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: formula_1. 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.