= Solution
{c}
The payoff is $(S_0S_T-S_T^2)\mathbf1_{\{S_T<S_0\}}$. Subtract the upper-tail <truncated lognormal moments> from the full moments in part <2/c>. With current spot $s=S_t$, the price is
$$
\boxed{e^{-r\tau}\left[S_0s\{M_1(\tau)-V_1(t,s,S_0)\}
-s^2\{M_2(\tau)-V_2(t,s,S_0)\}\right],}
$$
where $M_1=e^{r\tau}$ 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 $t<T$.
Back to article page
