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...




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