 Solution

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 $t<T$.