$(a)$ Suppose that you can trade calls and puts $in$ the market. You want $to$ generate the following payoff structure, where $S_{T}is$ the underlying stock price $in$ year $T.$

$(i)$ Create a trading strategy that replicates the payoff $by$ trading calls only.

$(ii)$ Create a trading strategy that replicates the payoff $by$ trading puts only.

$(b)$ Let $C(x)$ denote the call price with strike price $x.$ Which $of$ the following statements $is$ correct? Briefly explain.

$(A)C(100)+C(200)>C(120)+C(180)$

$(B)C(100)+C(200)=C(120)+C(180)C(100)+C(200)C(120)+C(180)C(100)+C(200)$

$(C)C(100)+C(200)=C(120)+C(180)$

$(D)$ Cannot tell whether $C(100)+C(200)orC(120)+C(180)is$ higher.