= Solution
For the non-dividend-paying <stock>, a <self-financing portfolio> replicating a smooth <European option> has $\Delta=V_S$ shares and $V-SV_S$ in the <bank account>. Its gain satisfies
$$
dV=V_S\,dS+r(V-SV_S)\,dt.
$$
The <Itô formula> gives $dV=(V_t+\mu SV_S+\sigma^2S^2V_{SS}/2)dt+\sigma SV_SdW$. Matching its diffusion and drift with the portfolio gains cancels $\mu$, yielding
$$
\boxed{V_t+\frac12\sigma^2S^2V_{SS}+rSV_S-rV=0}.
$$
The <financial payoff> supplies the terminal value. This is the <Black-Scholes equation>; the associated <risk-neutral measure> replaces the physical stock drift by $r$.
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