Stock-numeraire measure in the Black-Scholes model ($d{Q}^S/dQ=e^{-\rho T}{S}_T/S_0$)

For a non-dividend-paying stock in the Black-Scholes model, use density $d{Q}^S/dQ=e^{-\rho T}{S}_T/S_0$. Under $Q^S$, $W_t^S=W_t^Q-\sigma t$ is Brownian motion, and $\log S_t=\log S_0+(\rho +{\sigma }^2/2)t+\sigma W_t^S$. Thus $e^{-\rho T}{\mathbb{E}}_Q[S_{T}H]=S_0{\mathbb{E}}_{Q^S}H$. This change of numeraire is particularly useful for payments delivered in stock units.