 Solution

Apply part (a) under the risk-neutral log-price drift $r-{\sigma }^2/2$, integrate the maximum and minimum tails using the supplied identities, and simplify the two exponential moments. Fubini then identifies the residual term as $({\sigma }^2/2){\int }_0^T{S}_{t}dt$. Discounted expectations on both sides give exactly the stated pricing identity.