For events $E_n$, the Borel-Cantelli first lemma states that ${\sum }_n\mathbb{P}(E_n)<\infty$ implies $\mathbb{P}(E_n\text{ infinitely often})=0$. Indeed,These unions decrease to $\mathrm{limsup}{E}_n$, so continuity from above proves the assertion.

The Borel-Cantelli second lemma states that independent events with ${\sum }_n\mathbb{P}(E_n)=\infty$ satisfy $\mathbb{P}(E_n\text{ infinitely often})=1$. For every fixed $m$, independence and $1-x\le e^{-x}$ giveContinuity from above shows there is probability zero of no event after $m$. A countable union over $m$ gives probability zero of only finitely many occurrences. Independence is needed for this second conclusion, but not for the first.

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 $\delta \tau$.

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: $K_{\ast }=S_t{e}^{r\tau }+\delta (e^{r\tau }-1)/r$.

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 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.

Put $\tau =T-t$. Conditional on present information, $J={\int }_t^T(W_u-W_t)du$ is independent of the past and is normal with mean zero and variance ${\tau }^3/3$. Part (c) givesThe future drift $r$ in this auxiliary stock is precisely the risk-neutral drift. Discounting the conditional expectation and applying the normal exponential-moment formula yieldsAt expiry this reduces to $A_T^2$; at time zero it gives $S_0^2{e}^{{\sigma }^{2}T/6}$. This is a special case of conditional power moments of a continuous geometric Asian average.

For $v\le t$, the two stock paths agree, so their geometric averages agree. For $v>t$, the future log-price isSplit the average's log-integral into $[0,t]$ and $[t,v]$. The first part is $t\log A_t$, while integration of the displayed formula givesDividing by $v$ and exponentiating proves