= Black-Scholes equation with constant cash storage costs
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{title2=$V_t+(rS+\delta)V_S+\tfrac12\sigma^2S^2V_{SS}=rV$}
If holding one unit of a commodity costs a constant cash amount $\delta\,dt$ for storage, its gain is $dS-\delta\,dt$. A delta hedge long a derivative and short $V_S$ commodity units has riskless drift $V_t+\tfrac12\sigma^2S^2V_{SS}+\delta V_S$ and value $V-SV_S$. Equating the drift to the bank return gives the displayed <Black-Scholes equation>. This is an additive cash carry term, unlike a proportional dividend or storage yield.
The corresponding pricing diffusion is $d\widehat S=(r\widehat S+\delta)dt+\sigma\widehat S\,dW$. A fixed-strike forward has value $S+\delta(1-e^{-r\tau})/r-Ke^{-r\tau}$, and <put-call parity> gives that same value for a matching call minus put. Replication buys the commodity and a bank reserve equal to the present value of its future storage payments. For $r=0$, the reserve is $\delta\tau$.
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