# Cross Currency Swap Pricing in nowadays environment

Multicurve setting has now become the new paradigm for vanilla swap valuation. For the record I give here (without getting into too much details) the methodoloy for pricing Euribor3M swaps in this setting.

Step 1-Get your discounting curve from OIS Swap Curve quotes, by standard bootstrapping method just like in the old days.

Step 2-Using Euribor3M swaps ATM quotes and the OIS discount curve of step 1, bootstrap a "forward" curve of discount factors that can be used to get "adjusted" Euribor3M forward rates such that those "adjusted" forwards discounted over OIS curve gives the Euribor3M ATM Swap quotes (the fixed rate cashflows are discounted with OIS Curve).

(I hope I'm sufficiently clear if not let me know I will add extra explanations)

I was wondering about the corresponding setting (i.e. curve construction) for Cross Currency Swaps (CCY Swaps) products.

Here is a setting for Cross Currency Swaps Euribor3M +Margin vs Libusd3M collateralized in USD (at Fed Fund rate) that I believe is correct. So here it is :

Step 1,2- Construct the USD curves as in Step 1 and Step 2 with Fed Fund Swap Rates and Libusd3M ATM Swaps quotes.

Step 3- Then use Forward FX quotes over USD/EUR currency pair to bootstrap (together with the Fed Fund discounting curve) an $OIS^{usd}$ discount curve thank's to the spot/forward arbitrage argument where the unknowns are of course the $OIS^{usd}$ discount factors.

Step 4- Use Cross Currency ATM Swap Quotes, to build a "$Euribor3M^{usd}$ adjusted" curve using the 3 curves already built (with the spot Eur/USD exchange rate) so that all discounted cashflows (valued in USD) of those CCY ATM Swaps get a zero PV (in USD).

If anyone implied with CCY Swap valuation on a daily basis read this post, I would be grateful if he could confirm/infirm the relevance of this setting.

Best regards

PS 1 : This is not really an option pricing question nevertheless such a setting is the basis for any Multicurve option derivative model for any Cross Curreny Interest Rate Options so it seems important to me to establish first the market practice over underlying instruments before going further.

PS 2 : You can rebuild step 1 to 4 using EUR as collateral (with EONIA rate) then you would shortly discover that ( even if all ATM instruments give the same ATM margin and rates by construction), when pricing OTM or ITM Cross currency swaps you get different Net PVs if you use one setting or the other. This does not entails arbitrage as this difference actually originate from the different collateral settings.

Best regards

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## 2 Answers

I have a little more informations, so let me share it with you.

Even though I think that the frameworks I presented in my question are both corrects (i.e. aribtrage free), it happens to be the case that the market seems to have more "structure".

Here is a methodology that allows to retreive market quotes and which is the same as BBG (which is the best reference I could get in terms of market practice), so for a USD based investor :

Use step 1 and 2 to build multicurve US discount curve (based on Fed Funds) and forward curve for (LibUSD3M)

Use again step 1 and 2 to build multicurve EUR discount curve (based on OIS) and forward curve for Euribor3M.

Then forget about EUR discount curve and build a new discounting EUR curve in the following way, use :
-Cross Currency ATM Swap
-EUR forward Curve
-USD Discount
-USD forward curve
-Spot FX rate

So that you can build a new EUR Discount curve that allows you to retrieve Cross Currency Swaps ATM Quotes.

I have made a few tests to compare my USD based framework and this one and the results are quite similar.

What this really says IMO, is that in whatever economy you are based you have $E^{FF_T}[Euribor3m(T-3m,T)]$ and $E^{OIS_T}[Euribor3m(T-3m,T)]$ that are almost equals for any $T$, where $E^{FF_T}$ is the USD-FF T-forward pricing measure and $E^{OIS_T}$ is the EUR-OIS T-forward pricing measure (this is what I meant by "the market seems to have more structure" at the beginning of my post).

This was quite surprising to me and if anyone has more insight on this subject I am all ears.

NB : The methodology presented here can be applied for an EUR Based investor aswell with the same conclusions.

Best regards

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Assume you have an USD-EUR Cross Currency Swap (3M-FloatUSD+SpreadUSD vs 3M-FloatEUR+SpreadEUR) (spread on USD side is usually zero), collateralized by USD-OIS (Fed Fund)

I assume you know the USD-OIS discount curve, then you know the discount curve for USD cash flows.

I further assume that you know the USD-3M forwards collateralized w.r.t. USD-OIS (from corresponding single-currency swaps).

Then the cross currency swap has still two unknowns:

• the discount curve for EUR cash flows collateralized w.r.t. USD-OIS
• the forward of the EUR-3M index collateralized w.r.t. USD-OIS

It is important to understand that the forward of an index may depend on its collateralization.

In theory you can determine both from

• a) the cross currency swap quotes and
• b) a single currency EUR swap, collateralized in USD-OIS

Usually you do not have a quote for the second instrument, but sometimes you have, e.g. if an USD institution trades EUR swaps with another USD institution and collateralizes this in USD.

If you do not have instrument b) you can use some approximating assumption, e.g.

• the EUR-3M forward w.r.t. to USD-OIS collateral is approximated by the EUR-3M forward w.r.t. EUR-OIS collateralization, or,
• the EUR-3M market rate w.r.t. to USD-OIS collateral is approximated by the EUR-3M market rate w.r.t. EUR-OIS collateralization.

The different assumptions will lead to slightly different results.

(Remark: For curve calibration (bootstrapping) implementing both ways see http://www.finmath.net/topics/curvecalibration/ - the spreadsheet there depicts the difference between the two approximations).

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