Barrier Options are non$-$vanilla options contracts whose payoffs also depend on

whether the price attained by the underlying crossed a certain barrier level before expiration.

For example, an up$-$and$-$out call option is similar to a vanilla call option except that the payoff

will be zero if the price of the underlying security ever goes above the barrier level before

expiration. Other barrier options are up$-$and$-$in$,$ down$-$and$-$out, and down$-$and$-$in$.$ Assume that we have the same underlying stock $$100$

$($a$)$ Why would you want to buy any of the barrier call option variants instead of a vanilla

call option?

$($b$)$ Suppose you are interested in an European up$-$and$-$out call option with a barrier of

$$109$ and a strike of $$95.$

$($i$)$ What is the payoff of this option?

$($ii$)$ What is the price of this option?

$($iii$)$ What is the price of the corresponding vanilla option $($i$.$e$.,$ without the barrier$)?$

$($iv$)$ What should the price of the corresponding up$-$and$-$in call be$?$ An up$-$and$-$in has

the same payoff as the vanilla option if the price crossed the barrier, otherwise

the payoff is zero.