= Solution
{c}
The commodity's total gain from holding one unit is $dS-\delta\,dt$, because the storage cost is a fixed cash rate, not a proportional dividend yield. Form a locally self-financing delta hedge long the derivative and short $\Delta=V_S$ units of the commodity. <Itô's formula> eliminates Brownian risk and leaves drift
$$
V_t+\tfrac12\sigma^2S^2V_{SS}+\delta V_S.
$$
Its value is $V-SV_S$, so absence of arbitrage makes that drift equal to $r(V-SV_S)$. Therefore the <Black-Scholes equation with constant cash storage costs> is
$$
\boxed{V_t+(rS+\delta)V_S+\tfrac12\sigma^2S^2V_{SS}=rV,\qquad V(T,S)=\Psi(S).}
$$
The cash cost enters as $\delta V_S$, rather than as $\delta SV_S$, because one unit costs $\delta$ per unit time regardless of its current price.
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