A barrier option is a vanilla option with a trigger: a price level (the barrier, B) that, if touched by the underlying, either activates the option (knock-in) or kills it (knock-out). Four basic shapes exist depending on whether the barrier sits below or above spot and whether it switches the option on or off: down-and-out, down-and-in, up-and-out, up-and-in. Because a knock-out can expire worthless even when it finishes in the money, and a knock-in may never activate at all, a barrier option is always *cheaper* than the otherwise-identical vanilla. You are selling away some of the payoff distribution in exchange for a lower premium.