 Black-Scholes equation with constant cash storage costs

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 $V_t+\frac{1}{2}{\sigma }^2{S}^2{V}_{SS}+\delta V_S$ and value $V-S{V}_S$. 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 $d\hat{S}=(r\hat{S}+\delta )dt+\sigma \hat{S}dW$. A fixed-strike forward has value $S+\delta (1-e^{-r\tau })/r-K{e}^{-r\tau }$, 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 $\delta \tau$.