Change of numeraire (${\frac{d{Q}^M}{d{Q}^N}|}_{{\mathcal{F}}_t}=\frac{M_t/N_t}{M_0/N_0}$)

If $M,N$ are strictly positive traded numeraires and $M/N$ is a mean-one martingale after normalizing its initial value under $Q^N$, this ratio defines $Q^M$. Conditional change of measure then makes every $Q^N$-martingale asset price $S/N$ into the corresponding $Q^M$-martingale $S/M$, when the required integrability holds. In particular a martingale exchange rate under one currency's bank-account measure has a reciprocal martingale under the other currency's measure when their interest rates coincide.

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.