Problem $1.$ Consider a one period binomial model. Suppose $S_0=1$ at $t=T_0$; and $S_u=2$ and $S_d=\frac{1}{2}$ at time $T_1.$ If we assume the risk free rate $R$ is $1\mathrm{.}2,$ compute the current value of a European put with strike $K=1.$ Please round your answer to $2$ decimals. Give your answer in $\%$ rounded to $2$ decimal places.

Answer: Compute the probability of the two scenarios by $P_u=\frac{R-S_d}{S_u-S_d}=\frac{7}{15},P_d=1-P_u=\frac{8}{15}.$ Then calculate the expected payoff of the put option by $\frac{7}{15}0+\frac{8}{15}\frac{1}{2}=\frac{4}{15}.$ Finally, discount the expected payoff to current value by $\frac{\text{p}\text{a}\text{y}\text{o}\text{f}\text{f}}{R}=\frac{1}{1\mathrm{.}2}\frac{4}{15}=0\mathrm{.}2222$~~$$0\mathrm{.}22.$

Problem $2.$ Compute the number of units of stock we need to short in order to replicate the option in the previous question. Please round your answer to $2$ decimals.

Answer: $0\mathrm{.}33.$ Replicate the option using stock and cash. Consider the two market scenarios, we can list two equations as "option payoff $=$ replication payoff." Then utilizing the equations, we can derive the units of stock.