= Solution
The terminal <European put option> payoffs are zero at forty-five, thirty-six and sixteen, and five at ten. At the good first node its value is zero. At the bad node its continuation value is $(4/5)[(5/6)0+(1/6)5]=2/3$, below the exercise payoff three, so optimal exercise occurs there. At time zero the continuation value is $(4/5)[(3/8)0+(5/8)3]=\boxed{3/2}$, exceeding the immediate payoff zero.
To hedge one sold <American put option>, short $1/6$ of a share and put four in the bank at time zero. The portfolio costs $-15/6+4=3/2$, exactly the premium. At time one the bank holds five. After a good move the short position costs five to close, exhausting the portfolio and liability. After a bad move it costs two to close, leaving three for the exercised put. Thus after two bad periods the rational buyer already exercised at time one, and \b[the hedge pays three and closes then; there is no remaining time-two obligation]. Physical delivery gives the same accounting by immediately selling the stock received on exercise.
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