Solution
= Solution
Risk-neutral valuation gives
$$
V_0=e^{-rT}\mathbb E_Q[S_T^p].
$$
Since $W_T^Q\sim N(0,T)$,
$$
\mathbb E_Q[S_T^p]
=S_0^p
\exp\left[p\left(r-\frac12\sigma^2\right)T
+\frac12p^2\sigma^2T\right].
$$
Hence the <Power payoff in the Black-Scholes model> has price
$$
\boxed{
V_0=S_0^p
\exp\left(\left((p-1)r+\frac12p(p-1)\sigma^2\right)T\right)
}.
$$