 Down-and-out digital call

A digital call option that pays its terminal indicator only if a lower barrier was avoided. When its strike equals the lower barrier, survival already implies the terminal indicator is one. It is then a no-touch option, with alive price $C_d(s,t;B)-(s/B{)}^{1-2r/{\sigma }^2}{C}_d(B^2/s,t;B)$ in the dividend-free Black-Scholes model.