- Stochastic Interest Rates 251
We can see that the no-arbitrage conditions are satisfied: 0.84%< 0 .99%<
1 .25%, 0 .27%< 0 .52%< 0 .58% and 0.84%< 0 .87%< 1 .01%.
Finally, we find the desired probabilities by a direct application of (11.6):
p∗(0) = 0. 3813 ,p∗(1; u) = 0. 8159 ,p∗(1; d) = 0. 1811 ,see Figure 11.13.
Figure 11.13 Risk-neutral probabilities in Example 11.7A crucial observation about the model is this: Pricing via replication is
equivalent to pricing by means of the risk-neutral probability. This follows from
the No-Arbitrage Principle and applies to any cash flow, even a random one,
where the amounts depend on the states. This opens a way to pricing absolutely
any security by means of the expectation with respect to the probabilities
p∗(n, N;sn).The expectation is computed step-by-step, starting at the last
one and proceeding backwards through the tree.
Example 11.8
Consider a coupon bond maturing atN = 2 with face valueF = 100 and
with coupons equal to 5% of the current value of the bond, paid at times 1
and 2.In particular, the coupon at maturity isC 2 = 5 in each state. Using the
risk-neutral probabilities from Example 11.7, we can find the bond values at
time 1. In the up state we have the discounted value of 105 due at maturity
using the short rater(1; u)∼= 6 .26%,which gives 104.4540. In the same way in
the down state we obtain 104.0865. Now we add 5% coupons, so the amounts
due at time 1 become 109.6767 in the up state and 109.2908 in the down state.
Using the risk-neutral probabilities, we find the present value of the bond:
108. 3545 ∼=(0. 3813 × 109 .6767 + 0. 6187 × 109 .2908)/ 1 .01.
Exercise 11.7
Use the risk-neutral probabilities in Example 11.7 to find the present