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The payoff is $(S_0{S}_T-S_T^2)1_{\{S_T<S_0\}}$. Subtract the upper-tail truncated lognormal moments from the full moments in part c. With current spot $s=S_t$, the price iswhere $M_1=e^{r\tau }$ and $M_2=e^{(2r+{\sigma }^2)\tau }$. The threshold is the original $S_0$, not the current spot $s$. Since the conditional lognormal distribution has no atoms, the strict endpoint convention does not affect the price for $t<T$.

The logarithm of the geometric average isBy stochastic Fubini theorem, ${\int }_0^T{W}_{u}du={\int }_0^T(T-v)d{W}_v$, a centered Gaussian random variable with variance $T^3/3$. ThereforeThe Geometric Asian option is consequently a call on this lognormal random variable. Completing the square as in part c givesThe 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.