248 Mathematics for Finance
replaced by
k(n, N;sn− 1 u)>τr(n−1;sn− 1 )>k(n, N;sn− 1 d). (11.2)Any future cash flow can be replicated in a similar fashion. Consider, for
example, a coupon bond with fixed coupons.
Example 11.6
Take a coupon bond maturing at time 2 with face valueF = 100,paying
couponsC= 10 at times 1 and 2. We price the future cash flow by using the
zero-coupon bond maturing at time 3 as the underlying security. The coupon
bond priceP at a particular time will not include the coupon due (the so-
called ex-coupon price). Assume that the structure of the bond prices is as in
Figure 11.10.
Consider time 1.In state u the short rate is determined by the price
B(1,2; u) = 0. 9947 ,so we haver(1; u)∼= 6 .38%.HenceP(1; u)∼= 109. 4170.
In state d we useB(1,2; d) = 0.9913 to findr(1; d)∼= 10 .49% andP(1; d)∼=
109. 0485.
Consider time 0.The cash flow at time 1 which we are to replicate includes
the coupon due, so it is given byP(1; u) + 10∼= 119 .417 andP(1; d) + 10∼=
119. 0485 .The short rater(0)∼= 11 .94% determines the money market account
as in Example 11.5,A(1) = 1. 01 ,and we findx∼= 92. 1337 ,y∼= 28. 3998 .Hence
P(0)∼= 118 .009 is the present price of the coupon bond.
An alternative is to use the spot yields:y(0,1)∼= 11 .94% andy(0,2)∼=
10 .41% to discount the future payments with the same result: 118. 009 ∼= 10 ×
exp(− 121 × 11 .94%) + 110×exp(− 122 × 10 .41%).
In general,
P(0) =C 1 exp{−τy(0,1)}+C 2 exp{− 2 τy(0,2)}
+···+(CN+F)exp{−Nτy(0,N)}. (11.3)(For simplicity we include all time steps, soCk= 0 at the time stepskwhen
no coupon is paid.) At each timekwhen a coupon is paid, the cash flow is the
sum of the (deterministic) coupon and the (stochastic) price of the remaining
bond:
Ck+P(k;sk)=Ck+Ck+1exp{−τy(k, k+1;sk)}
+···+(Cn+F)exp{−τ(n−k)y(k, n;sk)}.
Quite often the coupons depend on other quantities. In this way a coupon
bond may become a derivative security. An important benchmark case is de-
scribed below, where the coupons are computed as fractions of the face value.