Can you please recommend a paper/book that discusses the attribution of the mark to market of a cross currency swap (rates, basis, fx, etc.)?

As for the book, the best one I have come across is Pricing and Trading Interest Rate Derivatives by Darbyshire, although it's a bit pricey (indeed as most finance books are) (https://www.amazon.com/Pricing-Trading-Interest-Rate-Derivatives/dp/099545552X).

I used to trade Xccy Basis Swaps (which is just another name for Cross-Currency Swaps): let me try to answer on their valuation.

SHORT STORY: Cross-Currency Swaps are valued at zero MtM, even if there is a cross-currency basis component added onto one of the floating legs. That is because the discount curves, which set the Cross-Currency Swap MtM to zero, are implied by the liquid Cross-Currency Swaps: the discount curves are solved for precisely in such a way that the MtM of the Xccy Swap is always zero at inception. Even if the theoretical discount curves implied by OIS Swaps differ from these discount curves implied by Cross-Currency Swaps, the OIS Swaps (except for USD) are not liquid enough to "arbitrage" the difference away.

LONG STORY:

You are probably familiar with FX Forwards. In an FX forward transaction, you agree to exchange one currency for another in the future at a predetermined exchange rate. Let's consider the EUR/USD currency pair. The "valuation" formula for an FX Forward expiring at time $$T$$ would be as follows ($$F$$ stands for the forward rate, $$S$$ stands for the spot rate, $$r$$ stands for the interest rates, $$\tau$$ is the annual fraction):

$$F_{EUR/USD}(t_0, T)=S_{EUR/USD}(t_0)\frac{1+\tau * r_{USD}}{1+\tau * r_{EUR}}$$

Above, $$r_{EUR}$$ and $$r_{USD}$$ would be respective "discount" rates implied from the OIS curves. When you look at the formula above mathematically, you might be tempted to compute the forward rate from the spot rate and the two respective OIS rates. But actually in real markets, the EUR/USD spot and the EUR/USD forwards are the liquid (quoted) products, with the $$r_{EUR}$$ being the most illiquid (not directly traded) component in the equation. So basically, assuming you know the spot, the forward and the USD rate, you can back out the implied EUR discounting rate (can be different to the discount rate implied from EUR OIS swaps: particularly if the tenor of the EUR OIS is not very liquid!).

Now let's discuss a related product, called an FX Swap. This is the same product as the FX Forward, but you also exchange notional at trade inception, not just at maturity. So in the EUR/USD case, suppose that today I lend you 1-million EUR and in exchange I borrow 1-million * EUR/USD spot from you. In one year, I will pay the 1-year EUR/USD forward rate and you'd return the 1-million EUR to me.

Beyond a certain time-horizon (i.e. 5 years), the EUR/USD FX Swaps and FX Forwards become illiquid, and instead, another product picks up in liquidity: the Cross-Currency Swap. Here, the principle is the same as with the FX Swap: but rather than just exchanging notionals at inception and at maturity, there are also periodic coupons on both legs. Let's consider a 6-year Cross-Currency EUR/USD Swap.

The Cross-Currency swaps (by convention) have both legs semi-annual floating, indexed to the respective Libors (so USD Libor and Euribor in our case). In addition, there is a Cross-Currency basis component added onto the EUR leg (this is again by convention, because the Cross-Currency swaps against USD are always quoted with the basis on the other non-USD leg: so the basis can be both positive or negative, depending on the currency pair. It can also be zero, although this is rare).

Last time I traded these, the 6-year EUR/USD basis was about -17 bps (so if I borrow dollars for six years and lend out EUR against it, I will be "penalized" and my counterparty will only pay me 6m Euribor - 17 bps, instead of just 6-m Euribor: the negative 17 bps basis on the EUR leg shows you that there is more demand to fund USD borrowing via EUR, than the other way around).

Now, one might be tempted to say that the 6-year EUR/USD cross-currency swap, which is 6-month USD Libor against 6-month Euribor should have zero MtM and one might argue that the cross-currency basis "adds MtM* onto the swap. But how do you compute this MtM? You would need both the USD as well as the EUR OIS rates up to 6 years, and you'd then use these to compute the discount factors which you'd apply to each coupon on the swap.

