Solution
= Solution
A <power option> need not have bounded payoff, but all real powers of a positive lognormal stock have finite moments, so risk-neutral valuation still applies. The <Gaussian> moment-generating identity gives $\mathbb E[e^{n\sigma B_T}]=e^{n^2\sigma^2T/2}$. Therefore
$$
\boxed{V(0,S_0)=S_0^n\exp\left(\left[(n-1)r+\tfrac12n(n-1)\sigma^2\right]T\right).}
$$
For $n=1$ this reduces to $S_0$, and for $n=0$ to the time-zero value $e^{-rT}$ of one unit paid at expiry.