A stock price is currently $$100.$ Over each of the next $1-$month period, it is expected to go up by $10\%$ or down by $10\%.$ Assuming a $2-$month American put written on the stock with a strike price of $$101,$ would it be optimal to exercise it early in one month? What is the price of the American option? Assume the risk$-$free rate r is $5\%.$