= Solution
{c}
Subtract the reflected solution from part <2/c>. On the alive side of the <down-and-out barrier option>,
$$
\boxed{V_B(S,t)=V(S,t)-(S/B)^{1-2r/\sigma^2}V(B^2/S,t),\qquad S>B.}
$$
Both terms satisfy the <Black-Scholes equation>, and at $S=B$ they cancel. At expiry, $S\geq B$ gives $B^2/S\leq B<K$, so the reflected <log-call option> has zero terminal <financial payoff>; the first term gives the required payoff. With the usual growth condition, uniqueness for the killed pricing problem identifies the expression as the <risk-neutral price>. After the barrier has been hit the contract value is zero. The algebraic image expression on $S<B$ is not its financial value.
Back to article page
