muRisQ Advisory offers quantitative finance advisory services. Its services focus on trainings and workshops, benchmark transitions, interest rate model development and validation, product design and risk management strategies.
It has been 10 years since Marc published the first edition of his book Interest Rate Modelling in the Multi-Curve Framework: Foundations, Evolution and Implementation. He is now preparing a new edition as announced on his personal blog:
As a preview of the book, we have created a new page on muRisQ Advisory site with the content of the "convention" appendix.
The page is available at:
The title may sound prosperous as SOFR as been the main benchmark in USD interest rate (at least in term of PV01) for more than six months.
SOFR has been manipulated (Note: "manipulate: to use something, often with a lot of skill", Cambridge Dictionary ) by the Federal Reserve Bank by intervening in the repo market, up to USD 1,000 billions a day. If you go back to the last quarter of 2019, there were large variations in SOFR between the “normal” days and the days influenced by government actions (month end for balance sheet measurement and 15th of the month for tax payments). After the Fed interventions, the market impact of those influences disappeared; the public sector compensated one intervention by another.
This blog title refers to that “life” of SOFR, its reaction to external influences. In 2019 we proposed a blog on how to include that seasonal life into curve calibration. After a couple of years where the SOFR daily reading (above the Fed target rate) were completely flat, there has been a little bit of action over the last months. Figure 1 represents the spread of overnight benchmarks (SOFR and EFFR) above the target rate. EFFR is completely flat at 8 bps. SOFR shows more life. The days have been divided in a certain number of buckets: 1st (business) day of the month, 2nd day of the month, 15th of the month, last day of the month, (overnight period containing the mentioned dates) and the rest.
Figure 1: Spread over target rate for overnight benchmarks.
Over the last 6 months displayed in the above figure, the mean spread for the “rest” bucket is 6.42 bps. For the other periods, we measured the spread above that mean (i.e. spread of spread). The results are
| Day of the month | Spread (bps) |
|---|---|
| 1st | 2.92 |
| 2nd | 2.75 |
| 15th | 0.08 |
| Last | 2.08 |
There is indeed some impact around month-end/start. It is largely impacted by year end. We will see if the impact continues through the year.
Marc will present at the Quant Insights Annual Conference on 1 November 2023 (free online tickets available).
Marc's presentation will be based on his recent paper
Don't hesitate to contact us if you you are interested by implemented this approach or other term structure modelling.
The post title may sound a little bit oxymoronic. It is not the fault of this author but the fault of the regulators and the market…
Derivatives are split in different categories: ETD, OTC cleared and OTC uncleared. The meaning of “LIBOR” in each of those categories is different and is also different from LIBOR fixing. This usage of the same name to describe different things is the source of the confusion in the post’s title.
Note that this usage of “same name; different things” description is fundamentally different to the
Mathematics is the art of giving the same name to different things.Henri Poincaré
Poincaré meant different things with the same properties, to which similar truths can be applied. Here we have things that should be the same in a decent world but have different properties and to which different (legal) truth apply.
LIBOR fixing in USD have ceased as of 30 June 2023 (not really ceased, but that will discussed later).
The ETD derivatives went through a forced (arbitrary) conversion well ahead of the milestone date (on 14 April 2023 for CME). They have completely disappeared and are not the subject of this post.
The OTC uncleared derivatives are mainly traded under ISDA master agreement. We discuss only those here. The derivatives traded before January 2021 had uncertain fallback mechanisms under the ISDA definitions, or more exactly, there is a certain fallback mechanism (call 4 baks for indicative LIBOR quotes) but it is certain that the mechanism cannot be implemented in practice. Some of those derivatives have been transferred, through a protocol mechanism, to fall under the post January 2021 definitions; this case is described below.
The pre January 2021 do not yet have a legal issue, because LIBOR fixing still exists! This may seem in contradiction to the post title that claim an absence of LIBOR fixing; but again this is a contradiction only because the same name is used for different things (again not in the Poincaré sens). Maybe we will be in trouble with EU Digital Service Act (DSA) that may try to censure this as “disinformation” or “misinformation”!
The “true” LIBOR, representing the interbank lending rate, as ceased on 30 June 2023. But a “fake” LIBOR fixing has been imposed by the regulator, the FCA. This LIBOR is based on an arbitrary formula replacing the market based definition in use since 1986. The regulator imposing an arbitrary definition for a critical benchmark is possible thanks to post-Brexit change of the UK BMR regulation, in particular Article 23A. That change gives the super-power to the regulator to change a benchmark definition. If tomorrow you see that the temperature in London 20 degree Celsius for the while month of the December, this is most likely not a result of the climate changes but a result of the FCA decision to change the meaning of “temperature”. This “fake” LIBOR should stop on 28 March 2024. We will only see then the true legal impact of the transition on USD LIBOR OTC derivatives (and bonds, and loans, and structure products).
