- Stochastic Interest Rates 247
Exercise 11.5
Evaluate the prices of a bond maturing at time 2 given a tree of prices
of a bond maturing at time 3 and short rates as shown in Figure 11.12,
withτ=1/12.Figure 11.12 Bond prices and short rates in Exercise 11.5We can readily generalize Example 11.5. The underlying bond matures at
timeNand we can find the structure of prices of any bond maturing atM<N.
The replication proceeds backwards step-by-step starting from timeM,for
whichB(M, M;sM) = 1 in each statesM. The first step is easy: for each
statesM− 1 we take a portfolio withx=0andy=1/A(M;sM− 1 ),since the
bond becomes risk free one step prior to maturity.
Next, consider timeM−2. For any statesM− 2 we findx=x(M−1;sM− 2 ),
the number of bonds maturing atN,andy=y(M−1;sM− 2 ), the position in
the money market, by solving the system
xB(M− 1 ,N;sM− 2 u) +yA(M−1;sM− 2 )=B(M− 1 ,M;sM− 2 u),
xB(M− 1 ,N;sM− 2 d) +yA(M−1;sM− 2 )=B(M− 1 ,M;sM− 2 d).In this way we can find the prices at timeM−2 of the bond maturing at
timeM,
B(M− 2 ,M;sM− 3 u) =xB(M− 2 ,N;sM− 3 u) +yA(M−2;sM− 3 ),
B(M− 2 ,M;sM− 3 d) =xB(M− 2 ,N;sM− 3 d) +yA(M−2;sM− 3 ).We can iterate the replication process moving backwards through the tree.
Remark 11.3
Replication is possible if a no-arbitrage condition analogous to Condition 3.2
is satisfied for the binomial tree. Here the conditionu>r>dof Chapter 3 is