= Solution
{c}
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):
$$
\boxed{C(t,S)-P(t,S)=S+\frac\delta r(1-e^{-r(T-t)})-Ke^{-r(T-t)}.}
$$
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 $Ke^{-r\tau}$ 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 $\delta\tau$.
Back to article page
