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Short-End Curve Bootstrapping and Missing Overnight Rate Nodes

Article Quant Q&A · Author: Frido
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Summary

The discussion examines why a USD SOFR overnight-indexed swap curve may show discount factors referenced to the curve date even when the quoted swaps settle two business days later. In the example, the one-week instrument’s discount factor is calculated using the interval beginning on the curve date, rather than only the period after spot settlement. The answer explains that curve nodes are placed at instrument maturities and that, when overnight and tomorrow-next instruments are absent, the first two days’ rates are effectively extrapolated backward from the one-week swap rate.

This treatment is described as a standard modeling choice when curve construction limits the available degrees of freedom. Its importance depends on the use case: small short-end errors generally have little effect on medium-tenor pricing, while forward- or spot-starting swaps are unaffected in the described setup. The caveat is that unusual overnight turns, such as year-end effects in some markets, can make accurate short-rate nodes material. Curve design should therefore reflect the purpose and required precision of the resulting curve.

Key ideas

  • Curve discount-factor nodes may be placed at instrument maturities even when instruments settle after the curve date.
  • Omitting overnight and tomorrow-next instruments can leave the first days’ rates to backward extrapolation.
  • The impact of short-end extrapolation depends on the product and pricing purpose.
  • Unusual overnight rate turns can make explicit short-rate modeling important.

Tags

Fixed incomeDerivatives pricingRisk management
Full text
# ICVS 490 curve date vs swap settlement date


# ICVS 490 curve date vs swap settlement date












The more I look into curve construction the more questions I have. For example, the ICVS 490 curve (USD OIS SOFR vs Fixed):

The calibration/input instruments are USOSFR* instruments, i.e. OIS SOFR swaps with various tenors.

My understanding is that USOSFR* swaps settle at T+2. In other words the accrual period starts at T+2 and at maturity date of the swap, i.e. T+2+tenor fixed vs floating is exchanged.

Next, if you use these swaps as bootstrapping instruments then the discount factors (DFs) you back out are DFs for the period [T+2,T+2+tenor].

However it seems to me that ICVS 490 doesn't do this.

For example, for those who have access to BBG, let's look at curve date 29th Oct 2025. Then the 1 wk swap used in the curve has market rate 4.04870. It settles at T+2 (31st Oct) and expires on Nov 7.

The discount factor corresponding to Nov 7 is 0.998989. It is easy to verify that this is 0.998989 = 1/(1+0.0404870*(nov 7 - oct 29)/360). Put differently 0.998989 is not equal to 1/(1+0.0404870*(nov 7 - oct 31)/360)

My question:

Why are they not using any assumptions of O/N and T/N rate? As written above, my understanding has always been that if you use swaps as bootstrap instruments with settle date T+2 then the DFs you impute are for the intervals [T+2,T+2+tenor] and you still need the discount factor D(T,T+2) to have a curve with settle date T. Or have I been wrong all along?

## Answer by Attack68 (score 2, accepted)

https://quant.stackexchange.com/a/85175

If you look at https://rateslib.com/py/en/latest/z_swpm.html I believe this is a replication of the 490 BBG Curve tenor points. It happens to be as of 27th Sep '23.

Yes the 1W instrument runs from (spot) 29th Sep to 6th October.

The discount factor nodes (parameters for the curve) are set as the maturity of the instruments.

As you rightly point out this curve does not have the O/N or T/N instruments, so the result is that the first two days' rates on the curve are backwards extrapolated from the known data of the 1W rate. This is quite standard. In the link above, it is because the Curve is allowed only a set number of degrees of freedom and in order to satisfy the model the first two rates of the curve are forced to match those of the 1W Instrument.

Most of the time this will not matter: probably 99% of the time these backwards extrapolations will be sufficiently accurate for these rates, and the 1% of the time when they are not accurate, pricing a medium tenor instrument (>9M) this mis-pricing of 1 or 2 days' rates will not have any significant impact. If the swap is forward, or spot, starting then it will have no impact at all.

In Sweden, there are two overnight indexes, SWESTR, which is the O/N RFR type, and STIBOR-T/N. SWESTR has a year-end turn on one particular day often about -10% so this really does matter. Most days in the year we can get away with curves that omit or are poorly constructed regarding these super short rates, but ahead of Christmas we review and ensure that these curves will be accurately constructed and will capture these rates at the appropriate time (there are other elements that go into curve construction, particularly if MPC meeting dates are also fairly close to these dates).

So you are not wrong, I guess it just depends upon the purpose of the constructed Curve and the level of accuracy required for that purpose.

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.