SERVING THE QUANTITATIVE FINANCE COMMUNITY

thedoc
Topic Author
Posts: 61
Joined: July 27th, 2010, 6:53 am

### Multiple curves for Cross Currency trades

Hi,Does anybody have any thoughts on whether one should have more than 1 cross currency (xccy) discounting curve curve per currency? My current problem is this:For each currency, I have just 1 xccy curve. So for USD this is just the Fed Funds curve, but for non-USD currencies this is a curve built from the domestic OIS plus a xccy spread against USD. So for example, my £ xccy curve is built from: SONIA + £/$xccy spdand my EUR xccy curve is built from: EONIA + EUR/$ xccy spread.On this basis, when I'm pricing a £/$xccy trade, I use my standard (non-xccy) curves to project out my forward rates, and use the relevant xccy curves to discount the resultant cash flows. For an on-market spread, when exchange of nominals are included, both sides of my trade PV to zero, and everyone is happy.Similarly, for an on-market EUR/$ xccy trade, both sides of my trade PV to zero.So far, so good. Unfortunately, if I now trade a EUR/£ xccy, the fact that I'm paying EURIBOR flat on the EUR leg means that I have a non-zero PV on this leg. This non-zero PV is negated by an equal and opposite PV on the £ leg, and so the total mark-to-market on my trade is zero. However, the fact that I have equal and opposite m-t-ms on each side of the trade, in two different currencies, means that I effectively have an FX position in EUR/£. For normal size trades, this is a material position, which I have to hedge.I find all of this conceptually troubling. I've done an on-market EUR/£ trade, which has given me a spurious FX position (if I had defined my £ xccy curve against EURs instead of $s, this position would not exist!). I am forced to conclude that I need several xccy curves for each ccy, e.g. for £ I would have to have a £/$ curve, a £/EUR curve, a £/JPY curve etc.. Yet most people in the market that I've spoken to don't do this.Any thoughts?

Kerkabanac
Posts: 51
Joined: July 20th, 2006, 3:31 am

### Multiple curves for Cross Currency trades

hi thedoc,not sure i understand what your issue is? is it that you have an fx position in none usd when trading no usd vs. non usd swap?if so what position are you talking about?:floating vs. floating?fix vs. floating?fix vs. fix?in each case you can find quite easily the equivalent hedge in the market3s6s if neededFX fwd (very common for this type of pos) when only looking at fixed risk.i dont know... maybe i dont understand your issue very well.K.

thedoc
Topic Author
Posts: 61
Joined: July 27th, 2010, 6:53 am

Hi Kerkababac,Thanks for the reply.My problem is that I have traded an Interest Rate swap (EURGBP cross currency), but ended up with an FX position (EURGBP). But even worse than that, the FX position seems to be dependant on my choice of curves (if I had defined my £ xccy curve via EUR/£ xccy spreads rather than £/$xccy spreads, then the position wouldn't exist).From what I can see, the problem arises from the fact that I have chosen USD as a 'base' curve for all my xccy curves, and all other non-USD xccy curves are defined in reference to this. However, though this is standard market practice, it is entirely arbitrary, and therefore conceptually troubling.In answer to your questions, all the deals I'm talking about are all floating vs floating. There is no basis involved (eg 3s6s).Hope that makes sense.Thanksthedoc thedoc Topic Author Posts: 61 Joined: July 27th, 2010, 6:53 am ### Multiple curves for Cross Currency trades Anybody else have any thoughts? cfaleontan Posts: 22 Joined: July 21st, 2011, 6:26 pm ### Multiple curves for Cross Currency trades QuoteOriginally posted by: thedocHi Kerkababac,Thanks for the reply.My problem is that I have traded an Interest Rate swap (EURGBP cross currency), but ended up with an FX position (EURGBP). But even worse than that, the FX position seems to be dependant on my choice of curves (if I had defined my £ xccy curve via EUR/£ xccy spreads rather than £/$ xccy spreads, then the position wouldn't exist).From what I can see, the problem arises from the fact that I have chosen USD as a 'base' curve for all my xccy curves, and all other non-USD xccy curves are defined in reference to this. However, though this is standard market practice, it is entirely arbitrary, and therefore conceptually troubling.In answer to your questions, all the deals I'm talking about are all floating vs floating. There is no basis involved (eg 3s6s).Hope that makes sense.ThanksthedocHi.Even you make a EUR/GBP xccy curve, it most likely just equal to spread of EUR/USD - spread of GBP/USD. Why does direct xccy curve and indirect xccy curves make a difference?

