# Value of an option to exchange an asset for another

I'm working out the examples in the paper "Changes of Numeraire, Changes of Probability Measure and Option Pricing", corollary 3. An option of exchanging asset 2 against asset 1 at time T, its time-0 value under r.n measure $Q$, taking constant rate and K=1, is $$C(0)=E^Q[e^{-rT}(S^1_T-KS^2_T)1_A]$$ where A is the event $S^1_T>KS^2_T$. Then (and I guess it may be the step that makes no sense : namely splitting the max function in 2 pieces...) $$C(0)=E^Q[e^{-rT}S^1_T1_A]-E^Q[e^{-rT}S^2_T1_A]$$ Define $\frac{dQ}{dQ^{S1}}=\frac{e^{rT}/1}{S^1_T/S^1_0}$ and $\frac{dQ}{dQ^{S2}}=\frac{e^{rT}/1}{S^2_T/S^2_0}$ $$C(0)=E^{Q^{S1}}[\frac{dQ}{dQ^{S1}}e^{-rT}S^1_T1_A]-E^{Q^{S2}}[\frac{dQ}{dQ^{S2}}e^{-rT}S^2_T1_A]=S^1_0Q^{S1}(A)-KS^2_0Q^{S2}(A)$$ However in the paper, the result is $S^1_0Q^{S2}(A)-KS^2_0Q^{S1}(A)$... Also on the Wikipedia about the Margrabe formula it is said that the derivation is done using only $S^2_t$ as the numeraire. If I were to guess I'd say that my reckless splitting of the expectation is erroneous, but I'm not overly sure about it either... so any help is appreciated

*Edit: After actually deriving the result for the first term $Q^{S2}(A)$ (with 0 correlation and constant rate for simplicity) I get the same result as the Margrabe formula... so I would have to say, either those simplifications lead to artifacts that make my result agree with Margrabe, or there is just a typo in the original paper...

Thanks