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Building FX Forward Curves with Forward Quotes and Cross-Currency Basis Swaps

Article Quant Q&A · Author: Boubakar
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Summary

The document explains why local overnight indexed swap curves for two currencies alone may not reproduce market FX forward rates. The difference arises because market participants cannot necessarily borrow or lend at those rates in both currencies. A cross-currency basis captures the funding imbalance between the currencies and is needed to fit the observed market pricing.

For maturities under one year, FX forward contracts can provide curve inputs; for longer maturities, cross-currency swaps supply additional points. The described approach uses swap cashflows and the market basis to solve for the forward rate at a maturity so that the swap has zero value, then bootstraps across maturities. The answer notes that swap conventions differ, especially for non-deliverable currencies, and that the basis is market-driven. The discussion is explanatory rather than a full curve-building specification: it does not detail discounting, collateral, interpolation, or implementation conventions, all of which matter in practice.

Key ideas

  • OIS curves alone may not capture the funding conditions embedded in FX forwards.
  • Cross-currency basis reflects differences in access to funding between currencies.
  • Short-dated FX forwards and longer-dated cross-currency swaps provide curve inputs.
  • Use swap cashflows and observed basis to solve forward rates, then bootstrap to longer maturities.
  • Conventions can vary, particularly for non-deliverable currencies.

Tags

ForexFixed incomeDerivatives pricing
Full text
# FX forward curve building


# FX forward curve building












Can someone explain which curves are used to calculate FX forward rates? I have the idea that it involves using the local OIS curves for both currencies, but my calculation shows that it is not the case.

Thanks!

## Answer by dm63 (score 6)

https://quant.stackexchange.com/a/31554

Your method assumes you can borrow or lend at OIS in both currencies, but in practice you cannot. That's why there is a current basis swap market , where you lend at OIS in one currency versus borrowing at OIS + X in the other currency , where X is not zero. That is the missing piece of your calculation.

Why, you may ask , is X not zero , as many textbooks assume? Apparently because most institutions cannot actually borrow and lend at OIS in all currencies. For example , only certain banks in the US have access to the Federal Funds market.

## Answer by jaehyukchoi49 (score 6)

https://quant.stackexchange.com/a/32610

I agree with dm63 in that cross-currency swap (CCS) is essential for building FX forward curve. Let me add/correct two things:

- FX curve < 1 year can be backed out by FX forward contract. CCS is typically longer than 1 year, so you need it for the long-end of the FX curve.

- CCS swap is typically exchange of 3m USD LIBOR vs 3m FOREIGN LIBOR (or equivalent) + BASIS on top of the notional exchange at the start and end of the swap. (CCS against non-deliverable currencies are a bit different.) Anyway, CCS doesn't involve central bank rate like OIS. The BASIS is determined by market force depending on the need for the USD funding from the foreign counterpart (or vice versa). Given the BASIS, you solve the FX forward at the swap maturity which make the exchange of the two cashflows zero value. Starting from the shorter maturity CCS, you can boot-strap to the longer end.

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