When CTD Discount-Factor Ratios Match Across Collateral Currencies

Summary

The document asks why a ratio of discount factors for a generic currency and the euro can be equal under euro-collateralized and US-dollar-collateralized agreements. The answer interprets this relationship through forward foreign-exchange pricing: it amounts to assuming that the generic-currency/euro forward rate is the same under either collateral convention.

That equality is an approximation rather than an unconditional identity. The answer says it holds when the theoretical convexity adjustment is ignored. Such an adjustment would arise from nonzero covariance between the generic-currency/euro exchange rate and the euro/US-dollar basis. The explanation reports that ignoring this effect is standard market practice, but gives no derivation, market data, or estimate of the adjustment’s size. Readers should therefore treat the relationship as a simplifying convention whose accuracy depends on the relevance of that covariance in the setting being modeled. The discussion is brief and assumes familiarity with collateral discounting, cross-currency basis, and forward FX rates.

Key ideas

• The discount-factor ratio equality corresponds to equal generic-currency/euro forward rates under the two collateral conventions.
• The equality relies on ignoring a theoretical convexity adjustment.
• The adjustment is associated with covariance between the exchange rate and the euro/US-dollar basis.
• The answer describes omission of that adjustment as standard market practice, without quantifying its impact.

Tags

# Cheapest-to-deliver (CTD) discount curve II


# Cheapest-to-deliver (CTD) discount curve II












This is a follow up question on this thread

I have come across the following relationship in a CTD curve bootstrapping routine:

$$\frac{DF_{XXX}^{CSA.EUR}}{DF_{EUR}^{CSA.EUR}} = \frac{DF_{XXX}^{CSA.USD}}{DF_{EUR}^{CSA.USD}}$$

where $XXX$ is a generic currency (e.g. CAD) and $DF_{XXX}^{CSA.EUR}$ denotes a discount factor collateralized in EUR, etc.

Would some be able to explain why(and under what conditions) is this true?

## Answer by Antoine Conze (score 2, accepted)

https://quant.stackexchange.com/a/63516

The formula simply states that the XXXEUR forward FX are the same under CSA.EUR collateralization and under CSA.USD collateralization.

It holds if disregarding the theoretical convexity adjustment that would result from non zero covariance between the XXXEUR FX and the EURUSD basis. Disregarding the adjustment is standard market practice.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.