A stock price is currently $$40.$ Over each of the next two $3-$month periods it is expected to go up by $10\%$ or down by $10\%.$ The risk$-$free interest rate is $12\%$ per annum with continuous compounding. $($a$)$ What is the value of a $6-$month European put option with a strike price of $$42?($b$)$ What is the value of a $6-$month American put option with a strike price of $$42?$