- Stochastic Interest Rates 245
11.2 Arbitrage Pricing of Bonds
Suppose that we are given the binomial tree of bond pricesB(n, N;sn)for
a bond maturing at the fixed time horizonN.In addition, we are given the
money market processA(n;sn− 1 ). As was mentioned in the introduction to
this chapter, the prices of other bonds cannot be completely arbitrary. We
shall show that the pricesB(n, M;sn)forM<Ncan be replicated by means
of bonds with maturityNand the money market. As a consequence of the
No-Arbitrage Principle, the prices ofB(n, M;sn) will have to be equal to the
values of the corresponding replicating strategies.
Example 11.5
Consider the data in Example 11.1. At the first step the short rate is deter-
ministic, being implied by the priceB(0,1).The first two values of the money
market account areA(0) = 1 andA(1) = 1. 01 .As the underlying instrument
we take the bond maturing at time 3. The prices of this bond at time 0 and 1
are given in Figure 11.9, along with the prices of the bond maturing at time 2.
We can find a portfolio (x, y),withxbeing the number of bonds of maturity 3
Figure 11.9 Bond prices from Example 11.1andythe position in the money market, such that the value of this portfolio
matches the time 1 prices of the bond maturing at time 2. To this end we solve
the following system of equations
0. 9848 x+1. 01 y=0. 9948 ,
0. 9808 x+1. 01 y=0. 9907 ,obtainingx=1andy∼= 0 .0098. The value of this portfolio at time 0 is
1 ×B(0,3) + 0. 0098 ×A(0)∼= 0 .9824, which isnotequal toB(0,2). The prices
in Figure 11.9 provide an arbitrage opportunity:
- Sell a bond maturing at time 2 for $0.9828 and buy the portfolio constructed
above for $0.9824. - Whatever happens at time 1, the value of the portfolio will be sufficient to
buy the bond back, the initial balance $0.0004 being the arbitrage profit.