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 standard barrier option, it comes in four different forms: up-and-in cumulative Parisian option, up-and-out cumulative Parisian option, down-and-in cumulative Parisian option, and down-and-out cumulative Parisian 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, cumulative Parisian 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.

The cumulative Parisian option is also called a delayed barrier option.