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Dual-Curve Swap Pricing and Collateral-Dependent Forward Rates

Article Quant Q&A · Author: Skrrrrrtttt
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Summary

The document explains how to interpret forecasting rates when pricing collateralized interest rate swaps with separate discounting and projection curves. Under OIS discounting, market prices for at-the-money Libor swaps imply expected forward Libor rates under the measure associated with that collateral arrangement. Those inferred rates therefore correspond to the collateralization of the swaps used to derive them, much like a forward rate agreement under the same terms.

To infer an uncollateralized rate, the answer says one would theoretically need a change of measure that accounts for differences in the distributions of future paths implied by the two market conventions. In practice, market participants often use the expectation inferred under the original measure without a separate adjustment, on the view that changing the discount curve has a larger effect on valuation. The response offers a conceptual explanation rather than a derivation or numerical example, and it characterizes practice as a common approximation rather than a universal rule.

Key ideas

  • OIS discounting and Libor forecasting use distinct curves in the dual-curve framework.
  • Swap prices imply forward Libor expectations under the measure associated with the swaps’ collateral terms.
  • Uncollateralized expectations theoretically require a change of measure that reflects different future-path distributions.
  • Practitioners often omit that adjustment, treating the discount curve change as the more material pricing effect.

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Fixed incomeDerivatives pricingStatistics
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# Proper Method for pricing Interest rate swaps using dual curves


# Proper Method for pricing Interest rate swaps using dual curves












I am aware that under the dual curve method for pricing standard collateralized fixed floating interest rate swaps, that first a discounting curve should be constructed e.g. OIS Discounting curve, as well as a separate forecasting curve that is used to forecast the cash flows e.g. 6m LIBOR.

And as far as I understand the existing swaps that are used to bootstrap the forecasting curve are themselves collateralized, will those rates be the same implied rates as if we were to bootstrap the curve using uncollateralized instruments

## Answer by Phil H (score 1, accepted)

https://quant.stackexchange.com/a/51537

The price of something under OIS discounting is (supposed to be) the expectation of its value under a particular measure, which specifies the measure and the interest rate of the collateral account etc.

So given the price of a set of ATM Libor IRS and their relevant discount curve, the immediate thing to do is to infer the markets' expectation of forward Libor, under that measure. So we're still looking at that rate e.g. as a FRA with the same collateralisation as the swap.

To see the rate priced without collateralisation, we have to somehow account for the change to that measure. The theoretical approach is a change of measure, which accounts for the differences between the distributions of future paths built into the measures by the market etc.

In reality people seem to usually essentially treat the expectations derived under a given measure as though they didn't need adjusting because the change of discount curve makes more difference to the final price.

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.