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Building EUR Curves for GBP-Collateralized Cross-Currency Swaps

Article Quant Q&A · Author: sam
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Summary

The document explains how to value the EUR leg of a EUR-GBP cross-currency swap when GBP collateral is specified. The EUR leg needs both a three-month EURIBOR forward curve and a EUR discount curve that are consistent with GBP collateral. In theory, these curves are jointly calibrated using GBP-collateralized EURIBOR swaps, FX swaps, and cross-currency swaps. When collateralized EURIBOR swap data is unavailable, the response describes using EUR-collateralized swaps as a practical proxy for the forward curve.

That proxy assumes the difference between EURIBOR forward curves under the two collateral currencies can be neglected; theoretically, the difference involves covariance between EURIBOR and the EUR/GBP basis. The discount curve can be built by adjusting EUR OIS for the EUR/GBP OIS basis, which must be bootstrapped rather than read directly from a cross-currency swap spread. Alternatively, an available forward FX curve can imply the adjusted EUR discount curve from GBP OIS discounting. These are curve-construction principles, and the answer gives no market data or worked calibration example.

Key ideas

  • Both the EURIBOR forward curve and EUR discount curve must reflect GBP collateral.
  • Joint calibration is the theoretical approach when suitable GBP-collateralized market instruments are available.
  • Using EUR-collateralized EURIBOR swaps as a proxy neglects a covariance effect involving EURIBOR and the currency basis.
  • The EUR discount curve can be adjusted using a bootstrapped EUR/GBP OIS basis.
  • A forward FX curve can also be used to derive the adjusted EUR discount curve from GBP OIS discounting.

Tags

Fixed incomeForexDerivatives pricing
Full text
# Forward and discount curves for cross currency swaps


# Forward and discount curves for cross currency swaps












I have a EUR-GBP cross currency swap, collateralised in GBP, each leg is paying 3m EURIBOR/LIBOR respectively. I know GBP leg can be modelled with 3m LIBOR forward curve and GBP OIS discount curve. How about the EUR leg?

I heard there are two approaches: 1. 3m EURIBOR as forward curve, and the discount curve should be EUR IOS adjusted with EUR vs GBP spread. 2. 3m EURIBOR adjusted with EUR vs GBP spread as the forward curve, and the discount curve should be GBP OIS

I am not sure if both above are accurately described, as I am not familiar with such trades. Can anyone share some insights and explain if any/both of them are correct? Many thanks.

## Answer by Antoine Conze (score 3)

https://quant.stackexchange.com/a/33881

The EUR leg should be valued in EUR but in a manner consistent with GBP collateral:

This means:

- a 3m EURIBOR forward curve consistent with GBP collateral

- a EUR discount curve consistent with GBP collateral

In theory both curves should be jointly bootstrapped to fixed vs EURIBOR swaps collateralized in GBP and EURGBP FX swaps and cross currency swaps collateralized in GBP.

In practice if market data for fixed vs EURIBOR swaps collateralized in GBP is unavailable, you should use fixed vs EURIBOR swaps collateralized in EUR. The theoretical difference between the 3m EURIBOR forward curve under GBP collateral and the 3m EURIBOR forward curve under EUR collateral is a covariance term between EURIBOR and the EUR/GBP basis spread, so assuming that the two curves are the same is equivalent to neglecting the EURIBOR, EUR/GBP basis correlation.

As for the EUR discount curve it should be the EUR OIS adjusted with EUR/GBP OIS basis spread. Beware that the OIS basis spread needs to be bootstrapped: the spread in a zero PV cross currency swap cannot be used directly as it is (roughly) equal to the EUR/GBP OIS basis spread plus the difference between the OIS-Libor spreads in both currencies.

As an alternative if you already have the forward EURGBP FX curve available, you can obtain the adjusted EUR OIS curve as GBP OIS discount x spot EURGBP / forward EURGBP.

Hope this helps.

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