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This is a follow up question on this thread

I have come across the following relationship in a CTD curve bootstrapping routine:

$$\frac{DF_{XXX}^{CSA.EUR}}{DF_{EUR}^{CSA.EUR}} = \frac{DF_{XXX}^{CSA.USD}}{DF_{EUR}^{CSA.USD}}$$

where $XXX$ is a generic currency (e.g. CAD) and $DF_{XXX}^{CSA.EUR}$ denotes a discount factor collateralized in EUR, etc.

Would some be able to explain why(and under what conditions) is this true?

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The formula simply states that the XXXEUR forward FX are the same under CSA.EUR collateralization and under CSA.USD collateralization.

It holds if disregarding the theoretical convexity adjustment that would result from non zero covariance between the XXXEUR FX and the EURUSD basis. Disregarding the adjustment is standard market practice.

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  • $\begingroup$ Thanks Antoine, but it doesn't answer the question. I understand what is the equation expressing, the question is why is that true? Is there an underlying no arb argument? If so, what? $\endgroup$
    – Frank Cho
    Commented Apr 23, 2021 at 5:54
  • $\begingroup$ Funding a transaction at CSA.USD when your own rate is CSA.EUR means that you will incur a daily cost equal to the deal PV times the daily EURUSD basis. When the EURUSD basis and the deal PV are uncorrelated the expected daily cost is the product of the expected EURUSD basis and the expected deal PV, the latter being equal to zero in the case of a XXXEUR forward FX (or any forward for that matter). When there is a non zero correlation the expected cost is non zero. You can hedge the expected cost so you can think of it as a cross gamma effect in the case of non zero correlation. $\endgroup$
    – Antoine Conze
    Commented Apr 24, 2021 at 15:04

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