 Solution

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.