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FX Swap Valuation Across Discount Curves and FX Forwards

Article Quant Q&A · Author: PBD10017
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Summary

The document compares two ways to value a fixed-for-fixed FX swap under multiple curves. One approach discounts the foreign-currency leg using a curve adjusted for cross-currency basis, then converts its present value at spot. The other converts each foreign cash flow using FX forward rates and discounts the resulting domestic-currency cash flows on the domestic curve. Both approaches are expected to agree when they represent the same instrument and use consistent market inputs, because otherwise an arbitrage opportunity could arise.

The response explains the intuition: forward rates embed the interest-rate differential, while the adjusted foreign discounting approach reflects that differential through the curves and basis. It does not give a derivation, numerical example, or diagnosis for the valuation discrepancy raised by the question. Agreement depends on consistent conventions, curves, basis, cash-flow details, and inputs; the answer asks for worked examples and the size of the difference before assessing the cause.

Key ideas

  • FX swap legs can be valued by discounting foreign cash flows and converting at spot.
  • Alternatively, FX forwards can convert those cash flows before domestic discounting.
  • The approaches should agree under consistent inputs when they represent the same instrument.
  • Forward rates incorporate interest-rate differentials, while basis-adjusted curves reflect them through discounting.
  • The response gives no numerical reconciliation and leaves the reported discrepancy unexplained.

Tags

ForexFixed incomeDerivatives pricingArbitrage
Full text
# Equivalency of FX forwards and FX fixed for fixed swaps? Are they still the same under multiple curves environment?


# Equivalency of FX forwards and FX fixed for fixed swaps? Are they still the same under multiple curves environment?












I am encountering two approaches for valuation of FX swaps (fixed for fixed, e.g. fixed USD payments for fixed EUR payments) which seem to result into different values although in theory they should be the same.

- Bloomberg's SWPM Bloomberg's SWPM takes EUR cash flows, discounts them at USD discount curve adjusted by EURUSD basis curve and then converts the EUR NPV into USD using EURUSD spot.

- Using FX forwards some choose to convert EUR cash flows into USD cash flows using EURUSD forward rates. Then discount the converted USD cash flows using a USD curve and this gives the NPV of the EUR leg in USD. Subtracting the other leg (USD) gives the NPV of the entire swap.

I always thought these two approaches should result into the same value, but they don't. Is there a reason for this? Is one better than the other?

## Answer by rupweb (score 1)

https://quant.stackexchange.com/a/23173

Yes I guess in theory they should be the same value, but only because in practise there is no arbitrage between the 2 approaches presuming you end up with the same instrument. I guess you mean that.

- getting a EUR net present value from USD curves adjusted for EURUSD basis is perhaps 1 way of calculating interest rate differentials. Presumably it gives a correct differential for EUR NPV which you then convert to USD at spot.

- Alternatively, getting a USD net present value from FX forward prices that have the interest rate differential already calculated, and then calculating an NPV from a USD curve.

Yeah, so the use of the different curves is like, priced under competition, and should be non arbitrageable otherwise we'd all be working that trade. So how much difference are you seeing in the values? Can you give 2 worked examples?

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