Consider a setup in which you're calibrating two currencies (USD and GBP), and you want to bootstrap the following curves:for USD: the discounting curve and the 3M Libor curvefor GBP: the discounting curve and the 3M AND 6M Libor curvesWe have the following calibration instruments (for simplicity): OIS swaps, USD vanilla 3M swaps, GBP vanilla 6M swaps, GBP 3M-6M basis swaps and USD/GBP 3M cross-currency swaps. Since all swaps are collateralised, we calibrate the discounting curves to OIS swaps. The 3M USD Libor curve is calibrated then to 3M vanilla USD swaps, the 6M GBP Libor curve - to 6M vanilla GBP swaps. But how can I calibrate the 3M GBP Libor curve simultaneously to the single- and cross-currency basis swaps? What am I missing here?

QuoteOriginally posted by: katastrofaConsider a setup in which you're calibrating two currencies (USD and GBP), and you want to bootstrap the following curves:for USD: the discounting curve and the 3M Libor curvefor GBP: the discounting curve and the 3M AND 6M Libor curvesWe have the following calibration instruments (for simplicity): OIS swaps, USD vanilla 3M swaps, GBP vanilla 6M swaps, GBP 3M-6M basis swaps and USD/GBP 3M cross-currency swaps. Since all swaps are collateralised, we calibrate the discounting curves to OIS swaps. The 3M USD Libor curve is calibrated then to 3M vanilla USD swaps, the 6M GBP Libor curve - to 6M vanilla GBP swaps. But how can I calibrate the 3M GBP Libor curve simultaneously to the single- and cross-currency basis swaps? What am I missing here?To Value a xccy swap you need a modified discount curve in one currency (usually your domestic currency). That means for xccy swaps you dont use your usual domestic ois discount curve. But a curve that reprices the xccy swap correctly. If you just used the both ois curves in the valuation of a (really existing and quoted) xccy overnight basis swap you would find there is no basis spread. But there is a basis spread even on the xccy overnight swap. So simply speaking you are missing one cuve (a special discount curve) on the xccy basis swap.

So I should be using different discounting curves depending on whether the swap is single- or cross-currency? Is it related to the currency in which the collateral is paid, by any chance?

Discount curves are driven by funding assumptions on book/portfolio level, e.g. currency of collateral, collateral type etc. Xccy basis swap pricing is one example where domestic discount curve is constructed out of foreign funding discount curve and domestic & foreign projection curves for floating rate fixings. So one may find a situation where a vanilla IRS is discounted at xccy funding curve because domestic funding is not cheap enough and posted collateral is assumed to be in another currency.

QuoteOriginally posted by: katastrofaSo I should be using different discounting curves depending on whether the swap is single- or cross-currency? Is it related to the currency in which the collateral is paid, by any chance?Yes.XCCY are special anyway with respect to "collateral". They normally have a reset feature that i would call "old stly collateral". So the movement of fx spot is realized at the end of each period. and the notional (the usd leg in a usd/euro xccy swap) is adjusted. This feels like usd collateral. But in fact some of the usd/eur xccy Swaps are collateralized in euro. Still they have this fx reset feature resulting in usd cashflows to reduce counterparty risk

QuoteOriginally posted by: berndLQuoteOriginally posted by: katastrofaSo I should be using different discounting curves depending on whether the swap is single- or cross-currency? Is it related to the currency in which the collateral is paid, by any chance?Yes.Thanks. Am I right in assuming that if the collateral is posted in USD, I should be using the OIS discounting curve for USD (assuming the overnight rate is being paid on the USD collateral)?QuoteXCCY are special anyway with respect to "collateral". They normally have a reset feature that i would call "old stly collateral". So the movement of fx spot is realized at the end of each period. and the notional (the usd leg in a usd/euro xccy swap) is adjusted. This feels like usd collateral. But in fact some of the usd/eur xccy Swaps are collateralized in euro. Still they have this fx reset feature resulting in usd cashflows to reduce counterparty riskSo cross-currency swaps quoted by brokers like ICAP don't have fixed notional?

QuoteOriginally posted by: katastrofaQuoteOriginally posted by: berndLQuoteOriginally posted by: katastrofaSo I should be using different discounting curves depending on whether the swap is single- or cross-currency? Is it related to the currency in which the collateral is paid, by any chance?Yes.Thanks. Am I right in assuming that if the collateral is posted in USD, I should be using the OIS discounting curve for USD (assuming the overnight rate is being paid on the USD collateral)?QuoteXCCY are special anyway with respect to "collateral". They normally have a reset feature that i would call "old stly collateral". So the movement of fx spot is realized at the end of each period. and the notional (the usd leg in a usd/euro xccy swap) is adjusted. This feels like usd collateral. But in fact some of the usd/eur xccy Swaps are collateralized in euro. Still they have this fx reset feature resulting in usd cashflows to reduce counterparty riskSo cross-currency swaps quoted by brokers like ICAP don't have fixed notional?Hi,no.sorry but i would say you have to use a modified usd discount curve. Not OIS. For sure if u are a usd bank. You wont get the xccy basis spread right if you use your usual dicount curves on both sides of the xccy swap. The basis spread for a 3mE vs 3mUSDL swap would result exclusivly from inner currency basis spreads. Which is not true. And Yes. Standard xccy swaps have the so called mtm reset feature. At least this was true a while ago (i havent dealt with them a while).

Thanks.So to price USD/GBP cross-currency derivatives correctly from the point of view of a GBP investor, what do I need to calibrate my discount/prediction curves to?

QuoteOriginally posted by: katastrofaThanks.So to price USD/GBP cross-currency derivatives correctly from the point of view of a GBP investor, what do I need to calibrate my discount/prediction curves to?Start with GBP OIS curve as standard discount curve. Calibrate forecast curves for different GBP Libors to inner Currency Basis Swaps or 2-swaps. Then do the same for usd.For XCCY Swaps GBP/USD from a GBP investor bootstrap a new GBP discount curve (to be used only to value fx cashflows like the xccy swap GBP Cashflows). Very likely you want to assume constant notional xccy swaps for this. I dont know at the moment how to adjust this for the mtm feature. There are a few papers but maybe you start with the constant notional assumption?

Thank you! That's immensely helpful. If I understand correctly, this new curve would be calibrated to xccy swaps while keeping the LIBOR prediction and USD discounting curves intact?

QuoteOriginally posted by: katastrofaThank you! That's immensely helpful. If I understand correctly, this new curve would be calibrated to xccy swaps while keeping the LIBOR prediction and USD discounting curves intact?you are welcome.yes i would do so.

Thanks again.

One more question: where do the FX forward come into play? I understand one should calibrate the cross-currency curve to them?

For the valuation of the xccy basis swap to be consistent with valuation of single currency irs and fx swap points you would require the following alternative valuation method to give the same result as the direct valuation via multiple curves:You can add fix float vanilla swaps as well as basis swaps to the xccy swap. The resulting cashflows should just be a string of fx outrights (fix amounts in USD exchanged vs fixed amounts in GBP). In other words you replicate the xccy swap with vanilla fix float swaps, basis swaps and fx outrights. This should give the same result as the multiple curve valuation of this swap.Assuming you value a vanilla fx outright via fx spot and the foreign and domestic curves you would calibrate (in your case) the GBP discount curve (the short end) to swap points GBP/USD and/or xccy basis swap quotes (the long end).

How would you do that if it was USD vs BRL?