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Pricing an Exchange Option with Asset-Specific Numeraires

Article Quant Q&A · Author: zebullon
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Summary

The document raises a derivation question for an option that pays the positive difference between two assets at maturity. Starting from the risk-neutral discounted payoff, it splits the payoff into two asset-weighted indicator terms and applies a change of numeraire to each. That calculation assigns each probability to the corresponding asset numeraire, while the cited Margrabe result appears to pair the event probabilities with the opposite asset values.

The writer checks the first term under simplifying assumptions and reports agreement with the Margrabe formula, then wonders whether the paper has a typo or the simplifications hide an error. The text presents an unresolved question rather than a verified derivation: it gives no full resolution or independent evidence that settles the apparent mismatch. Its useful lesson is that change-of-numeraire density conventions and the event probability under each measure need careful checking when pricing exchange options.

Key ideas

  • An exchange option pays the positive difference between two assets at maturity.
  • The payoff can be expressed as a difference of two asset-weighted event terms.
  • Changing numeraire changes the probability measure used to evaluate each term.
  • The document leaves a mismatch between its derivation and the cited formula unresolved.

Tags

OptionsDerivatives pricingMulti-assetStatistics
Full text
# Value of an option to exchange an asset for another


# 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

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.