The post January 2021 trade do not have a problem either, because ISDA forced a “non-representative” clause in the new definition. The LIBOR trade do not reference LIBOR anymore, even if LIBOR still exists. This is because the regulator that have created a new definition for LIBOR have immediately declared that what they have created is not representative of the market it is suppose to represent — while the previous version was representative. Those trades have been converted to SOFR-linked derivatives that are not really OIS (see fallback transformers and subsequent posts). Unfortunately, there is no detailed public information about the volumes of those; we can not report on it.
We come to the next type: OTC cleared. For that category, some data is available and we describe below some figures from LCH.
The existing LIBOR swaps have been converted to SOFR swaps. The conversion has been done but spliting the existing swaps in 2, one LIBOR part that continues to the last payment of the LIBOR fixed before 30 June and the SOFR part that starts after that. That conversion has been done mid-May 2023. Since that date, the outstanding LIBOR swaps have decreased from 48 trillions to 273 billions. The decrease is almost linear from the amount on the conversion date to 0 three months later. Most of the LIBOR swaps have a LIBOR-3M benchmark, hence that period. There is a little bit of 6-month tenors and 12-month tenors, so the small residual that continues to disappear.
Swaps linked to LIBOR continue to be cleared at LCH. From 3 July 2023 to 13 October 2023, roughly 400 billions of USD-LIBOR swaps have been cleared. Those swaps are most probably coming from USD-LIBOR swaption with physical exercise. This is a non-negligible amount for a non-existing benchmark. Those swaps are immediately converted to SOFR and do not enter into the outstanding amounts described in the previous section.
If we look at the total outstanding amount in USD, it decreased only by 9.3 trillions over the period, even with the 47.7 trillions of LIBOR swaps disappearing. This is somehow expected. The LIBOR swaps converted to SOFR-linked swaps are not standard OISs. So no direct offset between the converted swaps and the standard one is possible. It would be possible to eliminate them with some compression cycles, but not all participants are part of the those cycles.
Note that GBP, JPY an CHF LIBOR swaps (also coming from swaptions) are still cleared 22 months after the discontinuation of their respective LIBORs. The amounts (in USD equivalent) are GBP 1.330 billions, JPY 4.54 billions and CHF 0.17 billions.
There is also 13 trillions of outstanding basis swaps. With the disappearance of LIBOR, those basis are certainly SOFR v EFFR. The US SOFR RFR adoption indicator for September is down to 68.3% from 76.2%, another sign that EFFR is still playing an important role in USD derivatives.
The short conclusion is that LIBOR transition is still in progress even in the absence of LIBOR!
We are pleased to announce a new working paper titled
is available in our muRisQ Advisory Model Development series of research papers. The paper is available on SSRN preprint repository at http://ssrn.com/abstract=4542275.
Bond futures are characterised by a set of underlying bonds; the short party has the option to deliver at expiry any of those underlying bonds. Consequently, bond futures embed a choice option between bonds with different maturities and coupons. The delivery mechanism also incorporates conversion factors that create an implicit strike. The option is impacted by different maturities and different moneyness for each bond. It is important to take into account the full term structure of volatility with smile. A recent paper Bang and Daboussi (2022) developed such an approach for swap rate based products like CMS. In this paper we extend their approach to cover futures and apply it to the specific case of bond futures. The method allows the analysis of the impact of smile, term structure of volatility and correlations between rates on the delivery option and convexity adjustment values. All of them have an impact on the valuation and risk management of bond futures.
The paper has been submitted for publication in an international peer-reviewed journal.
All the results presented in the paper are based on a proprietary implementation extending a production grade open source quantitative finance library. We would be glad to provide model development or model validation advisory services based on the theoretical and practical development described in the paper. Don't hesitate to contact us if you require advisory services related to the interest rate modelling or benchmark transitions.
LIBOR swaps were converted to SOFR last Friday (2023-05-19 after EOD). The amount outstanding before conversion was around 48 trn, this Friday (2023-05-26) the amount outstanding was down to 15 trn.
Even after conversion, there is still a non-negligible amount outstanding. This is coming from the LIBOR swaps that have still a fixing between the conversion and the last LIBOR representative fixing (2023-06-30). Most of the USD LIBOR swaps are based on USD-LIBOR-3M, with one month to go it is not surprising that around one third of the LIBOR swaps (15 out of 48) still have a representative fixing.
