246 Mathematics for Finance
The model in Example 11.1 turns out to beinconsistent with the No-Arbitrage
Principleand has to be rectified. We can only adjust some of the future prices,
since the present prices of all bonds are dictated by the market. It is easy to
see that by takingB(1,2; u) = 0.9958 withB(1,2; d) unchanged, or by letting
B(1,2; d) = 0.9913 withB(1,2; u) unchanged, we can eliminate the arbitrage
opportunity. Of course, there are many other ways of repairing the model by a
simultaneous change of both values ofB(1,2). Let us putB(1,2; d) = 0. 9913
and leaveB(1,2; u) unchanged. The rectified tree of bond prices is shown in
Figure 11.10 and the corresponding yields in Figure 11.11.
Figure 11.10 Rectified tree of bond prices in Example 11.5Figure 11.11 Rectified tree of yields in Example 11.5Remark 11.2
The process of rectifying bond prices in Example 11.5 bears some resemblance
to the pricing of general derivative securities described in Chapter 8. The role
of the derivative security is played by the bond maturing at time 2. The bond
maturing at time 3 plays the role of the underlying security. The difference
is that the present price of the bond of maturity 2 is fixed and we can only
adjust the future prices in the model to eliminate arbitrage. At this stage we
are concerned only with building consistent models rather than with pricing
securities.