= Solution
{c}
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:
$$
\boxed{H(S,t)=D_K(S,t)-(S/B)^\gamma D_K(B^2/S,t),\qquad0<S<B,\quad B>K.}
$$
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.
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