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Par Swap Rates Under OIS Discounting and Dual Curves

Article Quant Q&A · Author: Sinbad The Sailor
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Summary

The document raises a pricing question about initiating an interest rate swap when discounting and floating-rate projection use different curves. Under a single-curve setup, the floating leg is treated as having value equal to notional at inception, so the fixed coupon can be chosen as the par swap rate that makes the two legs equal.

With OIS discounting and a separate LIBOR projection curve, that equality between the floating-leg value and notional no longer follows automatically. The question therefore asks how to reconcile a coupon that equates the fixed and floating legs with the market’s quoted par swap rate. The document presents the issue but contains no answer, valuation procedure, numerical example, or evidence resolving it. It is useful as a statement of the dual-curve pricing distinction and of the specific point that needs clarification, rather than as a complete method for calculating the inception rate.

Key ideas

  • In a single-curve setup, the floating leg is commonly treated as worth par at inception.
  • OIS discounting and a separate LIBOR projection curve break that automatic par relationship.
  • The fixed coupon must be selected so the discounted fixed and projected floating cash flows have equal value.
  • The document poses the pricing issue but does not provide its resolution.

Tags

Fixed incomeDerivatives pricingStatistics
Full text
# Confusion about Initial Pricing IRS with Dual Curves


# Confusion about Initial Pricing IRS with Dual Curves












This is my first time delving into dual curves, or multiple yield curves. A question struck me about using OIS discounting when choosing the swap rate of a new IRS.

Without multiple yield curves I thought it went something like this

$$ V_{swap} = V_{fixed}-V_{float} ~~ \text{with $N$ as par}. $$

where at the time of inception, $V_{float} = N$ due to the fact that the discount and projection rate were the same. This way, we simply put the coupon $C$ of $V_{fixed}$ such that it equals par. Thus $C$ is a par-yield and can be used as such when looking at market data.

However, if we discount the cash flows of the instruments with an OIS rate, it mustn't hold that $V_{float}=N$ initially. If it doesn't, how can we choose a $C$ such that $V_{fixed}$ both becomes valued at the float value, but also becomes a par-yield.

To put it more explicitly, we want

$$ N(\sum_{i=1}^{n-1}Cd(T_i)+d(T_n)) = N(\sum_{j=1}^{m-1}\text{LIBOR}_jd(T_j) + d(T_m)) $$ where $d(T_i)$ are the discount factors, $c_j$ is the appropriate LIBOR rate, n is the number of fixed cash flows and m is the number of floating cash flows. Again, a coupon cannot be chosen so that it is both a par-yield and necessarily equates to the FRN initially.

Is there something I am missing here? I've searched through some articles but all they say is that "something has to be adjusted" in this scenario. But never what that exactly might be.

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.