$(a)$ Suppose you observe the following information regarding several $securities$:

A stock, which pays $no$ dividends over the next two years, trades $at$ $$100$ per share.

A call option with a maturity $of2$ years and a strike price $$95$ trades $at$ $$10.$

A put option with a maturity $of2$ years and a strike price $$95$ trades $at$ $$1.$

A zero$-$coupon bond with $1$ year maturity and face value $of$ $$100$ trades $at$ $$97.$

$In$ the $absenceof$ arbitrage, what would $be$ the price $of$ a two$-$year coupon bond with a coupon

rate $of5\%$ and face value $$100?$ Assume that the coupons are paid annually.

$(b)$ A stock price currently trades $at$ $$100.Itis$ known that next year $it$ will $be$ either $$110or$

$$90.$ The yield curve $is$ flat, and the risk$-$free rate $of$ interest with annual compounding $is$

$10\%$ per annum. Using the binomial model, calculate the value $of$ a derivative, whose payoff

$is$ max$\{2(S_T-$$$95),3($$$102-S_T)\},$ where $S_{T}is$ the stock price next year.