256 Mathematics for Finance
Example 11.10
Consider a fixed-coupon bond and a floating-coupon bond, both with annual
coupons, trading at par and maturing after two years with face valueF= 100.
Given that the tree of one- and two-year zero-coupon bond prices is
B(1,2; u) = 0. 9101
B(0,1) = 0. 9123
B(0,2) = 0. 8256<
B(1,2; d) = 0. 8987where a time step is taken to be one year,τ= 1, we can evaluate the coupons
of the fixed- and floating-coupon bonds. The size of the floating coupons can
be found from (11.4),
C 1 =(B(0,1)−^1 −1)F∼= 9. 6131 ,
C 2 (u) = (B(1,2; u)−^1 −1)F∼= 9. 8780 ,
C 2 (d) = (B(1,2; d)−^1 −1)F∼= 11. 2718.The fixed couponsCcan be found by solving equation (11.3), which takes the
form
F=CB(0,1) + (C+F)B(0,2).
This gives
C∼= 10. 0351.
By buying a fixed-coupon bond and selling a floating-coupon one (or the other
way round, selling a fixed-coupon bond and buying a floating-coupon one) an
investor can create an random cash flow with present value zero, since the two
kinds of bond have the same initial price.
A company who has sold fixed-coupon bonds and is paying fixed interest
may sometimes wish to switch into paying the floating rate instead. This can
be realised by writing a floating-coupon bond and buying a fixed-coupon bond
with the same present value. In practice, a financial intermediary will provide
this service by offering a contract called aswap. Clearly, a swap of this kind
will cost nothing to enter. Here is an example of a practical situation, where
the role of the intermediary is to match the needs of two particular companies.
Example 11.11
Suppose that company A wishes to borrow at a variable rate, whereas B prefers
a fixed rate. Banks offer the following effective rates (that is, rates referring to