I can say that the valuation of barrier options can be tricky, it is due to the reason that unlike other simpler options they are path-dependent, what it means is that the value of the option at any time not just depends on the underlying at that point, but it also depends on the path that is taken by the underlying (since, if it has crossed the barrier, a barrier event has occurred). Although the classical Black-Scholes approach can not be directly applied, there are several more complex methods that can be used:
-
The simplest way to value barrier options is to use a static replicating portfolio of vanilla options (which can be valued with Black-Scholes), so chosen so as to become identical to the value of the barrier at expiry and at selected discrete points in time along with the barrier. This approach has been first used by Peter Carr and for all types of barrier options it gives closed form prices and replication strategies.
