108156.pdf

11. Stochastic Interest Rates 259

The price of the bond is reduced by the value of the caplet, that is, by 0.32773.
For a caplet at time 2 with the same strike rate the maximum size of the
coupon is 0.66889, as before. In the up state we pay the original interest,
exercising the caplet in the down state. The value of the bond at time 1 is
not 100, since the final coupons are no longer the same as for the par bond.
The time 1 prices are obtained by discounting the time 2 values. At time 0 we
find the bond price by evaluating the risk-neutral expectation of the discounted
values of the bond at time 1. The resulting cash flow is

n=0 n=1 n=2

− 0 .99990 — − 100. 52272

99. 87323 <

− 0 .99990 — − 100. 66889

This fixes the price of this caplet at 0.12677.
Finally, consider a cap for both times 1 and 2 with the same strike rate as
above. The cash flow can be obtained in a similar manner:

n=0 n=1 n=2

− 0 .66889 — − 100. 52272

99. 54550 <

− 0 .66889 — − 100. 66889

We can see that the value of the cap, 0.45450, is the sum of the values of the
caplets.

Analogously, aflooris a provision limiting the coupon from below. This
will be of value for a bond holder. It is composed from a series offloorlets,each
referring to a single period.

Exercise 11.12

In the framework of the above example, value a floor expiring at time 2

with strike rate 8%, based on the bond prices in Example 11.5.

11.4 Final Remarks

We conclude this chapter with some informal remarks on possible ways in which
models of the structure of bond prices can be built. This is a complex area and
all we can do here is to make some general comments.
As we have seen, the theory of interest rates is more complicated than the
theory of stock prices. In order to be able to price interest rate derivatives,