$(5)(20$ points$)$ Compute widetilde$(E{)}_1[S_3],$widetilde$(E{)}_2[S_3]$ and $.$ Verify that widetilde$(E{)}_1[S_3]=$

widetilde$(E{)}_1[widetilde(E{)}_2[S_3]]$

$(6)(20$ points$)$ Suppose $N=2.$

$($a$)$ You have a $($European$)$ security which pays you $$5$ if the stock price has

jumped up twice and $$0$ otherwise. Use the backward algorithm to find

$V_n$ and ${}_n$ for $n=0,1.$

$($b$)$ You have a $($European$)$ security which pays you $$5$ if the stock price

has jumped up twice and $$0$ otherwise. Use the conditional expectation

method to find $V_n$ for $n=0,1.$

Solve $(5)$ and $(6)$