(i) Consider a single-period binomial model of a riskless asset (asset 0 ), worth 1 at time 0 and $1+r$ at time 1 , and a risky asset (asset 1 ), worth 1 at time 0 and worth $u$ at time 1 if the period was good, otherwise worth $d$. Assuming that

$$d<1+r<u$$

show how any contingent claim $Y$ to be paid at time 1 can be priced and exactly replicated. Briefly explain the significance of the condition $(*)$, and indicate how the analysis of the single-period model extends to many periods.

(ii) Now suppose that $u=5 / 3, d=2 / 3, r=1 / 3$, and that the risky asset is worth $S_{0}=864=2^{5} \times 3^{3}$ at time zero. Show that the time- 0 value of an American put option with strike $K=S_{0}$ and expiry at time $t=3$ is equal to 79 , and find the optimal exercise policy.