(a) Complete the proof of the no arbitrage lemma for the equality cases.

(b) We know for the put-call-parity that an European call is equivalent to an European put plus a future that have the same strike price and maturity assuming the underlying stock pays no dividends. Write down an explicit portfolio to take advantage of the arbitrage opportunity when ct - pt < St - Kexp(r(T-t). Also, what would the put-call parity be if the stock pays dividend with Dt being the present value of all known dividends paid between now (i.e. time t) and the expiration date. In fact, this also tells us the price of a future contract at time t that expires at a later time T on a stock with price St that paying dividends in a continuous way with annual rate D and the riskless annual rate is r.