$9.($i$)$ The price of an American call on a non$-$dividend$-$paying stock is $$4.$ The stock price is

$$31,$ the strike price is $$30,$ and the expiration date is in $3$ months. The risk$-$free interest rate is $8\%.$ Derive upper and lower bounds for the price of an American put on the same stock with the same strike price and expiration date.

$($ii$)$ Explain carefully the arbitrage opportunities in Problem $9($i$)$ if the American put price is greater than the calculated upper bound.