A Double Knock-Out Is Worth Less Than Half the Two One-Sided Options
The sharp statement is stronger than the usual one: on every path, missing the ceiling plus missing the floor counts the paths that miss both exactly twice, so the pair is twice the double plus the value of the one-sided survivors. That is a polynomial identity in indicators, so it holds under every pricing measure with no volatility anywhere, and one half is the tight bound. In a worked instance the pair is 10.317 against a double of 1.494, and the fastest refutation of the trap is that the pair exceeds the plain call with no barriers at all.