= Solution
At the upper time-one node, both exercise and continuation values are zero. At the lower node, immediate exercise gives three while continuation gives $2/3$, so exercise is strictly better. At time zero immediate exercise gives zero, whereas continuing gives
$$
\boxed{P_A(0)=\tfrac45\bigl(\tfrac38\cdot0+\tfrac58\cdot3\bigr)=\tfrac32.}
$$
Thus \b[continue at time zero and exercise at time one after a down move]; at the upper node, exercise or continued holding is immaterial. One initial replicating portfolio for these optimal stopping values is $\Delta=-1/6$, $B=4$, costing $3/2$ and producing zero at stock 30 and three at stock 12.
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