Return to the USD swap pricing basics in a post-LIBOR world
Many of our clients have noticed some pricing differences in basic USD interest rate swaps. The differences were almost entirely in swaps with maturities less than 2 years.
The differences are typically:
During the transition from LIBOR to SOFR, we were regularly seeing the quarterly adjustment spread of 24.8 basis points for the rolls after 30 June 2023. This seemed somewhat strange because the ISDA spread was fixed in March 2021 at 26.161 basis points.
More generally, we see forward-starting swap pricing differing considerably between banks.
The interbank SOFR markets are based on a 2-day settlement delay whereas many corporate trades (to match underlying assets and liabilities) have 2 or 5-day lookbacks. This results in pricing differences especially in steeply inverted rate curves.
This paper uses a simple example to demonstrate how these differences can arise in a pricing model. While this example uses linear interpolation, my testing in cubic spline interpolation gives similar results in the current USD curve.
The maturity model and algorithm
In a LIBOR pricing model, the curve for the first 2 years was typically built from LIBOR and the Eurodollar futures prices. The straight dates (e.g., the quarterly roll dates) for the spot swaps were calculated by interpolating between the implied futures yields and building the curve.
In the SOFR pricing model there is actually a choice. The SOFR markets developed from inputs more related to deposits and loans so shorter dated swaps, say less than 2-years, had basic pillar points from 1 – 12 months and then 18 and 24 months. SOFR futures did not exist when these trading conventions were developed.
As expected, the SOFR swap rates derived from the SOFR futures align with those published on screens with pillar points as described in the previous paragraph. This is unsurprising as the two markets (OTC and futures) are (and should be) very closely connected and are regularly used by many participants.
A rate for a series of one-roll, quarterly swaps can be calculated by interpolating 3-month SOFR OIS and the SOFR futures out to 2 years. The following chart and table shows this calculation based on the closes of 19 June 2023.
The curve is shown in the following chart.
I use the notation of, for example, 0X3 as a one roll swap stating spot and maturing in 3 months. The rates in the following table are derived from those shown in the chart above.
IMM Swaps when spot date is the first IMM date
I notice greater differences in forward start swaps and particularly IMM dates, i.e., swaps rolling on the IMM dates.
If the IMM dates are within a few days of the spot-start swaps, then things line up very well. This can be seen in the following table. In this case, I am assuming the current spot rate is 21 June 2023 to correspond with the June 2023 IMM date.
Note that the rates do differ slightly. This is because IMM dates are not exactly the same as the straight swap dates, so small interpolation differences will be present.
But the rates are generally quite similar and eventually solve to very similar swap rates in columns 3 and 5.
IMM Swaps when spot date is not the first IMM date
This is where the differences become meaningful. I have moved the spot date to 8 May 2023 and kept all the rates the same. This time the rates derived from the swap rates are often quite different to those of the SOFR futures which will not change.
The results are in the following table.
When I move the spot date back to mid-way between IMM dates, the one-roll swap rates and IMM date rates start to differ as seen in columns 2 and 4. Similarly, the maturity swap rates differ as well.
How can this be happening?
The explanation
This effect arises from the fact that the rate curve (as seen in the chart above) is not a smooth curve. It has a few non-linearities (i.e., it is not a straight line) which causes differences if the pillar points of the underlying curve and the swap being priced do not perfectly align.
This is also known as a double interpolation problem. It occurs quite often in swap pricing and is generally exacerbated by ‘bumpy’ input prices from futures when the spot date is between IMM dates.
I also note that the interpolation method is critical. Whether you assume linear, cubic spline or the myriad of more exotic methods, you must understand the positive and negative aspects of interpolation in short-dated swaps where differences are not averaged out over many rolls.
If you ignore the interpolation characteristics (especially for complex methodologies) then you may pay a significant ‘tuition fee’ in a pricing error.
Implications for buy-side firms
The fact that different banks can and do price forward swaps differently means you will likely see different prices for these swaps from banks. While this seems unusual, it is quite expected as they all use different interpolation and pricing methodologies.
Buy-side firms could consider forward-start swaps rather than spot-start swaps. The forward nature of the swap will often create the price differences. The best dates are usually between the IMM dates where most deviations will be maximised
Implications for banks
The golden rule is start with the underlying hedge. If you are using futures, then be aware of a possible double interpolation problem and always revert to using IMM dates to build the curve.
But if you do not use SOFR futures, then price off spot-start swaps and hedge with other banks using the same inputs. You will likely work this out by simply asking for prices and back solving to the pricing methodology.
Summary
There is no ‘absolutely correct’ methodology for pricing swaps with maturities less than 2 years. But all participants should consider:
Where is the start date of the swap relative to the IMM dates.
Whether you use SOFR futures for pricing and hedging. If so, then align the pillar points accordingly.
For buy-side, consider using forward-start swaps to achieve your best price and find the banks with the methodology which will give that outcome.
For banks, consider the hedging strategy and include this in your pricing methodology.
The interpolation method needs to be well understood. All have positive and negative features which should be considered in any short-dates swap price.
Over my many years of pricing and running books, the pricing and management of short-dated swaps is the most complex skill to acquire. Revaluation systems and pricing systems tend to be similar at banks (often the same) so a divergence between the hedging and pricing assumption will not be immediately apparent.
But they will converge at some point and the P&L will reflect the correct or incorrect assumptions.
As always, there is no free lunch.