Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 3 / 27I / c / 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 $\mathbb{E}[e^{n\sigma B_T}]=e^{n^2{\sigma }^{2}T/2}$. ThereforeFor $n=1$ this reduces to $S_0$, and for $n=0$ to the time-zero value $e^{-rT}$ of one unit paid at expiry.