Solution
= Solution
{c}
If $p>0$ and $K>0$, the payoff is positive exactly when $S_T>k:=K^{1/p}$. Splitting it into two truncated moments gives
$$
\boxed{e^{-r\tau}\left[s^pV_p(t,s,K^{1/p})-K V_0(t,s,K^{1/p})\right].}
$$
For $p=0$, the payoff is the constant $(1-K)^+$, so its price is $e^{-r\tau}(1-K)^+$. For $K=0$ and $p>0$, its price is $e^{-r\tau}s^pM_p(\tau)$. These boundary cases avoid taking $K^{1/p}$ when $p=0$.