Suppose that we know that a certain stock for which will rise or drop in

price by exactly $10\%$ every $6$ months. Today this stock is trading at exactly $$100$ and the

continuously compounded interest rate is $4\%.$

$($e$)$ What is the price of an at$-$the$-$money American Call Option on this stock expiring in

$1$ year?

$($f$)$ What is the price of an at$-$the$-$money American Put Option on this stock expiring in

$1$ year?