An arbitrage is a self-financing strategy with zero initial value and terminal value nonnegative almost surely and strictly positive with positive probability. If an equivalent martingale measure exists, a self-financing portfolio in the stated constant-numeraire setting is a -martingale. Its terminal expectation therefore equals zero. Equivalence preserves the positive-probability event of positive profit, which would make that expectation strictly positive. This contradiction proves no arbitrage in this finite-time finite-state setting.
For the numerical tree use the bank account growing by and discount stock prices accordingly. The risk-neutral up probabilities solve , givingThus the four path probabilities are , respectively.
The American put option has terminal payoffs . At the upper time-one node, both immediate exercise and continuation are zero. At the lower node, exercise gives three while discounted continuation gives . Hence the option values at time one are , and backward induction givesThe lower node should indeed be exercised immediately, since .
The seller's initial replicating hedge holds share and four units in the bank account, costing . At the bad node it is worth , enough for immediate exercise. If the buyer fails to exercise, the remaining European liability costs only : hold share and in the bank account, whose final payoffs are zero at stock price16 and five at stock price10. Switching to that hedge freesas certain profit at time one, worth if banked until time two.
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