The lower bound for the price of an American put option is P(S(t), t) ≥ max(E−S(t), 0), while the lower bound for the price of a European put option is p(S(t), t) ≥ max(E e−^r(T −t)− S(t), 0).
(a) Suppose that E > S(t) and that P(S(t), t) < E − S(t) where P(S(t), t) is the price of an American put option. Show that arbitrage profits are possible by describing the arbitrage trades. (This is intended to be easy – no borrowing or lending is needed, and the profits occur immediately.)
(b) Suppose instead that E e−^r(T −t) > S(t) and that p(S(t), t) < E e−^r(T −t) − S(t) where p(S(t), t) is the price of a European put option. Show that arbitrage profits are possible by describing the arbitrage trades. (It is important to remember that the European put can be exercised only at maturity T.