108156.pdf

11. Stochastic Interest Rates 241

describes the random evolution of a single bond purchased at time 0 for 0.9726.
The returns are easy to compute, for instance

k(2,3; ud) = ln

B(2,3; ud)

B(1,3; u)

∼= 0 .27%.

The results are gathered in Figure 11.4. (Recall that the length of each step is
one month.)

Figure 11.4 Returns on the bond maturing at time 3 in Example 11.2

Exercise 11.1

For the tree of weekly returns shown in Figure 11.5 construct the tree of

bond prices and fill in the missing returns.

Figure 11.5 Returns in Exercise 11.1

The evolution of bond prices is in perfect correspondence with the evolution
of implied yields to maturity. Namely,

y(n, m;sn)=

1

τ(m−n)

ln

1

B(n, m;sn)

with the same tree structure as for bond prices. In particular, the final yields are
non-random given that the statesn− 1 at the penultimate step is known. Note