American Knock In & Knock Out

A knock in & knock out (akiko) option is a European vanilla with two American barriers, one a knock out and one a knock in.

There are two types of KIKO options:

Knock out until expiration

In this KIKO option, the knock in barrier must be hit to activate the underlying vanilla option. In addition, the knock out barrier (which if hit causes the option to be automatically terminated) is valid throughout the lifetime of the option. That is, even if the underlying option is activated (by the knock in barrier being hit first), it can still be extinguished at any time until expiration by the knock out barrier being hit.

The two barriers can be any combination of regular and reverse barriers, i.e., rki+rko, rki+ko, rko+ki or ki+ko. However, if both barriers are on the same side of the underlying asset, the knock in barrier must be defined so as to be between the underlying asset and the knock out barrier.

If during the lifetime of the option, the knock in barrier is hit first (activating the option) and the knock out barrier is not hit, the payout is that of the underlying vanilla option.

Knock out until knock in

In this kiko option, if the knock out barrier is hit first (or the knock in barrier is not hit during the lifetime of the option) the underlying vanilla option is automatically extinguished and there is no payout; if the knock in barrier is hit first, the knock out barrier is no longer valid and the payout is that of the underlying vanilla option.

One of the barriers is regular and one is reverse (i.e., ki+rko or ko+rki), and they must be defined so that the underlying asset is between them.