254 Mathematics for Finance
Figure 11.15 Data for Exercise 11.10model the cash flow associated with the derivative security can be replicated
using the money market account and a bond with sufficiently long maturity.
The bond does not even have to be the underlying security, since the prices
of various bonds must be consistent. An alternative is to use the risk-neutral
probabilities. The latter approach is often preferable to replication because of
its simplicity. The equivalence of both methods should be clear in view of what
has been said before.
The pricing of complex securities can essentially be reduced to finding the
associated cash flows. Below we present examples of some classical interest rate
contingent claims. We begin with the simplest case of options.
11.3.1 Options
The underlying securities for interest rate options are bonds of various kinds.
Example 11.9
With the bond prices as in Example 11.5 (Figure 11.10), consider a call option
with exercise time 2 and strike priceX=0.99 on a zero-coupon bond maturing
at time 3. Starting with the final payoffs shown in the last column in the table
below, we move back step-by-step, computing the risk-neutral expectations of