Problem $1.($Knock$-$out Option$)$ Consider a derivative contract that gives the

holder of this contract the option to buy the stock for $20$ dollars at time $t=3,$

provided that the stock price never reaches $20$ dollars or lower, at any time from

$t=0$ to $t=3.$ If the stock price ever reaches $20$ dollars or lower, the option becomes

worthless, and that means the holder of the contract will no longer have the option

to buy the stock at time $t=3.$