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Hull and White (HW)

The underlying stochastic differential equation for the HW model is:

 

Equation Template

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 Ho-Lee model with mean reversion, or as the Vasicek model with a time-dependent 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 closed-form solutions available for the value of zero-coupon bonds and European options written on both zero-coupon 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

Building HW Trinomial Trees with Changing Timestep

Improving Efficiency of the HW Model

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