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A put with a prescribed exercise rule: exercise at the first touch of $B<K$, or at expiry if no touch occurs. Its value is bounded above by the American put option, which optimizes over exercise times. With zero interest rate, its value equals the down-and-out put value plus $(K-B)$ times a cash-at-expiry one-touch option value. At positive interest rate the rebate is paid at touch, so replacing it by an expiry payment would generally be incorrect.

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.

Let ${\tau }_B=\inf \{u\ge t:S_u\le B\}$. While the contract is alive, the down-and-out put pays $(K-S_T{)}^{+}{1}_{\{{\tau }_B>T\}}$ at maturity; a prior touch has already knocked it out.

The truncated terminal payoff can be represented by the ordinary European-price combinationAt maturity its three regions areIndeed it is zero below $B$, equals $K-S$ between $B$ and $K$, and is zero above $K$. This supplies the terminal building block for reflection.

The zero-below-barrier property is a property of the terminal payoff. Before maturity, $F(S,t)$ need not vanish for $S<B$, because the ordinary European contracts can still finish above $B$. The killed barrier option price, rather than this unrestricted European combination, is zero after a touch.

If the barrier is reached, path continuity gives $S_{{\tau }_B}=B$ and the early payment is $K-B$. If it is not reached, the terminal payment is exactly the down-and-out put payoff. When $r=0$, moving a fixed payment from the hitting time to expiry does not change its value. Hence the barrier-triggered put price isHere $G,H$ are evaluated with zero interest rate. If the barrier has already been hit, the exploding put has been exercised and has no remaining value after its payment; the displayed replication concerns an initially alive contract and its cash flows.