Cross-Gamma Risk in Cross-Currency Swaps with Different CSAs

# Cross-Gamma from XCCY booked on different CSAs


# Cross-Gamma from XCCY booked on different CSAs












This question is about the cross-gamma arising from hedging trades on different CSAs.

As an example imagine the following set-up:

- We are paid EUR/USD Xccy Basis against a specific counterparty on an EUR CSA

(being paid the basis means that we execute a cross-currency swap where at inception of the trade, we post USD notional whilst receiving EUR notional; subsequently, until maturity of the swap, we pay EUR ESTR + Basis whilst receiving USD SOFR. At maturity, the nationals are returned).

- We hedge the first trade against a different counterparty on a USD CSA (so we do the opposite trade and we are received the basis on the USD CSA).

Does this set-up generate cross-gamma on the Xccy basis and if yes, is it positive or negative?

I think the answer can be approached by arguing about the collateral that needs to be received and posted as the basis goes up (or down), as I explain below. But I have also heard arguments based on discounting which give a different answer, so curious to hear what others here think.

Here is my reasoning:

> Imagine the XCCY Basis moves up. Then on the EUR CSA where we are paid the basis, we are in the money and receive EUR collateral. On the USD CSA, we are out of the money and need to post USD collateral. I believe (assuming rehypothecation) that the EUR collateral that we received can be exchanged for some USD via a new EUR/USD Xccy swap: this swap will be a basis receiver swap (we post EUR and receive USD notional at inception) -> so naturally, we'd end up being more received the basis as the basis increases, giving rise to "wrong way risk", i.e. "negative cross-gamma".

## Answer by Attack68 (score 2, accepted)

https://quant.stackexchange.com/a/81276

Pricing and Trading Interest Rate Derivatives discusses 4 types of similar combinations that give rise to cross-gamma risks:

- Identical trades with different CSAs,

- IRSs hedged by ZCSs,

- Non-MtM-XCSs hedged by Mtm-XCSs,

- Cash settled swaptions hedged by physically settled swaptions or IRSs

The reasons and profiles of the cross-gamma are slightly different in each case and the material was originally written in 2015, which was a few years before 2 major changes: IBOR transition and Mandatory Clearing. These regulation changes eliminated some of these effects, but it did not eliminate it all.

For example it is still possible, but less common, to trade IRSs bilaterally for various exemption reasons. XCSs are still not cleared so a fixed-fixed XCS which represents 2 IRSs and a float-float XCS is not cleared and hence implicitly contains 2 uncleared IRSs.

To your question, does your portfolio contain cross-gamma? Yes. Since cross-currency basis does not have a whole lot of volatility the net PnL one might accrue, that causes the cross-gamma, is probably quite limited. And in general cross-gamma risks are typically of the order 1/10,000'th the size of the delta risks multiplied by some factor relating to the tenor.

On the other hand for IRSs with different CSAs the cross-gamma is more relevant since outright interest rates are much more volatile.

Here is your suggested scenario properly setup:

```
from rateslib import *  # Python 3.12, rateslib 1.5.0

# Setup a Calibrating Market and a Hegding Solver
eur = Curve({dt(2024, 11, 28): 1.0, dt(2029, 12, 5): 1.0}, calendar="tgt", convention="act360", id="eur")
usd = Curve({dt(2024, 11, 28): 1.0, dt(2029, 12, 5): 1.0}, calendar="nyc", convention="act360", id="usd")
eurusd = Curve({dt(2024, 11, 28): 1.0, dt(2029, 12, 5): 1.0}, convention="act360", id="eurusd")

fxr = FXRates({"eurusd": 1.08}, settlement=dt(2024, 12, 2))
fxf = FXForwards(fx_rates=fxr, fx_curves={"eureur": eur, "eurusd": eurusd, "usdusd": usd})

solver = Solver(
    curves=[eur, usd, eurusd],
    instruments=[
        IRS(dt(2024, 12, 2), "5Y", spec="eur_irs", curves="eur"),
        IRS(dt(2024, 12, 2), "5Y", spec="usd_irs", curves="usd"),
        XCS(dt(2024, 12, 2), "5y", spec="eurusd_xcs", curves=["eur", "eurusd", "usd", "usd"])
    ],
    s=[2.05, 3.81, -1.75],
    instrument_labels=["eur5y", "usd5y", "xcs5y"],
    fx=fxf,
    id="solver"
)
```

With this in place I'll show you the output for your constructed trades and portfolio:

The gamma is negative and in this case it is almost exactly: -xccy delta / 10,000 * 5y = -25