Assume we are working in an arbitrage$-$free $N=10-$period binomial model with parame$-$

ters $u=2,d=\frac{1}{2},r=\frac{1}{4}$ and $S_0=8.$

Let $V$ be an American option with intrinsic value process $\{G_n{\}}_n=1^N$ given by

$G_n=\sqrt[\phantom{2}]{S_n}$

for nin$\{1,2,$dots,$N\}.$ The option can not be exercised at time $t=0.$

Find the arbitrage$-$free price $V_0$ of this option. You may assume that the exercise policy

${}^{**}=$inf$\{nN$:$V_n=G_n\}$ discussed in class is indeed the optimal exercise policy, and

that $V_0=V_0({}^{**})$ is$,$ in fact, the arbitrage$-$free price.