= Solution
{c}
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) gives
$$
\widetilde A_T^2=A_t^{2t/T}S_t^{2\tau/T}
\exp\left[(r-\sigma^2/2)\frac{\tau^2}{T}+\frac{2\sigma}{T}J\right].
$$
The future drift $r$ in this auxiliary stock is precisely the risk-neutral drift. Discounting the conditional expectation and applying the normal exponential-moment formula yields
$$
\boxed{V^A(t,S_t,A_t)=e^{-r\tau}A_t^{2t/T}S_t^{2\tau/T}
\exp\left[\left(r-\frac{\sigma^2}{2}\right)\frac{\tau^2}{T}
+\frac{2\sigma^2\tau^3}{3T^2}\right].}
$$
At expiry this reduces to $A_T^2$; at time zero it gives $S_0^2e^{\sigma^2T/6}$. This is a special case of <conditional power moments of a continuous geometric Asian average>.
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