Cross-Currency Basis and Discount Curves in Multi-Curve Pricing

# Cross-currency Basis Adjustment for Multi-curve Models


# Cross-currency Basis Adjustment for Multi-curve Models












I am studying Pricing and Trading Interest Rate Derivatives - A Practical Guide to Swaps, and I have troubles to really understand the use of cross-ccy swaps (XCS) and why the cross-ccy basis enter the way it does in a multi-curve model (Chapter 7). I try to present an example and make my questions as precise as possible.

Suppose that today ($t$) I have access to USD and have a USD cashflow payable to B at $T>t$, collateralized in EUR at ESTR. If $X$ is the collateral I should post to B, my cashflows should be the following:

$S=$ spot EURUSD; $d_\text{ccy}=$ daycount fraction for currency ccy; $r_\text{ccy}=$ rate for currency ccy; $Z=$ EURUSD cross-ccy basis;

- Why is the EUR OIS curve adjusted to reprice the EURUSD XCS precisely the EUR:USD-CSA curve? I think because from the above I should adjust the ESTR curve $r_\text{eur}$ used in the collateral to B to include the basis and remove arbitrage. Is this right?

- I've understood that the argument in 1. is based on the fact that we derive the discount factor from $r_\text{eur}+Z$. But $r_\text{eur}$ in $X\cdot d_\text{eur}\cdot (r_\text{eur} + Z)$ refers on the forward curve in the XCS, what if this is not ESTR but EURIBOR_3M for example? How does the argument of adjusting the discount curve and get EUR:USD-CSA work?

- In this, how does the XCS-CSA enter into play? The book says that "XCS usually trade with benchmark CSA based on SOFR, this is why we have always adjusted the non-USD discount curve." I think this refers to the fact that the discount factors derived from $r_\text{eur},\,r_\text{usd}$ in the XCS refer to the XCS collateralization, so we get the formula $$ \text{FX Fwd EURUSD} = \frac{df\text{(EUR:USD-CSA)}}{df\text{(USD:USD-CSA)}}\cdot (\text{FX $T+0$ EURUSD}) $$ but again, I am not sure about this.

It's quite some time I have been thinking about this but (as you can tell) I'm still very confused. Many thanks for any help!

### Edit After @Attack68 Answer and Comments

Following @Attack86 suggestion, I've added the discount factor column to the table above, showing that cashflow's PVs don't directly offset as they have different collateral:

In fact, I am hedged if and only if \begin{align*} &d_\text{eur}\, r_\text{eur}\, v_\text{ee} - d_\text{eur}\,(r_\text{eur}+Z)\, w_\text{eu} + v_\text{ee} - w_\text{eu} = 0\\[2mm] \quad\iff\quad &w_\text{eu} = \frac{1 + d_\text{eur}\, r_\text{eur}}{1 + d_\text{eur}\,(r_\text{eur}+Z)}\,v_\text{ee}. \end{align*} It appears clear how the discount factor that we are looking for is obtained from a single currency model and the XCBasis adjustment. As answered above, it is also clear that nothing changes if the forward curves are based on LIBOR instead of RFR.

## Answer by Attack68 (score 5, accepted)

https://quant.stackexchange.com/a/83990

Instead of going directly to your hypothesis, consider the simpler variety:

> Suppose that today (t) I have a fixed USD cashflow, of size 1, payable to counterparty B at T>t, collateralized in USD at USD.

What are the general points of interest here? Firstly, what is the present value of this and, secondly, is it deterministic? The first point refers to how much money you need to borrow to post as collateral, and the second point determines if you will need to borrow more (or less) in the future, i.e. market sensitivity.

The answers here are that the present value of this cashflow is equal to the discount factor for USD cashflows collateralized by USD. These values are calibrated by local currency interbank market traded SOFR IRS. The specific discount factor we require is $v_{USD:USD}(T)$. If I pay fixed on a SOFR IRS between time 0 and T in a notional $v_{USD:USD}(T)$ my portfolio is completely hedged and deterministic

Now consider,

> Suppose that today (t) I have a fixed USD cashflow, of size 1, payable to counterparty B at T>t, collateralized in EUR at ESTR.

What's different? Suppose that we define, or label, the value of this portfolio as $w_{USD:EUR}(T)$ in USD. We acknowledge that if we transact a mid-market, forward USDEUR FX transaction (collateralized in EUR) we can convert the 1 USD at T to $f_{usdeur}^{eur}$ EUR. The portfolio now has three cashflows whose values are:

$$ w_{usd:eur}(T) = (Y - Y + f_{usdeur}^{eur} v_{eureur}(T))F_{eurusd} $$

where the $Y$ terms here represent equal and offsetting 1USD flows collateralized in EUR of some value, which just cancel. And $F_{eurusd}$ is the immediate FX rate to convert the EUR value back into units of USD.

The same argument applied to a 1 EUR payable cashflow collateralized in USD results in:

$$ w_{eur:usd}(T) = f_{eurusd}^{usd} v_{usdusd}(T) F_{usdeur} $$

(switching to shorter notation) The $v_{ee}$ and $v_{uu}$ are known. This means that $w_{eu}$ and $w_{ue}$, which represent the discount factors for these specific cashflows are determined from the FX forwards instruments directly.

With the assumption that the FX forward rate is universal across collateral types we have:

$$ f_{ue}^{u} = f_{ue}^{e} = \frac{1}{f_{eu}^u} = \frac{1}{f_{eu}^e} $$

and this leads to,

$$ \frac{f_{ue}}{F_{ue}} = \frac{w_{ue}}{v_{ee}} = \frac{v_{uu}}{w_{eu}} $$

which means that $w_{ue}$ and $w_{eu}$ are entirely determined from the FX market. The FX market comprises FX forwards, FX Swaps and XCS all of different tenors. These instruments are used to calibrate these values.

> Why is the EUR OIS curve adjusted to reprice the EURUSD XCS precisely the EUR:USD-CSA curve?

The curve $w_{eu}$ is the EUR:USD-CSA curve. It is an object in its own right. The term "adjustment" is used to relate it to the local $v_{ee}$. A better term might be "deviation to", since when the XCS basis is permanently zero the curves $w_{eu}$ and $v_{ee}$ are equal and this is arrived at via covered interest parity.

> what if this is not ESTR but EURIBOR_3M for example?

It doesn't matter. IBOR and RFR curves are constructed and known in local currency markets. The purpose of XCS prices is to derive the forward FX rate which in turn derive the unknown curves. You just configure a curve configuration framework to match the numeraire (i.e. interbank traded) instruments.

> "XCS usually trade with benchmark CSA based on SOFR, this is why we have always adjusted the non-USD discount curve."

Interbank markets generally trade quoting prices based on USD CSA. Therefore when configuring the framework you match your calibrating instruments, and their associated prices, to this setting. I.e. the USD leg of a XCS will be forecast and discount with the $v_{uu}$ curve whilst the EUR leg will be forecast with the $v_{ee}$ and discounted by the $w_{eu}$ curve.