In a two$-$step binomial tree model with a one$-$step interest rate of $r=0\mathrm{.}05,$

$S_0=100,u=1\mathrm{.}2,d=0\mathrm{.}8,$ and $t=1,$ consider a contingent claim that expires

after two years. The payoff is $S_T^2*1_{\{S_T)}>90.$

$($a$)$ Find the initial price and the replication strategy of the European version

of the option.

$($b$)$ Find the initial price and the replication strategy of the American version

of the option.