On the OIS side, the notional outstanding increased from 76 to 114 trillions. For OTC products, we don’t see the same offset as in futures that we described in US - Eurodollar futures a USD 8 million rounding error. There is no exact netting as it is same conventions for SOFR vanilla swaps (annual/annual money market) and the converted LIBOR swaps (semi-bond / quarterly money market). Maybe some compression run will reduce the outstanding notional.
The total for IRS plus OIS increased from 124 trn to 129 trn, probably due to the split of former IRS into the LIBOR part and the SOFR part.
Note that over the week, there were still 62 bn IRS and 48 bn FRAs traded. On the IRS side, it is not clear if it is the result of swaption exercise or maybe some BSBY swaps. For the FRA, we don’t have a direct explanation. On the FRA side, not that there is still 376 bn outstanding (unchanged with respect to last week).
We are pleased to announce a new working paper titled
is available in our muRisQ Advisory Model Development series of research papers. The paper is available on SSRN preprint repository at http://ssrn.com/abstract=4461418.
The ISDA designed fallback for cash-settled swaptions with collateral discounting generates swap rate and term structure dependent exotics. To analyse precisely the fallback impact a full term structure of rates and volatility modelling is required. A recent paper Bang and Daboussi (2022) developed such an approach for swap rate based products like CMS. In this paper we apply those techniques to the valuation and risk management of the instruments resulting from the cash-settled and physical delivery swaption fallback. In doing so, we provide some model validation for the approximations previously proposed in this context.
All the results presented in the paper are based on a proprietary implementation extending a production grade open source quantitative finance library. We would be glad to provide model development or model validation advisory services based on the theoretical and practical development described in the paper. Don't hesitate to contact us if you require advisory services related to the interest rate modelling or benchmark transitions.
Last week, Marc presented a workshop on the ”LIBOR transition: What could we have done better from a quantitative perspective?” at the CEETA Benchmark Congress in Warsaw.
Among the elements presented were the results of the fallback on euro-dollar futures at CME that were converted to SOFR futures. An interesting issue with the conversion is the number of decimals. The conversion was done using the ISDA/Bloomberg designed spread between USD-LIBOR-3M and the replacement SOFR of 26.161 bps. This is a number in basis points with three decimals. But the futures are quoted with only 0.5 basis point precision. A natural question is what happen to the extrta decimals? Simply it created a profit for those who understood it.
In the conversion, roughly 4 millions LIBOR futures were converted to SOFR futures, but the open interests in SOFR futures were almost unchanged. The 4 millions futures did not simply disappear, there were offsetting positions between the LIBOR and SOFR futures that had cancelling effects on the next day. The offsets were not between different market participants, but for the same participant having offsetting positions between LIBOR and SOFR futures. The spread between the two was known since 5 March 2021. The amount involved is 4,000,000 contracts with USD 1,000,000 notional each, this means USD 4,000,000,000,000 (4 trillions), not a notional resulting from an ignorant retail trader!
The closing price for the different futures on 14 April 2023 (the conversion date) were:
| Closing prices | Sep 23 | Dec 23 | Mar 24 | |
| ED | 94.965 | 95.33 | 95.785 | |
| SR3 | 95.225 | 95.59 | 96.045 | |
| Spread (bps) | 26 | 26 | 26 |
Take the strategy to buy SR3 at 95.225 and Sell ED at 94.965. The next day, the position is Long SR3 at 95.225 and Short SR3 at 95.22626. The total P/L is, irrespective of the price the next day: (P - 95.225) - (P - 95.22661) = 0.00161%.
The impact of rounding is (number of contracts x notional / basis points / quarterly / offsets * spread)
4,000,000 x 1,000,000 / 10,000 / 4 / 2 * 0.161 ~ 8,000,000
The rounding had an impact of USD 8,000,000.
Given that the spread is known since March 2021, one can guess that market makers have used any opportunity in the last 2 years to add ofsseting positions each time the tradable spread moved away from the 26.161 basis point. The actual arbitrage from the conversion is probbaly well above the figure indicated above.
Hopefully our readers were on the profitable side of the arbitrage.
Don't hesitate to contact us if you require advisory services related to the benchmark transitions.
The implementation note related to the Swap Rates and Term Structure Modelling described in one of our previous post has now been published on SSRN.
The note is available as a muRisQ Advisory Implementation Notes:
The paper is available on SSRN
https://ssrn.com/abstract=4438524.
WThis document contains implementation notes related to Bang and Daboussi (2022). We have extended the original paper by allowing actual accrual factors (not all 1) and non-annual frequency on the fixed side. The note first describes the detailed formulas in this extended setting. In a second part some choices of the implementation and results obtained are provided.
Don't hesitate to contact us if you require advisory services related to interest rate modelling.