A barrier option that knocks in or out after the underlying spends a preset amount of time above or below the barrier. It belongs to a class of contingent claims based on occupation times. Like a standard barrier option, it comes in four different forms: up-and-in delayed barrier option, up-and-out delayed barrier option, down-and-in delayed barrier option, and down-and-out delayed barrier option.
In comparison with step option, which knocks in or out gradually, amortizing its principal for each unit of time the underlying is above or below the barrier, delayed barrier option allows its holder to receive either the full payoff from an otherwise identical vanilla option if the occupation time never exceeded the fraction of the option’s lifetime, or nothing.
Delayed barrier option is also called cumulative Parisian option.
