= Solution
{c}
A <down-and-out digital call> with strike at the barrier pays one exactly when the barrier is avoided until expiry. Reflection therefore gives its alive price
$$
D_{\mathrm{out}}(S,t)=C_d(S,t;B)-\left(\frac SB\right)^qC_d(B^2/S,t;B),\qquad S>B.
$$
The complementary contract pays one at expiry when the barrier has been hit. Its price is
$$
\boxed{H(t,S)=e^{-r\tau}-D_{\mathrm{out}}(S,t),\qquad S>B\text{ and no previous hit}.}
$$
This is a cash-at-expiry <one-touch option>, not cash paid at the first touch. After the barrier has already been hit, its price is $e^{-r\tau}$ regardless of the current spot. In particular this branch applies immediately for $S\leq B$. Current spot alone does not encode the full path history.
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