A barrier option is a family of options which only have value if the underlying stock does not cross some set price level.
There are several types of barrier option including: - Down-and-out options - Down-and-in options - Up-and-out options - Up-and-in options
A down-and-out European call with strike \(K\), maturity \(T\) and barrier level \(L\) has payoff function given by \[ h(S_{[0,T]})=(S_{T}-K)_{+}\mathbb{1}_{\{\min_{t\in[0,T]}S(t)>L\}} \]