The above graph presents a two$-$period binomial model, where the per$-$period risk$-$free

interest rate $r_f$ is ; NOT $\{$:$r_c).$ The strike price of European put option $P$ which

will expire at time $2$ is $40($i$.$e$.,x=40).$ For any given scenario $($UU$,$ UD$,$ DU or DD$),$ put option

payoff at time $2,P_2,$ is $ma{x}_{0,x-S_2}.$

$($a$)(4\%)$ Calculate the up parameter $u$ and the risk$-$neutral probability $p$ for the up move

$($risk$-$free savings account as numeraire$).$

$($b$)(3\%)$ Calculate stock$-$measure $($using stock price as numeraire$)$ for the up move.

$($c$)(3\%)$ Under stock$-$measure $($using stock price as numeraire$),$ calculate $E_1[\frac{P_2}{S_2}](U)$ at time $1.$

$($d$)(3\%)$ Under stock$-$measure $($using stock price as numeraire$),$ calculate $E_1[\frac{P_2}{S_2}](D)$ at time $1.$

$($e$)(4\%)$ Find $P_0($European put option price$)$ using $E_0[\frac{P_1}{S_1}]=\frac{P_0}{S_0}.$