= Solution
{c}
Using the <risk-neutral probability> $q=2/3$ from part <1/c>, the <stock> tree has values $9$ initially, $12,6$ after one step, and $16,8,4$ after two. The <American put option> has terminal <financial payoffs> $0,1,5$. Work backwards, comparing the immediate <exercise payoff> with its discounted continuation value.
At $S_1=12$, continuation is $(3/10)(2\cdot0+1)=3/10$, exceeding the zero exercise payoff. At $S_1=6$, continuation is $(3/10)(2\cdot1+5)=21/10$, while exercise pays three. Therefore \b[exercise at the down node $S_1=6$ and continue at $S_1=12$]. The initial exercise payoff is zero and the continuation value is
$$
\boxed{P_0=\frac3{10}\left(2\cdot\frac3{10}+3\right)=\frac{27}{25}=1.08.}
$$
The <Snell envelope> gives this backwards maximum recursion. For comparison, restricting exercise to expiry would give $81/100$, so early exercise genuinely adds value.
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