 Solution

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.