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.

• Then there is another approach in order to study the law of the maximum (or minimum) of the underlying. By this approach explicit (closed form) of prices to barrier options is given.

• The PDE approach is yet another method. The PDE that is satisfied by an out barrier options is the same one that is satisfied by a vanilla option under Black and Scholes assumptions, with extra boundary conditions by which it is demanded that the option becomes worthless when the underlying touches the barrier.

• When it is difficult to obtain an exact formula, Monte Carlo option model can be used to price the barrier options. However, computing the greeks (sensitivities) using this approach is numerically unstable.

• A faster approach to value barrier options is to use finite-differencing techniques in order to diffuse the PDE backwards from the boundary condition (which is the terminal payoff at expiry, in addition to this the condition that the value along the barrier is always 0 at any time). The advantages are given by both explicit finite-differencing methods and the Crank–Nicolson scheme.

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