Alechinsky and Margin

Today I visited an exposition on Pierre Alechinsky, a Belgian artist. Some of his works are visible at the MoMA.

Among the frames, there was the poster of a previous artist exposition in New York titled  
Margin and Center.

Margins are following me everywhere now, even in art shows!


Algorithmic Differentiation training

I will present a training session on AD in December.
Location: Warsaw, Poland
Date: 13 December 2017 (full day)
Organizer: CEETA - Tomasz Dendura

Algorithmic Differentiation in Finance

Algorithmic Differentiation (AD) has been used in engineering and computer science for a long time. The term Algorithmic Differentiation can be explained as the art of calculating the differentiation of functions with a computer. AD is now also a standard tool in quantitative finance.

The workshop presents AD from a practical point of view and targets quantitative analyst, risk manager and developers working in finance. The focus is on the foundation of the method and the idiosyncrasies of the applications in finance. Different implementation alternatives are presented, allowing each participant to adapt the general method to his needs.

  • Introduction
    • AD is magic (or not)!
    • Exact derivatives
    • Finite difference
    • Development time v running time
  • The Principles of Algorithmic Differentiation
    • Derivative
    • Composition
    • Algorithm: assignment
    • Algorithm: branches
    • Algorithm: loops
  • Application to Finance
    • Basics: Black formula and SABR
    • Interest rate sensitivities
    • Monte Carlo
  • Automatic Algorithmic Differentiation
    • Standard Algorithmic Differentiation by Operator Overloading
    • Adjoint Algorithmic Differentiation by Operator Overloading
    • Automatic Algorithmic Differentiation applied to finance
    • Mixed Algorithmic Differentiation implementations
  • Application to Finance (2)
    • Automatic Algorithmic Differentiation applied to finance
    • Non-derivatives with respect to inputs - sticky smile
  • Calibration
    • Curve calibration
    • Model calibration and implicit function theorem: exact calibration
    • Model calibration: least-square

The lecture notes of the workshop are provided in the form of the recently published book Algorithmic Differentiation in Finance Explained, Palgrave (2017).

The workshop is backed by open source code (freely available on Github). On one side, the code is composed of tutorials of increased complexity. Those tutorials present the different fundamental principles of AD and propose several implementation for each of them. Another side of the code used is a full production grade quantitative finance library using AD as one of its tools. That library is used in production by hedge funds, banks and clearing houses.

Learning outcomes:

The mathematical foundations of Algorithmic Differentiation methods.
The effective application and use of AD in finance.
Beyond vanilla implementation: further efficiency gains specific to finance.

I'm also propose the course as in-house training. Don't hesitate to contact me for more details.

Game of Benchmarks: Season 1 - Episode 4: Transition and transition team.

Transition to a different world is difficult, more difficult than inventing a better world from scratch. In this episode, I’m looking at the transition.

There are probably more questions than answers. But, as you know,

In mathematics, the art of posing a question is more important than the art of solving one.
Georg Cantor, 1867.

Is transition important?

Valuation of flow derivatives, and in particular Libor linked derivatives, and the associated collateral discounting framework are vanilla in name but certainly not simple if you have to implement them from first principles. The impact of features that were considered before the Great Financial Crisis (GFC) as “details” are nowadays very significant. You can not simply replace one benchmark by another and hope for the best. There are numerous items to take care of, like valuation framework, non linear effects, convexity adjustment, risk management strategies, liquidity, regulatory impacts, etc. I don’t know if, from a purely legal perspective, you can replace a benchmark with something else by using a legal trick - like an act of parliament, as suggested by some lawyer — but certainly not from a financial perspective.

From a financial perspective, changing a benchmark will create winners and losers. Selecting a new benchmark and imposing it on everybody (with or without spread) may look easier from an external eye, but is only removing some symptom on the short term, not the disease. Derivatives are not only a legal document or a value, there are mainly a risk management strategy, often spanning over several decades. To successfully deal with a change of benchmark, one needs to create new strategies for those decades of risk management.

Behind each derivative linked to a benchmark there is a risk management strategy (or risk taking one, but this is a very small part). If Libor is changed or discontinued, one has to review each contract individually, reassess the risk management strategy - including all the collateral, regulatory and valuation aspects - and have a bespoke new strategy. Imposing an external artificial solution is not helping on that review. If your review concludes that the proposed solution does not fit perfectly your needs, you have anyway to engage in discussion with the trade counterparts on what to do; if the proposed solution fit exactly the strategy you designed, there was no need of an external solution as you had created it already. Whatever the proposed default solution is, I don’t see how it could fit the requirement of the majority. Not that I know that the choice that still has to be made will be bad, it is that I know that there are many diverse reasons to use benchmark-linked derivatives and a unique solution can not cover the majority of those reasons.

