# Cross Currency Swap pricing

I have seen two methods for calculating the value of a xccy swap -

1) Convert the future foreign payments to the base currency using forward FX rates, net with the base currency payments and discount using the risk-free rate for the base currency.

2) Discount the foreign payments using the foreign risk free curves and convert to the base currency using the spot rate. Discount the base currency payments with the base/foreign basis curve and net with the foreign payments.

It seems to me that if I calculate the forward fx prices using a simple interest rate differential, then the basis curve should match the base risk free curve. Am I correct in this view and if so, how does one calculate forward fx rates to yield a result equivalent to the basis curve method?

• I am not an expert on FX and xccy. But the modern point of view (post 2008) is there is no such thing as 'risk free curves' any more. Every curve embodies some credit, liquidity or other risks. Therefore approaches (1) and (2) will in general give different results. Jun 19, 2016 at 21:04

You are correct; methods 1) and 2) will give you the same result but keeping two things in mind:

A) You need to make sure your FX forwards and xccy basis swaps are priced under the same collateral assumption.

B) The difference between the two methods is more a question of semantics because FX forwards and xccy basis swaps are mutually dependent, so it really depends what market data you have to start with.

A) By collateralisation I mean the amount regularly exchanged between the two parties of the swap/contract, on top of any coupons or notional payments scheduled, as the value of the swap/contract fluctuates with the market. In order to take out any credit risk issues, broker feeds assume perfect collateralisation (=> daily, no minimum transfer amount), and so should you for your pure "first-order" rates pricing purposes. But the currency which will be actually posted matters too since it will determine which discount curve and factors will ultimately be used. Xccy swaps vs USD are usually collateralised with US Dollars, which means the cashflows are discounted at Fedfunds. By saying "risk-free rate" I assume this is what you are referring to; the OIS discount rate of the corresponding currency.

B) You can bootstrap your FX forwards from the xccy basis curve or you can bootstrap you xccy bases from FX forwards, assuming you also have all the other curves needed for both discounting and libor forwards. Usually you would go from the xccy basis to FX forwards since the latter usually has market data only for short-term maturities (<1 year). More details on the methodology can be found in this answer to a related question:

https://quant.stackexchange.com/a/28255/22726

Assuming you are looking at EURUSD discounted at Fedfunds, in the end what you end up with is:

• Fedfunds discount factors: df(FF) -> those can be applied to any USD amount in the future to get its USD PV.
• FX forwards: fwd(EURUSD) -> those can be applied to any EUR amount in the future to get its USD-equivalent amount in the future
• Basis discount factors: df(EURUSD) -> those can be applied to any EUR amount in the future to get its EUR PV discounted at Fedfunds.

But:

df(EURUSD) = df(FF) x fwd(EURUSD) / spot(EURUSD)

So what I mean is that depending on how you combine these factors (associative property of multiplication...) you can either call them FX forwards, single-currency discount factors or basis discount factors but it really comes down to the same thing.

One caveat is if you use a different market feed, like the EONIA/FEDFUNDS basis curve, to get your basis discount factors, but whether or not you find the same result depends on the non-arbitrage property of your market data.

I think you're arriving at a value for a swap using 2 different expressions of the same thing because FX forward prices are calculated using spot rates and adding or subtracting forward points. The forward points for a currency pair express interest rate differentials between the 2 currencies in the pair.

I think your question then moves from arriving at a net value of an FX swap to looking at how interest rate / money market data, in other words rates for deposits and loans of a given currency for given timeframes, are used to calculate FX forward points?

Yeah in theory the curves should all be the same, but in practise as Alex C says above, the main factor is what rates, for deposits and loans, you can actually get from the money markets.

The thing about "risk free" curves - or money market data - is they only exist if you have an account with the central bank. In reality the curve you're able to obtain depends on your relationships with deposit and loan providers in a given currency. If you can then you shop around in the money markets for the best rates....

You calculate the forward FX rate by getting the forward cash flows adjusted for interest rates of the 2 currencies according to the interest rates you can get in the money markets. The forward points then express that "rate" as points from the spot rate.

Check out a worked example of interest rate arbitrage to see how swaps and rates relate together....

I think you're asking if say a EURUSD CCBS should be discounted on the USD curve (using FX to convert the EUR payments to USD) or on the EUR curve (adjusted for basis).

Once upon a time, the two would have been considered equivalent. That is, investing in USD at Libor flat would be equivalent to investing in EUR at Euribor+Basis, so the basis swap is par-valued.

Equivalently, you could say that the basis is the required premium (discount) to cover the increased (decreased) risk of lending in the EUR interbank market.

But! We are not in Kansas any more. With cash-collateralised, daily margined, OIS-accrued bilateral arrangements, as far as I'm aware, a CCBS executed as a single trade (in major currencies) is generally collateralised in a single currency (usually USD), and thus the right discounting curve to use is the OIS curve relevant to that currency (i.e. FedFund curve). CCBS can be traded with different collateralisation choices written into the CSA on the contract, so the price then depends on what that choice is for a given trade.

I'll take a guess that when such trades are cleared on LCH, it will depend on what the LCH curve looks like, possibly with other adjustments to take into account Initial Margin.