234 Mathematics for Finance
Proposition 10.3
The bond price is given by
B(n, N)=exp{−τ(f(n, n)+f(n, n+1)+···+f(n, N−1))}.Proof
For this purpose note that
f(n, n)=−lnB(n, n+1)
τ,
sinceB(n, n)=1,so
B(n, n+1)=exp{−τf(n, n)}.Next,
f(n, n+1)=−lnB(n, n+2)−lnB(n, n+1)
τ
and, after inserting the expression forB(n, n+1),
B(n, n+2)=exp{−τ(f(n, n)+f(n, n+1))}.Repeating this a number of times, we arrive at the required general formula.
We have a simple representation of the forward rates in terms of the yields:f(n, N)=(N+1−n)y(n, N+1)−(N−n)y(n, N). (10.7)In particular,
f(n, n)=y(n, n+1),
resulting in the intuitive formula
y(n, N)=f(n, n)+f(n, n+1)+···+f(n, N−1)
N−n.
Example 10.12
We can clearly see from the above formulae that if the term structure is flat,
that is,y(n, N) is independent ofN,thenf(n, N)=y(n, N). Now consider an
example off(n, N) increasing withNfor a fixedn,and compute the corre-
sponding yields
f(0,0) = 8.01%,
f(0,1) = 8.03%,
f(0,2) = 8.08%,
y(0,1) = 8.01%,
y(0,2) = 8.02%,
y(0,3) = 8.04%.