- Stochastic Interest Rates 255
the consecutive values discounted by the appropriate short rates:
n=0 n=1 n=2
0. 0005
0. 00041 <
/ 0
0. 00024
\ 0. 0008
0. 00014 <
0As a result, the price of the option is 0.00024.
Exercise 11.11
Assume the structure of bond prices in Example 11.5 (Figure 11.10).
Consider a coupon bond maturing at time 2 with face valueF = 100
and couponsC= 1 payable at each step. Find the price of an American
call option expiring at time 2 with strike priceX= 101.30. (Include the
coupon in the bond price at each step.)Call options on bonds can be used by institutions issuing bonds to include
the possibility of buying the bond back prior to maturity for a prescribed price.
A bond that carries such a provision is called acallable bond. Its price should
be reduced by the price of the attached option.
11.3.2 Swaps
Writing and selling a bond is a method of borrowing money. In the case of a
coupon bond trading at par the principal represents the sum borrowed and the
coupons represent the interest. This interest may be fixed or floating (variable).
The interest is fixed if all coupons are the same. Floating interest can be realised
in many ways. Here we assume that it is determined by the short rates as
in (11.4). The basis for our discussion is laid by Proposition 11.1, according to
which the market value of such a floating-coupon bond must be equal to its face
value, the bond trading at par. For a fixed-coupon bond trading at par the size
of the coupons can easily be found from (11.3). We could say that the resulting
fixed coupon rate is equivalent to the variable short rate over the lifetime of
the bond.