Down-and-out put ($e^{-r(T-t)}{\mathbb{E}}_{\mathbb{Q}}[(K-S_T{)}^{+}{1}_{\{{\tau }_B>T\}}\mid S_t=s]$)

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.