= Down-and-out put
{title2=$e^{-r(T-t)}\mathbb E_{\mathbb Q}[(K-S_T)^+\mathbf1_{\{\tau_B>T\}}\mid S_t=s]$}
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 $(K-S)^+\mathbf1_{S\geq B}$. 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.
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