- Stochastic Interest Rates 239
Next, we shall describe the evolution of bond prices. At time 0 we are given
the initial bond prices for all maturities up toN, that is, a sequence ofN
numbers
B(0,1),B(0,2),B(0,3),...,B(0,N−1),B(0,N).
At time 1 one of the prices becomes redundant, namely the first bond matures
and only the remainingN−1 bonds are still being traded. We introduce ran-
domness by allowing two possibilities distinguished by the states u and d, so
we have two sequences
B(1,2; u),B(1, 3 ,u),...,B(1,N−1; u),B(1,N;u),
B(1,2; d),B(1,3; d),...,B(1,N−1; d),B(1,N;d).At time 2 we have four states and four sequences of lengthN−2:
B(2, 3 ,uu),...,B(2,N−1; uu),B(2,N;uu),
B(2,3; ud),...,B(2,N−1; ud),B(2,N;ud),
B(2,3; du),...,B(2,N−1; du),B(2,N;du),
B(2,3; dd),...,B(2,N−1; dd),B(2,N;dd).We do not require that the ud and du prices coincide, which was the case for
stock prices movements in Section 3.2.
This process continues in the same manner. At each step the length of the
sequence decreases by one and the number of sequences doubles. At timeN− 1
we have just single numbers, 2N−^1 of them,
B(N− 1 ,N;sN− 1 )indexed by all possible statessN− 1 .The tree structure breaks down here be-
cause the last movement is certain: The last bond matures, becoming a sure
dollar at timeN, B(N, N;sN) = 1 for all states.
Example 11.1
A particular evolution of bond prices forN=3,with monthly steps (τ= 121 )
is given in Figure 11.2. The prices of three bonds with maturities 1, 2, and 3
are shown.
The evolution of bond prices can be described be means of returns. Suppose
we have reached statesn− 1 and the bond priceB(n− 1 ,N;sn− 1 ) becomes
known. Then we can write
B(n, N;sn− 1 u) =B(n− 1 ,N;sn− 1 )exp{k(n, N;sn− 1 u)},
B(n, N;sn− 1 d) =B(n− 1 ,N;sn− 1 )exp{k(n, N;sn− 1 d)},