Stock-numeraire measure in the Black-Scholes model
= Stock-numeraire measure in the Black-Scholes model
{title2=$dQ^S/dQ=e^{-\rho T}S_T/S_0$}
For a non-dividend-paying <stock> in the <Black-Scholes model>, use density $dQ^S/dQ=e^{-\rho T}S_T/S_0$. 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 $e^{-\rho T}\mathbb E_Q[S_TH]=S_0\mathbb E_{Q^S}H$. This <change of numeraire> is particularly useful for payments delivered in <stock> units.