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Suppose the following:

A xccy.basis swap EUR for USD. Having obtained relevant discounting and forwarding curves in both currencies respectively (i.e. OIS, LIBOR, EURIBOR), and spot exchange rate FX(t) quoted as EURUSD, I would calibrate to the xccy.basis collateralized in USD (I'm assuming that's what the market quotes) like this:

$\sum_{i=1}^N \delta_i \left(\texttt{fwrd.3m}^{\texttt{\$USD}}(0, t_{i-1}, t_i) \right) \texttt{DF}^{\texttt{\$USD,OIS}}(0, t_i) = \\ \texttt{FX}(t) \sum_{i=1}^N \Delta_i \left(\texttt{fwrd.3m}^{\texttt{\EUR}}(0, t_{i-1}, t_i) + \texttt{xccy.spread}_N^{\texttt{EUR}} \right) \texttt{DF}^{\texttt{\EUR,\$USD}}(0, t_i)$

The quantity $\texttt{DF}^{\texttt{\EUR,\$USD}}(0, t_i)$ is the unknown that allows calibration to the market.

Suppose now that everything stays the same but the collateral is exchanged in EUR, not USD. How would one go about pricing this? Anyone has a reference to a clear text/example?

Right now, my first instinct is to use all the information already obtained from the steps above and simply use the spot exchange rate to convert the value of the swap from USD to EUR and post this value as collateral. That is, apply the exchange rate to the left side (and thus value the swap in EUR) instead of the right side (valuing the swap in USD).

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