240 Mathematics for Finance
Figure 11.2 Evolution of bond prices in Example 11.1implicitly defining the logarithmic returns
k(n, N;sn− 1 u) = lnB(n, N;sn− 1 u)
B(n− 1 ,N;sn− 1 ),k(n, N;sn− 1 d) = lnB(n, N;sn− 1 d)
B(n− 1 ,N;sn− 1 ).
We assume here thatk(n, N;sn− 1 u)≥k(n, N;sn− 1 d).
Remark 11.1
Note that there are some places in the tree where the returns are non-random
given the statesn− 1 is known. Namely,
k(n, n;sn− 1 u) =k(n, n;sn− 1 d) = ln1
B(n− 1 ,n;sn− 1 ),
sinceB(n, n;sn) = 1 for allsn.
Example 11.2
From the data in Example 11.1 we extract the prices of bonds with maturity 3,
completing the picture with the final value 1. The tree shown in Figure 11.3
Figure 11.3 Prices of the bond maturing at time 3 in Example 11.2