{title2=$\left.\frac{dQ^M}{dQ^N}\right|_{\mathcal F_t}=\frac{M_t/N_t}{M_0/N_0}$}

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.


 Change of numeraire (source code)