At time zero, exercise pays , while continuation is worth
(1)
Therefore exercise at time zero is optimal, and pounds.
The discounted unexercised option-value process is not a Q-martingale: already . It is the Snell envelope of the discounted exercise payoff, hence a supermartingale in general, with strict decrease of expected value when immediate exercise beats continuation. Stopping at an optimal exercise time restores the stopped martingale property; that is a different assertion from the unexercised price being a martingale.
Figure 1.
The American arithmetic-average put: four terminal histories are retained because ud and du have different average prices. Backward induction selects immediate exercise at both time-one nodes and at time zero.
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