But do you have the EUR discount rates for every six months up to 6 years? You don't, because the EUR OIS swaps are not liquid enough (certainly not more liquid than the EUR/USD Cross-Currency swaps). You can imply the EUR discount factors from the FX Swaps up to 5-year maturity (using the formula I showed above for the FX Forwards). Then, you actually use the EUR/USD Cross-Currency Swaps to back out the EUR discount factors beyond the last liquid point on the EUR/USD FX Swap curve.

How do you back these EUR discount factors? These will be such that they solve for zero MtM on the EUR/USD Cross-Currency Swap. (btw, in all of this, I of course assume that the USD OIS curve is liquid for all maturities: which it is). So by definition, the EUR/USD Cross-Currency swap, including the basis, has zero MtM. That is because you solve for the discount factors on the EUR leg in such a way, that they yield zero MtM.

It took me a while to digest this at first, but when it sinks in, it starts making sense. As a quant, one might be tempted to "solve for the basis" on the Cross-Currency Swaps, by using OIS discount factors and the FX spot rate between two currencies as the inputs, getting the basis as the output (such that it "values" the swap at zero MtM): but you then realize that the markets don't price products according to illiquid "theoretical" curves. You might be able to find a theoretical EUR OIS curve up to 20 years, but when you try to use it to value Cross-Currency swaps, your valuation will be off: this shows you that the curve is not liquid enough for the markets to push the "off-valuation" onto the "correct theoretical valuation".

Hopefully, the above makes sense and was of some help.

I believe that cross currency basis swaps are marked to market always. The issue is that theoretical value for an xccy swap is always 0. but they don't trade at 0, that's why there is a premium for this kind of trade.

On the fixing side - you are receiving and paying float. So the value is 0. The issue is that the spread is based on market demand, and that's unrelated in any way to the floating fixes.

Usually brokers circulate runs of the different xccy swaps. Maybe someone here from MO can comment on what methodology they use in practice. I'll bet that ultimately is just comes from the basis curve, which is in turn from an average of broker runs.

One minor caveat. As you progress forward each day you will materialize/decay theta into your p/l. That comes from the fact that you are trading and hedging in libor. BUT you actually have to fund your position. And your treasury will charge you a rate that doesn't equal the libor fixings. Often a bank's treasury will add a premium to discourage people from using scarce dollars.

• So I guess the conclusion is that 100% of the P&L is due to "change in xccy basis"? No finer breakdown possible? Oct 2 '20 at 14:30
• Unless I'm forgetting something... You swap fx spot/fwd at a fixed rate. You then receive float on the other side. So that all hedges out. The only p/l in theory is the premium. I'll add one minor caveat. Oct 2 '20 at 14:32
• @noob2: 100% spot on. On the Cross-Currency Swaps, all the PnL swings are due to the fluctuations of the basis. In fact, the Cross-Currency swaps are quoted in terms of the basis. What I found stunning as a trader were days when the same-maturity IRS would move massively (implying forward changes in Libor rates), but the Cross-Currency basis didn't move at all: how come this would still generate zero MtM on the Currency swap? As per my answer below, this is due to the implied discount factors moving proportionally to the forward Libors. Oct 5 '20 at 9:39

The question is subjective.

Suppose you have a USD based accounting framework and an interest rate swap in NOK.

At the accounting period 1 the USDNOK is 10, and the IRS is worth 100 NOK (10 USD).

At the accounting period 2 the USDNOK is 11, and the IRS is worth 110 NOK (10 USD).

In your USD accounting framework there is no reported PnL, but clearly this is attributed to two components: FX rate and NOK interest rates.

Either:

a) we allocate (100/11 - 100/10 = -0.91\$) as the FX loss, and (110/11 - 100/11=0.91\$) as the IR gain.

b) we allocate ((110-100)/10=1\$) as the IR gain and -1\$ as the FX loss.

c) some linear combination between the two.

When it comes to cross currency swaps the FX rate does indeed have a direct impact on the mark-to-market of the instrument. This delta risk might actually be easier to separate out in terms of PnL explain or allocation (since it can be directly hedged) but the accounting regime will still generate problems particularly if the accounting currency is different to the two currency on the currency swap.