= Solution
{c}
The <logarithm> of the geometric average is
$$
\log\overline S_T=\log S_0+\frac12(r-\sigma^2/2)T+\frac\sigma T\int_0^T W_u^{\mathbb Q}\,du.
$$
By <stochastic Fubini theorem>, $\int_0^TW_u\,du=\int_0^T(T-v)\,dW_v$, a centered <Gaussian random variable> with <variance> $T^3/3$. Therefore
$$
\log((\overline S_T)^p)\sim N(m,v),\qquad
m=p\left[\log S_0+\frac12(r-\sigma^2/2)T\right],\quad v=\frac{p^2\sigma^2T}{3}.
$$
The <Geometric Asian option> is consequently a call on this <lognormal random variable>. Completing the square as in part <2/c> gives
$$
\boxed{e^{-rT}\left[e^{m+v/2}\Phi(d_1)-K\Phi(d_2)\right],\qquad
d_2=\frac{m-\log K}{\sqrt v},\quad d_1=d_2+\sqrt v.}
$$
The averaging changes both the log-mean and log-variance; replacing $S_T$ by an ordinary stock price at a shorter maturity without these adjustments would give the wrong formula.
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