- Stochastic Interest Rates 243
At time 2 we have four sequences ofN−2 forward rates, and so on. At timeN− 1
we have 2N−^1 single numbersf(N− 1 ,N−1;sN− 1 ).
Example 11.4
Using (11.1), we can evaluate the forward rates for the data in Example 11.1.
For instance,
f(1,2; u) =−lnB(1,3; u)−lnB(1,2; u)
τ.
Alternatively, we can use the yields found in Example 11.3 along with for-
mula (10.7):
f(1,2; u) = 2y(1,3; u)−y(1,2; u).
The results are gathered in Figure 11.7.
Figure 11.7 Forward rates in Example 11.4The information contained in forward rates is sufficient to reconstruct the
bond prices, as was shown in Proposition 10.3.
Exercise 11.3
Suppose a tree of forward rates is given as in Figure 11.8. Find the
corresponding bond prices (using one-month steps).The short rates are just special cases of forward rates,r(n;sn)=f(n, n;sn)forn≥ 1 ,with deterministicr(0) =f(0,0).The short rates are also given by
r(n;sn)=y(n, n+1;sn),n≥ 1 ,andr(0) =y(0,1),that is, by the rates of
return on a bond maturing at the next step. This is obvious from the relations
between the forward rates and yields.