Bounds on European and American puts
(a) Using put-call parity and bounds on a European call, or otherwise, prove that the price of a European put on a stock paying no dividends satisfies
(b) Prove that the price of an American put on the non-dividend paying stock satisfies
(c) Give an example to show that the American put can be worth more than the European put, that is, where immediate exercise of the American gives payout at time T strictly greater than the payout of the European.