= Solution
{c}
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 $(1/3)6+(2/3)27=20$. Consequently
$$
\boxed{V_0^{\mathrm{American}}=20\text{ pounds}.}
$$
It 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.
\Image[/past-exam-of-the-martingales-and-financial-mathematics-course-of-the-university-of-oxford/2011-american-put-tree.png]
{title=Backward induction for the American put when the risk-free multiplier is one and the up probability is one third. Each node shows the stock price and option value.}
Back to article page
