spot price falls below $120 again. By in-out parity, we mean that the combination of one "in" and one "out" barrier option with the same strikes and expirations yields the price of the corresponding vanilla option: . Note that before the knock-in/out event, both options have positive value, and hence both are strictly valued below the corresponding vanilla option. After the knock-in/out event, the knock-out option is worthless and the knock-in option's value coincides with that of the corresponding vanilla option. At maturity, exactly one of the two will pay off identically to the corresponding vanilla option, which of the two that depends on whether the knock-in/out event has occurred before maturity. Barrier events A barrier event occurs when the underlying crosses the barrier level. While it seems straightforward to define a barrier event as "underlying trades at or above a given level," in reality it's not so simple. What if the underlying only trades at the level for a single trade? How big would that trade have to be? Would it have to be on an exchange or could it be between private parties? When barrier options were first introduced to options markets, many banks had legal trouble resulting from a mismatched understanding with their counterparties regarding exactly what constituted a barrier event. Variations Barrier options are sometimes accompanied by a rebate, which is a payoff to the option holder in case of a barrier event. Rebates can either be paid at the time of the event or at expiration. A discrete barrier is one for which the barrier event is considered at discrete times, rather than the normal continuous barrier case. A Parisian option is a barrier option where the barrier condition applies only once the price of the underlying instrument has spent at least a given period of time on the wrong side of the barrier. A turbo warrant is a barrier option namely a knock out call that is initially in the money and with the barrier at the same level as the strike. Barrier options can have either American, Bermudan or European exercise style. Valuation The valuation of barrier options can be tricky, because unlike other simpler options they are path-dependent – that is, the value of the option at any time depends not just on the underlying at that point, but also on the path taken by the underlying (since, if it has crossed