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Suppose that in the Hull-White model a = 0.1 and a = 0.015 and we wish to value a 3-month European put option on a 15-month bond that pays a coupon of 12% semiannually. We suppose that both the bond principal and the strike price are 100. The continuously compounded zero rates for maturities of 3 months, 9 months, and 15 months are 9.5%, 10.5%, and 11.5%, respectively. The option under consideration is an option on a portfolio of two zero-coupon bonds. The first zero- coupon bond has a maturity of 9 months and a principal of $6. The second zero-coupon bond has a maturity of 15 months and a principal of $106.

Define R as the value of the 6-month rate at the maturity of the option. The value of the first zero-coupon bond underlying the option is 6e~Rx 0 ' 5 . The value of the second zero-coupon bond is, from equation (23.22),

where the St in equation (23.22) equals 0.5. In this case, B(0.25, 0.75) = 0.4877 and B(0.25, 1.25) = 0.9516, so that equation (23.24) gives B(0.25, 1.25) = 0.9756. Also, from equation (23.23), A(0.25, 1.25) = 0.9874. Let R K be the value of R for which the coupon-bearing bond price equals the strike price. It follows that

This can be solved using an iterative procedure such as Newton-Raphson to give R K = 10.675%. When R has this value, the zero-coupon bond maturing at time 0.75 is worth 5.68814 and the zero-coupon bond maturing at time 1.25 is worth 94.31186. The option on the coupon-bearing bond is therefore equivalent to:

• A European put option with a strike price of 5.68814 on a zero-coupon bond maturing at
time 0.75 with a principal of 6; and

• A European put option with a strike price of 94.31186 on a zero-coupon bond maturing at
time 1.25 with a principal of 106.

Equation (23.26) gives the prices of these options as 0.01 and 0.43. The price of the option on the zero-coupon bond is therefore 0.44.

As explained in Section 22.4, a European swap option can be viewed as an option on a coupon- bearing bond. It can therefore be valued analytically using the Ho-Lee or Hull-White model.

INTEREST RATE TREES

An interest rate tree is a discrete-time representation of the stochastic process for the short rate in much the same way as a stock price tree is a discrete-time representation of the process followed by This emphasizes the fact that the initial zero curve does not have to be differentiable to use the Ho-Lee and Hull-White models.

a stock price. If the time step on the tree is St, the rates on the tree are the continuously compounded <5r-period rates. The usual assumption when a tree is constructed is that the <$r-period rate, R, follows the same stochastic process as the instantaneous rate, r, in the corresponding continuous-time model. The main difference between interest rate trees and stock price trees is in the way that discounting is done. In a stock price tree, the discount rate is usually assumed to be the same at each node. In an interest rate tree, the discount rate varies from node to node.

It often proves to be convenient to use a trinomial rather than a binomial tree for interest rates. The main advantage of a trinomial tree is that it provides an extra degree of freedom, making it easier for the tree to represent features of the interest rate process such as mean reversion. As mentioned in Section 18.8, using a trinomial tree is equivalent to using the explicit finite difference method.