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!


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