### Which instruments to use in a FX/CcySwap curve

Posted:

**January 13th, 2022, 1:33 pm**Hi,

I'm wondering which instruments one should use to bootstrap in a curve used to discount cashflows in CCY1 collateralized in CCY2, e.g. EUR collateralized in USD. One could use either FxFwds or CcySwaps or a combination of both. When it comes to liquidity one should use FxFwds for the short maturities (maybe up to 18M?) and CcySwaps thereafter (maybe >= 2Y).

However, having this kind of a mixed curve introduces some problems when working with par-rate/benchmark sensitivities (zero-rate sensitivities would be fine). Traders usually look at the sensitivities on some aggregated level, i.e. defining sum buckets for which they want to be flat instead of looking at every individual delta. Also the parallel delta of a curve is been looked at. But adding up the (par-rate/benchmark) sensitivities of a curve bootstrapped from both FxFwds as well as CcySwaps doesn't make sense as shifting a FxFwd rate by 1bp is totally different to shifting a CcySwap Spread by 1 bp. This can be solved by adjusting the shift applied to FxFwds to have the same sign and dimension as shifting a CcySwap spread by 1bp.

But what about e.g. the Libor-curves that are naturally present in a CcySwap but not a FxFwd (the problem will be the same for an RFR-CcySwap but it's easier to explain in the good old Libor world)? Assume that I have a USD collateralized trade paying 1 EUR both at 1Y and at 2Y. Using the combined curve this trade will have a EUR-Libor and USD-Libor sensitivity at 2Y and none at 1Y. So the Libor-curve sensitivities will behave differently on the FxFwd segement than they do on the CcySwap segment and one should be careful when agregating the sensitivities over this discontinuity. Or one can look at a different example. Assume I'm looking at the risk of a 21M CcySwap with pv ~= 0. Using a pure CcySwap curve this trade will only be sensitive against this curve with deltas at 18M and 2Y that are roughly the same. Using a pure FxFwd curveĀ (with adjusted FxFwd shifts in the delta calculation) the trade will have an additional EUR-Libor and USD-Libor sensitivity at 18M and 2Y that are very similar to the ones from the FxFwd curve (when expressed in the same ccy). Using a mixed curve (with adjusted FxFwd shifts in the delta calculation) the FX/CcySwap curve deltas are as before, the EUR-Libor and USD-Libor delta at 2Y will be as it was using the pure CcySwap curve and the EUR-Libor and USD-Libor delta at 18M will be 0. The trade now decides to hedge the cross-currency risk of this trade solely with a 2Y CcySwap. If he than hedges his EUR-Libor and USD-Libor risk he has to manually ad some 18M risk to the numbers coming from the system.

I hope what I say makes some sort of sense . How do other players deal with these problems? I know that larger banks have a STIR desk and one for longer-dated rates trades. If the STIR desk handles all the risk <2Y it could live in a "FxFwd world" and the other desk could live in a "CcySwap world" and both could aggregate their sensitivities as they wish to. What does the longer-dated rate desk do with trades whose maturity becomes smaller than 2Y?

Thanks,

Bernd

I'm wondering which instruments one should use to bootstrap in a curve used to discount cashflows in CCY1 collateralized in CCY2, e.g. EUR collateralized in USD. One could use either FxFwds or CcySwaps or a combination of both. When it comes to liquidity one should use FxFwds for the short maturities (maybe up to 18M?) and CcySwaps thereafter (maybe >= 2Y).

However, having this kind of a mixed curve introduces some problems when working with par-rate/benchmark sensitivities (zero-rate sensitivities would be fine). Traders usually look at the sensitivities on some aggregated level, i.e. defining sum buckets for which they want to be flat instead of looking at every individual delta. Also the parallel delta of a curve is been looked at. But adding up the (par-rate/benchmark) sensitivities of a curve bootstrapped from both FxFwds as well as CcySwaps doesn't make sense as shifting a FxFwd rate by 1bp is totally different to shifting a CcySwap Spread by 1 bp. This can be solved by adjusting the shift applied to FxFwds to have the same sign and dimension as shifting a CcySwap spread by 1bp.

But what about e.g. the Libor-curves that are naturally present in a CcySwap but not a FxFwd (the problem will be the same for an RFR-CcySwap but it's easier to explain in the good old Libor world)? Assume that I have a USD collateralized trade paying 1 EUR both at 1Y and at 2Y. Using the combined curve this trade will have a EUR-Libor and USD-Libor sensitivity at 2Y and none at 1Y. So the Libor-curve sensitivities will behave differently on the FxFwd segement than they do on the CcySwap segment and one should be careful when agregating the sensitivities over this discontinuity. Or one can look at a different example. Assume I'm looking at the risk of a 21M CcySwap with pv ~= 0. Using a pure CcySwap curve this trade will only be sensitive against this curve with deltas at 18M and 2Y that are roughly the same. Using a pure FxFwd curveĀ (with adjusted FxFwd shifts in the delta calculation) the trade will have an additional EUR-Libor and USD-Libor sensitivity at 18M and 2Y that are very similar to the ones from the FxFwd curve (when expressed in the same ccy). Using a mixed curve (with adjusted FxFwd shifts in the delta calculation) the FX/CcySwap curve deltas are as before, the EUR-Libor and USD-Libor delta at 2Y will be as it was using the pure CcySwap curve and the EUR-Libor and USD-Libor delta at 18M will be 0. The trade now decides to hedge the cross-currency risk of this trade solely with a 2Y CcySwap. If he than hedges his EUR-Libor and USD-Libor risk he has to manually ad some 18M risk to the numbers coming from the system.

I hope what I say makes some sort of sense . How do other players deal with these problems? I know that larger banks have a STIR desk and one for longer-dated rates trades. If the STIR desk handles all the risk <2Y it could live in a "FxFwd world" and the other desk could live in a "CcySwap world" and both could aggregate their sensitivities as they wish to. What does the longer-dated rate desk do with trades whose maturity becomes smaller than 2Y?

Thanks,

Bernd