= Call delta equation for a driftless local volatility diffusion
{title2=$U=V_S$}
In the cash-numéraire market $dS_t=a(S_t)dW_t$, let a smooth call value solve $V_t+\tfrac12a^2V_{SS}=0$. Its <delta hedge> is $U=V_S$. Differentiating the pricing equation gives
$$
U_t+aa'U_S+\frac12a^2U_{SS}=0,\qquad U(T,S)=\mathbf1_{\{S\ge K\}},
$$
with the terminal value at the strike understood up to the null probability of the diffusion hitting that exact terminal value. By <Itô formula>, $dU(t,S_t)=-aa'U_Sdt+aU_SdW_t$, while $dV(t,S_t)=U(t,S_t)dS_t$. Thus holding $U(t,S_t)$ shares and cash $V-SU$ replicates the terminal call. The drift in the <option delta> equation comes from differentiating the spatially varying diffusion coefficient, not from a drift in the original <stock>.
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