15.12. 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 S0e−qT − K < C − P < S0 − Ke−rT, where S0 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−qT 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 Ke−rT invested at the risk-free rate Portfolio D: a European put option plus one stock with dividends being reinvested in the stock.]