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I'm asking for the curve construction of the discount curve in the case where payment currency and collateral currency are different. If I refer to BBG, in the case of a USD swap collateralized in EUR, they use the curve N°400 (MBB EUR Coll For USD). How do they construct this curve ?

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  • $\begingroup$ Is the collateral necessarily EUR, or does the CDA include a cheapest to deliver option, allowing sometimes USD and sometimes EUR? Bloomberg probably has documentation, but take a look at quant.stackexchange.com/questions/39886 $\endgroup$
    – Dimitri Vulis
    Mar 1 at 13:30
  • $\begingroup$ Yes is necessarily EUR. the link you provide is a more complex example than what I'm looking for. My question is simple: If I have a USD cash-flow in 5Y but the posted collateral is in EUR. How do I discount this cash-flow ? $\endgroup$
    – SIMO
    Mar 1 at 14:16
  • $\begingroup$ Apologies foe the typo above, CSA (Credit Support Annex), no "CDA". $\endgroup$
    – Dimitri Vulis
    Mar 2 at 14:01
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    $\begingroup$ Bloomberg - ICVS - Help Page (F1) - White Papers: direct link: {LPHP ICVS:0:1 2894955 <GO>} It explains how the CSA curves are constructed in some detail on several pages. If you press F1F1, you can also ask the help desk. They will likely only give you the white paper but it is explaining all relevant details anyways. $\endgroup$
    – AKdemy
    Mar 2 at 15:33
  • $\begingroup$ In the new documentation I have prepared for rateslib there is example of a curve construction process which systematically creates EUR and USD curves and finally as the last step on the page uses multi-currency derivatives (in this case cross currency swaps, but FX swaps can also be used) to construct cash-collateral curves as the final step: link rateslib.readthedocs.io/en/latest/c_solver.html $\endgroup$
    – Attack68 ♦
    May 1 at 14:43

3 Answers 3

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TO answer the question in the comment. Suppose you have a USD cash flow receivable in 5yrs and you are trying to calculate the PV. You need to know the interest rate that you are paying on the EUR cash collateral. Suppose this is Eonia flat. Then you execute a 5yr currency basis swap where you lend the Euro collateral out at Eonia flat, against borrowing USD on which you pay Fed funds + X, where X is determined by the basis swap market. Then you discount the USD receivable at Fed Funds + X (by which I mean, you construct a new USD curve X bp higher in rate than the standard Fed Funds curve).

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    $\begingroup$ The explanation is good, A side remark, EONIA itself was discontinued on 3 January 2022. $\endgroup$
    – AKdemy
    Mar 2 at 15:37
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    $\begingroup$ Good point. ESTR. $\endgroup$
    – dm63
    Mar 2 at 21:12
  • $\begingroup$ would the swap or fx fwd you use to construct the discount curves also be a function of collateral currencies? $\endgroup$
    – rd560
    May 1 at 20:26
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If I recall Cooking with collateral by Piterbarg (https://www.risk.net/derivatives/2194249/cooking-collateral) has the details but you effectively need to use FX swaps to get the basis adjust discount rate you want.

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In an arbitrage free framework you may be interested to know that a USD cashflow payable in 5y collateralised in EUR can either be valued (in USD) by discounting the cashflows with a USD-EUR discount curve (that is a discount curve for discounting USD cashflows under a EUR CSA).

Or, you can convert the USD cashflow to EUR with the 5y EURUSD FX forward rate, then discount the converted EUR cashflow to the present day with the EUR-EUR discount curve (that is the discount curve for EUR cashflows under a EUR CSA) and then convert the resultant EUR PV to USD with the immediate EURUSD FX rate.

The EURUSD FX Forward rate is essentially determined from the USD-EUR discount curve or vice versa within an arbitrage free framework, so this may practically not be helpful, but it may help with the concept.

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