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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 driftIts value is $V-S{V}_S$, so absence of arbitrage makes that drift equal to $r(V-S{V}_S)$. Therefore the Black-Scholes equation with constant cash storage costs isThe cash cost enters as $\delta V_S$, rather than as $\delta S{V}_S$, because one unit costs $\delta$ per unit time regardless of its current price.