a$)$ Consider a $3-$period binomial model of pricing American option. Let the initial stock price be

$S_0=10$ per share, $u=2$ be up factor, $d=0\mathrm{.}5$ be down factor, $r=0\mathrm{.}25$ be rate of interest per time

period, $K=12$ be strike price.

$($i$)$ Find the initial price of the American put option.

$($ii$)$ Compute the optimal stopping time ${}^{**}.$ Compute the expected pay$-$off of the put option with

${}^{**}$ exercise policy.

$($iii$)$ Let $$ denote a exercise policy and defined as follows: $(HHH)=,(HHT)=,(HTH)=$

$2,(HTT)=2,(THH)=1,(THT)=1,(TTH)=1,(TTT)=1.$ Check whether $$ is a

stopping time and compute the expected pay$-$off of the put option with $$ exercise policy.