Past exam of the martingales and financial mathematics course of the University of Oxford / 2010 / B / 6

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.

Use the auxiliary diffusion with generator $(rS+\delta ){\partial }_S+({\sigma }^2{S}^2/2){\partial }_{SS}$. Applying Itô's formula to $e^{-r(v-t)}V(v,{\hat{S}}_v)$ makes its drift zero by part (a), leaving only $e^{-r(v-t)}\sigma {\hat{S}}_v{V}_{S}d{W}_v$. Under the usual integrability conditions, localization and expectation give the Feynman-Kac formulaFor polynomial-growth payoffs, the linear diffusion has the finite moments needed for this martingale argument. Its pricing drift includes the cash storage cost as an additive term.

The conditional mean solves $m^{\prime }(v)=rm(v)+\delta$ with $m(t)=S_t$. Hence, for $r\ne 0$,The forward contract's terminal payoff is $S_T-K$, so discounting this expectation gives its current valueFor $r=0$, the continuous limits are $S_t+\delta \tau$ for the mean and $S_t+\delta \tau -K$ for the contract value. A fair delivery price, as distinct from the current contract value, is the strike making this value zero: $K_{\ast }=S_t{e}^{r\tau }+\delta (e^{r\tau }-1)/r$.

At expiry, the difference of a European call and put with matching strike is $(S_T-K{)}^{+}-(K-S_T{)}^{+}=S_T-K$. Linearity of pricing therefore makes their difference equal the forward-contract value from part (c):A direct put-call parity replication gives the same result. Buy one unit of the commodity and deposit $\delta {\int }_t^T{e}^{-r(u-t)}du$ in the bank. Withdraw the fixed cash storage payments from this reserve; the reserve is exhausted at expiry, leaving one commodity unit. Its initial cost is $S+\delta (1-e^{-r\tau })/r$. Borrow $K{e}^{-r\tau }$ as well, leaving terminal value $S_T-K$. The law of one price proves the parity identity. If $r=0$, replace the storage reserve by $\delta \tau$.