solutions available for the value of zerocoupon bonds and European options written on both zerocoupon and coupon bonds. Other more complex instruments, including those with an early exercise privilege, must however be valued using an interest rate lattice like that described above for the BDT model. Implementation of the HW (and BK) model differs from the BDT model in that the most efficient way to construct the lattice is to use a trinomial rather then binomial tree. 

Two slightly different approaches to the trinomial tree construction are discussed in the following sections. The first assumes that the branches in the tree are evenly spaced, while the second method is more general in that it allows for the time span between branches to vary. This feature is quite important when one attempts to accurately incorporate events such as coupon payments or exercise decisions directly onto the lattice, and is the main reason why the supported models that use trinomial trees are all based on varying time steps. Despite this, discussion of the procedure that is used to build trees with a constant timestep is still included because it greatly adds to the overall intuition behind the tree building process. 


In This Section Building HW Trinomial Trees with a Constant Timestep 
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