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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.

Let $D_K(S,t)=P_d(S,t)$ and $\gamma =1-2r/{\sigma }^2$. The Black-Scholes reflection symmetry gives another solution $(S/B{)}^{\gamma }{D}_K(B^2/S,t)$. Subtracting it enforces absorption:At $S=B$ the terms agree. At expiry the reflected argument satisfies $B^2/S>B>K$, so its digital-put payoff vanishes; the first term supplies the required terminal payoff. These facts and the bounded pricing solution's uniqueness prove the formula. The up-and-out digital put price is zero once knocked out.

If $0<B<K$, every surviving terminal price is already below the strike, so this is a barrier digital put paying one at expiry exactly when the upper barrier has been avoided. To use the Black-Scholes reflection symmetry, cap the free digital-put strike at $B$, rather than continuing to use $K$:Indeed, $D_B$ has terminal payoff one throughout the alive region, while its reflected argument exceeds $B$ and hence has zero terminal payoff. The barrier values again cancel. Using $D_K$ in both terms would generally leave an unwanted reflected terminal payoff near the barrier.

Writing $b=r-{\sigma }^2/2$ makes the discounted survival probability explicit:The price is independent of $K$ in this case. Both cases combine by replacing the free strike with $\min (K,B)$ in the up-and-out digital put formula.

The Black-Scholes reflection symmetry becomes a weighted reflection after taking a logarithm. Put $x=\log (S/B)$ and $v(x,t)=V(B{e}^x,t)$. The multivariable chain rule transforms the equation intoFor $w(x,t)=e^{\gamma x}v(-x,t)$, its left-hand side, after removing the factor $e^{\gamma x}$, isevaluated at $(-x,t)$. Taking $\gamma =-2b/{\sigma }^2$ makes the derivative coefficient $b$ and the additional zeroth-order term zero. Thussatisfies the same Black-Scholes equation.

Put $q=1-2r/{\sigma }^2$. The Black-Scholes reflection symmetry maps $F$ to $(S/B{)}^{q}F(B^2/S,t)$, which solves the same PDE. Their difference vanishes at $S=B$. For $S>B$ at maturity the reflected argument is below $B$, where the terminal function in part ii is zero. Thus the boundary and terminal data of the alive down-and-out put are satisfied:The killed-domain Feynman-Kac formula identifies this solution with the discounted payoff conditional on survival. For $S\le B$, or after any prior barrier hit, the actual price is zero. The reflected difference is a formula on the alive domain $S>B$; extending that algebraic expression to $S<B$ generally gives a nonzero, possibly negative function and is not a valid barrier-option price.

A digital put option extinguished if the stock reaches an upper barrier $B$. Under the dividend-free Black-Scholes model, conditional on not having hit the barrier before the current time, let $0<S<B$, $\tau =T-t>0$, $m=\min (K,B)$, $\gamma =1-2r/{\sigma }^2$, andThen its price isThe Black-Scholes reflection symmetry proves the pricing equation, and the two terms cancel at $S=B$. At expiry the reflected argument is above $B\ge m$, so its payoff is zero, while $S<m$ is equivalent to $S<K$ inside the alive region. Thus the correct free strike is capped at the barrier. For $K\ge B$, the value is the discounted upper-barrier survival probability, independent of $K$, as for a barrier digital put. For $K>B$ its terminal data have a corner discontinuity at the absorbing barrier. Set the price to zero after a hit; the reflected expression on the other side is not a valid contract value.