An option (American option) that automatically activates (KI: knocks in) or comes into life only when a certain price of its underlying (disadvantageous to the option seller) is hit at any time before, and including, expiration date. It is a barrier option whereby the holder has the right to exercise at any time insofar as the option has activated anytime over the lifetime of the option. In other words, the barrier option is embedded with an early exercise right. For example, an American up-and in call option becomes active when the up-barrier is breached by the underlying price or the option is early terminated by the holder exercising the right.

The option value of an American KI barrier option constitutes the value of a European KI barrier option and the early exercise premium (the additional price paid to the writer/ seller in return for the early exercise right).