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Improving Efficiency of the HW Model

In some cases, the closed-form solutions for zero coupon bonds can be used to improve the speed of the tree construction process for the Hull – White model. Consider valuing a 1 year American option written on a 20 year bond. The value of the option on the valuation date is determined with reference to the potential range of option payoffs 1 year from now, which are in turn dependent on the possible range of values for the underlying bond.

Without the closed-form solutions presented below, the interest rate lattice would need to be generated out to the maturity date of the underlying asset (20 years from now) and would therefore require a large number of branches. The alternative is to generate the tree out to the maturity date of the option itself, and to use the closed-form solutions for zero–coupon bonds to determine the value of the underlying bond at each of the nodes at the terminal branch in the tree. These values are then used to determine the value of the option.

For the discrete-time version of the Hull-White interest rate process presented in Equation Template(from Hull and White), it can be shown that the time t value of a zero coupon bond with maturity at time T and unit face value is given by:

 

Equation Template

where,

P(t, T) = Time t value of a zero coupon bond with maturity T

R(t) = the Dt - period interest rate at time t. This differs across the nodes at any given branch in the lattice.

And where the components A(t,T) and B(t,T) are defined as:

 

Equation Template

Equation Template

For the Ho-Lee model, the zero-coupon bond values are also computed using Equation Template (from above), but with the following redefinitions for A(t,T) and B(t,T):

 

Equation Template

Equation Template

 

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