 Solution

Return to $r=0$ and $q=1/3$ from part (b). An American put option is valued by the larger of immediate exercise and discounted continuation at each node. At time one with stock price $120$, exercise gives zero, while continuation gives $(1/3)0+(2/3)9=6$. With stock price $90$, exercise gives $27$, and continuation gives $(1/3)9+(2/3)36=27$; the holder is indifferent.

At time zero, exercise gives $17$, while continuation gives $(1/3)6+(2/3)27=20$. ConsequentlyIt is optimal to wait initially and at the up node; at the down node either exercise or waiting is optimal. At this zero interest rate, the European put also costs twenty pounds.