Suppose that NA stock now sells at $$100,$ and the price will either increase by a factor of u $=1\mathrm{.}2$ to $$120$ or fall by a factor d $=0\mathrm{.}8$ to $$80$ at every $6-$month. Suppose that the risk$-$free interest rate is $5\%$ per $6-$month period. Answer the following questions.

$($a$)$ Find the value of European put option with one year maturity and exercise price $$105$ on NA stock described in the two$-$period price tree.

$($b$)$ Find the value of European call option with one year maturity and exercise price $$105$ on NA by applying put$-$call parity relationship.

$($c$)$ If you believe the stock would stay in a narrow range around $$105$ per share in one year, what would be a simple options strategy using a put in $($a$)$ and a call in $($b$)$ to exploit your conviction about the stock price$$s future movement? Find the break$-$even points for that strategy.