Risk-neutral measure for the Black-Scholes model

For $S_t=S_0{e}^{\mu t+\sigma W_t}$, shifting Brownian drift bymakes $S_t=S_0{e}^{(r-{\sigma }^2/2)t+\sigma W_t^Q}$ under an equivalent martingale measure $Q$. The discounted stock is then a martingale.

For the payoff $S_T^p$, risk-neutral valuation givesIts delta hedge holds $p{V}_t/S_t$ units of the risky asset.

For a Brownian motion with drift $X$, the process $X_T-X_{T-t}$ on $[0,T]$ has the same law as $X_t$. Thereforehave the same distribution, which equates the corresponding geometric-Brownian lookback payoffs.