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.

(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.