A barrier option whereby the holder has the right to exercise at any time insofar as the option has not expired. In other words, the barrier option is embedded with an early exercise right. For example, an American up-and out call option vanishes away 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 barrier option constitutes the value of a European barrier option and the early exercise premium (the additional price paid to the writer/ seller in return for the early exercise right).