A stock price is currently $$50.$ Over each of the next two three$-$month periods it is expected to go up $6\%$ or down $5\%.$ The risk$-$free interest rate is $5\%$ per annum with continuous compounding. What is the value of a $6$ month european put with a strike price of $$51?$ Show the binomial tree and your main steps to calculate. If the put option were american, would it ever be optimal to exercise early?