Joshua2004
Posts: 100
Joined: September 16th, 2004, 5:41 pm

### Multiple curves for Cross Currency trades

I think what kind of curve you use should depend on what kind of asset you use as collateralsay for USD/GBP, because you are using USD as collateral, so your stripping should based on USD FED curve, and you solve the SONIA+spread as discount on GBP side to get par FX swap, but should keep the forward as the regular GBP swap rates.same for USD/EURnow for GBP/EUR, if your collateral is USD, then use above two discount curve to disocunt. and you can even solve what's the market par basis. if you are using EUR as collateral, you should use ENIOA as basis to solve the GBP+sperad curve. so conceptally there are different par basis for GBP/EUR fx based on different collateral and that is reasonable.that's just my guess, I do not know market do collateral or not for FX swap, and which asset they use as collateral for GBP/EUR trade.

kelang
Posts: 46
Joined: November 14th, 2011, 4:53 pm

### Multiple curves for Cross Currency trades

QuoteOriginally posted by: Joshua2004I think what kind of curve you use should depend on what kind of asset you use as collateralsay for USD/GBP, because you are using USD as collateral, so your stripping should based on USD FED curve, and you solve the SONIA+spread as discount on GBP side to get par FX swap, but should keep the forward as the regular GBP swap rates.same for USD/EURnow for GBP/EUR, if your collateral is USD, then use above two discount curve to disocunt. and you can even solve what's the market par basis. if you are using EUR as collateral, you should use ENIOA as basis to solve the GBP+sperad curve. so conceptally there are different par basis for GBP/EUR fx based on different collateral and that is reasonable.that's just my guess, I do not know market do collateral or not for FX swap, and which asset they use as collateral for GBP/EUR trade.you do have some inconsistency, say USD/GBP collateralized in USD, you solve FF curve and SONIA+Spread to recover FX swap and GBP swap, so the assumption behind is, GBP swap is also USD collateralized... The methodology will get itself confused if changing collateral currency.practically, i found approaches under different assumption will create 1~3 bps difference in far-end rates...

Joshua2004
Posts: 100
Joined: September 16th, 2004, 5:41 pm

### Multiple curves for Cross Currency trades

I think you misunderstand me: for GBP swap itself, it should collateral with GBP and then you use SONIA as discount which should get from OIS swap, and you solve the forward rate 6 month based on the GBP swap market.Then when you do to USD/GBP assume collateral is USD, then USD side is regular USD swap: OIS discount + 3month LIBORGBP side should be 6month LIBOR which you should before and SONIA+fxbasis. here you change the discount is because it is different collatal.In this idea, eveything make sense to me.

Joshua2004
Posts: 100
Joined: September 16th, 2004, 5:41 pm

### Multiple curves for Cross Currency trades

some typo before, correction:I think you misunderstand me: for GBP swap itself, it should collateral with GBP and then you use SONIA as discount, which should get from OIS swap market, and you solve the forward 6 month rates based on the GBP swap market.Then when you go to USD/GBP assuming collateral as USD, USD side is regular USD swap: OIS discount + 3month LIBORGBP side should be 6month LIBOR which you solved before in GBP market, and discount is SONIA+fxbasis. here you change only the discount, because it is using different collatal.In this idea, eveything make sense to me.

kelang
Posts: 46
Joined: November 14th, 2011, 4:53 pm

### Multiple curves for Cross Currency trades

QuoteOriginally posted by: Joshua2004some typo before, correction:I think you misunderstand me: for GBP swap itself, it should collateral with GBP and then you use SONIA as discount, which should get from OIS swap market, and you solve the forward 6 month rates based on the GBP swap market.Then when you go to USD/GBP assuming collateral as USD, USD side is regular USD swap: OIS discount + 3month LIBORGBP side should be 6month LIBOR which you solved before in GBP market, and discount is SONIA+fxbasis. here you change only the discount, because it is using different collatal.In this idea, eveything make sense to me.yes, that is another assumption..., but it also has inconsistency mathematically and only valid if you assume the funding spread difference accross different market is deterministic..., which is not true...again, different assumptions give at most 1~3 bps difference under "normal" market conditions...

Joshua2004
Posts: 100
Joined: September 16th, 2004, 5:41 pm

### Multiple curves for Cross Currency trades

QuoteOriginally posted by: kelang<bryes, that is another assumption..., but it also has inconsistency mathematically and only valid if you assume the funding spread difference accross different market is deterministic..., which is not true...again, different assumptions give at most 1~3 bps difference under "normal" market conditions...I get your point, to really solve this problem, you need quanto adjustment, which has to bring in option market...

GoGoFa
Posts: 47
Joined: March 3rd, 2005, 11:23 am