Down-and-out claim

A down-and-out claim expires worthless if the stock reaches a lower barrier before maturity. A matching down-and-in claim plus a down-and-out claim pays the ordinary terminal claim on every path; this is in-out parity.

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.

A European put option that becomes worthless when its continuously monitored lower barrier is touched. The alive state has $s>B$ and no earlier hit. A terminal building block for the Black-Scholes reflection symmetry is $F=C(K)-C(B)+(K-B)C_d(B)$ for $0<B<K$. Its terminal payoff is $(K-S{)}^{+}{1}_{S\ge B}$. The alive price is $F(s,t)-(s/B{)}^{1-2r/{\sigma }^2}F(B^2/s,t)$ and is set to zero after any hit.