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Building Discount Curves for Cross-Currency Swap Valuation

Article Quant Q&A · Author: Alisha
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Summary

The document explains why a curve is not defined by the fixed or floating legs of a swap. A curve instead supplies discount factors or forecast rates, and those functions must be distinguished when valuing a cross-currency swap. For a fixed-fixed USD-INR swap valued in USD, the answer calls for discount factors for each currency’s cash flows under the relevant USD collateral arrangement, together with the spot exchange rate to convert values.

Curve construction is model dependent. The USD discount curve can be inferred from USD interest rate swaps, while the INR leg may require a local forecast curve and discount curve, with the INR discounting adjusted to fit cross-currency market prices. The calibration instruments may be float-float, fixed-fixed, or fixed-float if the required market price adjustments are handled correctly; float-float swaps are commonly used because they are often more liquid. The discussion is a general framework, not a complete specification of collateral, conventions, or curve-building methodology.

Key ideas

  • A curve represents discount factors or forecast rates, rather than swap legs.
  • Cross-currency swap valuation needs discounting for cash flows in both currencies and an FX rate for conversion.
  • Curve construction depends on collateral assumptions and market calibration instruments.
  • Float-float swaps are often used to build cross-currency curves because they can be more liquid.

Tags

Fixed incomeForexDerivatives pricingRisk management
Full text
# For a fixed-fixed cross currency swap, can I use a curve with two floating legs to discount the cash flows?


# For a fixed-fixed cross currency swap, can I use a curve with two floating legs to discount the cash flows?












I'm doing a USD to INR fixed-to-fixed cross currency swap. The default curve has a fixed and a floating rate. However, the curve that I'm looking to use has floating rates on both legs. Would this curve be appropriate to use?

## Answer by Attack68 (score 3)

https://quant.stackexchange.com/a/41251

A pricing curve does not have legs.

A curve is a series of datapoints that defines either a discount factor (under some assumed model, e.g. USD cash collateral) for each business date, or a forecast interest rate (of some index e.g. 1M USD Libor).

Here is a curve: `GBP 3M Libor Forecast Rates: .. 13/Aug/19: 1.250%, 14/Aug/19: 1.252%, ..etc` Here is another curve: `GBP cashflows discounted under EUR Cash only CSA: .. 13/8/19: 0.99181, 14/Aug/19: 0.99112 ..etc`

To NPV your fixed-fixed cross-currency swap you need to have the `USD cashflows under a USD cash only CSA discount factor` curve and the `INR cashflows under a USD cash only CSA discount factor` curve and, to equate the values in USD, the USDINR spot FX rate.

How you generate these curves is model dependent. Always, the `USD cashflows under a USD cash only CSA discount factor` curve is calculated by analysing USD Interest rate swaps (IRSs). To get the `INR cashflows under a USD cash only CSA discount factor` curve you would typically need the local `INR IBOR forecast` curve and `INR discount curve`, calculated from local INR IRSs or otherwise and then adjust the `INR discount curve` to suit the mid-market cross-currency swap prices.

It would not necessarily matter whether the cross-currency swap prices were float-float or fixed-fixed or fixed-float, so long as you are able to make correct adjustments to accomadate those correct market prices. Most curve models are constructed with float-float cross-currency swaps since they are most liquid in most markets.

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.