258 Mathematics for Finance
the fixed rate of 13.40% it was offered. This means a gain of $360 in each year
on an $100,000 loan.
The result is that the $1,000 gain should be shared as $640 and $360 be-
tween companies A and B.
Finally, note that the value of the swap may vary with time and state,
departing from the initial value of zero. If a company wishes to enter into a
swap agreement at a later time, it may purchase aswaption, which is a call
option on the value of the swap (with prescribed strike price and expiry time).
11.3.3 Caps and Floors....................................
Acapis a provision attached to a variable-rate bond which specifies the maxi-
mum coupon rate paid in each period over the lifetime of a loan. Acapletis a
similar provision applied to a particular single period. In other words, a caplet
is a European option on the level of interest paid or received. A cap can be
thought of as a series of caplets.
Example 11.12
We take a loan by selling a par floating-coupon bond maturing at time 3. (That
is, a bond which always has the par value, the coupons being implied by the
short rates as in (11.4).) We use the bond prices and rates in Example 11.5,
see Figures 11.10 and 11.11. The cash flow shown below includes the initial
amount received for selling the bond together with the coupons and face value
to be paid:
n=0 n=1 n=2
− 0 .99990 — − 100. 52272
100 <
− 0 .99990 — − 100. 87764
Consider a caplet that applies at time 1 (one month) with strike interest rate of
8% (corresponding to 0.67% for a one-month period). The coupon determined
by the caplet rate is 0.66889 and we modify the cash flow accordingly. At time 0
we find the bond price by discounting its time 1 value, 100.66889 in each state,
that is 100 plus the coupon. This gives the following cash flow:
n=0 n=1 n=2
− 0 .66889 — − 100. 52272
99. 67227 <
− 0 .66889 — − 100. 87764