Show that, if \(C\) is the price of an American call with exercise price \(K\) and maturity \(T\) on a stock paying a dividend yield of \(q\), and \(P\) is the price of an American put on the same stock with the same strike price and exercise date, then

\[S_{0} e^{-q T}-K

where \(S_{0}\) is the stock price, \(r\) is the risk-free rate, and \(r>0\). (Hint: To obtain the first half of the inequality, consider possible values of:

Portfolio \(A\) : a European call option plus an amount \(K\) invested at the risk-free rate Portfolio B: an American put option plus \(e^{-q T}\) of stock with dividends being reinvested in the stock.

To obtain the second half of the inequality, consider possible values of:

Portfolio \(C\) : an American call option plus an amount \(K e^{-r T}\) invested at the riskfree rate Portfolio D : a European put option plus one stock with dividends being reinvested in the stock.)