# Change of numeraire/probability when asset pays dividends

So I was looking at Margrabe's formula for exchange call options in the book 'Mathematical Methods for Financial Markets' (Jeanblanc, Chesney, Yor), and I was having trouble justifying their change of numeraire in Section 2.7.

For two stocks: $$dS_t^i = (r - q_i)dt + \sigma_idW_t^i \quad \text{i = 1, 2}$$, a change of measure is made from the $$\mathbb Q$$ measure to a new measure $$\mathbb Q^*$$, where the numeraire is the asset $$S^2$$. Here $$r$$ si the risk-free rate and $$q_i$$ is the dividend yield of the $$i$$th asset.

So normally using the change of numeraire technique I would write: $$\left. \frac{d\Bbb{Q^*}}{d\Bbb{Q}} \right\vert_{\mathcal{F}_t} = \frac{B_0 S^2_t }{ B_t S^2_0 }$$, where $$B$$ is the money market account. So $$B_0 = 1$$ and $$B_t = e^{rt}$$, and this makes the Radon-Nikodym derivative as $$\left. \frac{d\Bbb{Q^*}}{d\Bbb{Q}} \right\vert_{\mathcal{F}_t} = e^{-rt}\frac{S^2_t }{S^2_0 }$$.

But according to the book the correct R-N derivative is actually: $$\left. \frac{d\Bbb{Q^*}}{d\Bbb{Q}} \right\vert_{\mathcal{F}_t} = e^{-(r-q_2)t}\frac{S^2_t }{S^2_0 }$$. So basically the dividend yield of $$S^2_t$$, $$q_2$$ is now involved. I couldn't figure out where it came from since the money market account doesn't involve $$q_2$$. I was thinking maybe they are using $$e^{q_2t}S^2_t$$ as the numeraire instead, but I feel like they would explicitly say that instead of just saying the numeraire is $$S^2$$.

I'm not sure what I'm missing here so any help would be appreciated. I checked a few books and a lot of them say to derive the dividend case as an exercise, or use a different method, etc... so I haven't been able to pinpoint what's going on. Thanks!