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
$$
S_0 e^{-q T}-K<C-P<S_0-K e^{-r T},
$$
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.)