One important aspect of benchmark discontinuation is the disappearance of an underlying. If the previous strategy can not be attained anymore, i.e. there is no perfect replicating strategy anymore, you cannot reason in term of generally agreed value anymore. The standard  valuation of derivatives is based on replication strategy. There is no accepted common price without hedging strategy. If you remove the hedging strategy, the risk preference of each user is coming into play. No price compensation for the change of risk can be externally calculated anymore. We are back to the fundamental and evanescent price of risk! For each user, the price is what he estimates it is, not what an external party, even an independent expert, estimates. We may be able to set absolute bounds on that price, but no absolute exact value.

Transition from LIBOR, if required, will be extremely important. It will be costly (at the very least in time). Multiple solution will be designed and multiple solutions will be used.

Transition team

Who need to make proposal? As described below, ISDA is running some working group on new benchmarks and fallback. This will be a consensus opinion from the derivative industry. A similar working group is probably required from the end-users perspective. End-users form a  more diverse group. They will probably not be able to come to a consensus, but certainly they require to be informed of the implications and plan their strategy. Regulators are also running some working groups on the subject. Those regulators are usually finance industry regulators, the point of view of end users may or may not be in their scope. Maybe some consultation should be organised and the authors of contribution to the consultation invited to present their findings in a conference. Such a gathering, would probably not solved the question of a general agreement, but would certainly be an efficient way to expose the problems associated to benchmarks and benchmark changes, it would expose the different opinions and possible solutions.

Value destruction?

When I said earlier that there will be winners and losers, it is likely that there will be more losers than winners. As much as a willing transaction is in theory a win-win situation, an imposed new transaction — and a change of benchmark is a new transaction  — can be a lose-lose situation.

With new overnight benchmarks, replacement of LIBOR, new CSA reference rate, all trades under that CSA are economically changed: this is a lot of new trades and trade amendments Should they be treated as new trades from a regulatory margining and clearing rules perspective? To my opinion, the answer is yes. But maybe regulators, who are currently reviewing the coherence of the regulation put in place since the GFC, will take this opportunity to review the rules and change them in the direction of principle based regulation which would allow the decrease of risk and not rigid rule-based regulations.

The next transition

Planning for the dismissal of the next benchmark need to the prepared when trades referring to that benchmark are entered into, not when the benchmark is about to be discontinued. The design of the new benchmark cannot would not be complete without their fallback provisions. This would be similar in idea to the European Single Resolution Board (SRB). To my opinion, some of the SRB foundations are good ideas, at least the “R” planning part. As a believer in diversification, I’m not a fan of the “S” part nor of the associated fund, but this is a different subject. We now have a USD 300 trillion problem with the potential discontinuation of LIBOR; as much we need to deal with this problem now, as much we need to design a solution that does not simply kick the can down the road but contained the structural solution to avoid the repetition of a similar problem. A good fallback procedure, defined and accepted by the trade counterparts before the trades are entered into is necessary.


The UK reform group announced some month ago the support for a new SONIA, over other alternatives like RONIA and SONET. Among the arguments for that choice, one was that it is a better benchmark for the future “because it is use for discounting”. That argument is confusing the cause and the consequence. Sonia is used for discounting because of collateral remuneration; Sonia discounting is OIS discounting because the OISs use Sonia as reference; the collateral is remunerated at SONIA because it was the only benchmark rate available; it is the reference rate for OIS because it was the only rate available. The argument can be summarized by “SONIA is a better benchmark for the future because it was the only benchmark available in the past”. If we create a new index (RONIA, SONET or RIR), we have the possibility to change the market. Changing the market is difficult, as I have argued over the last years (see here and here), but possible. The difficulty of the change is a valid argument. But if that argument is used, it should be used explicitly, no hiding behind a pretense of risk free, reform or best practice.

Summary: The Working Group in charge of reference rate reform has decided to keep SONIA because it was the benchmark used in the past, i.e. the reform group has decided not the reform.

From an economical perspective, using a secure rate for collateral will always make more sense. The working groups for USD (SOFR) and CHF (SARON) have decided in that direction. This is a change with respect to current market practice. We will discover how difficult it is in practice over the coming years.

Fallback rate

With the FCA announcement of removing its support to LIBOR, the discussion about a potential fallback rate reform in ISDA master agreement have accelerated. ISDA has created a fallback committee that is working on proposals. The committee has already proposed some potential approaches to the fallback. At this stage those are only technical discussions, but we can expect that the final proposal will not be too far from those initial discussions.

One proposal would be to fallback LIBOR on OIS benchmarks plus a deterministic spread.

And OIS is the swap, not directly the overnight benchmark underlying it. A new benchmark would be created, based on the term swap, itself referencing the overnight benchmark. This would be adding an extra layer of rate benchmark: the benchmark is based on a derivative referencing the composition of another benchmark over a certain term. The OIS benchmark would need itself to be created. On what data would that benchmark be based? On some market players estimations, as the LIBOR and ISDAFIX, or on actual trades? If on estimates, how do we avoid the same manipulation risk than the one existing for LIBOR? If based on actual transaction, what is the reference data repository? Is it public and to which extend are the underlying trade details available?

Once you have dealt with the “OIS” part, you have to deal with the “spread” part. The idea would be to look at the market spread between OIS and LIBOR in a period before the discontinuation of the LIBOR. That spread would be recorded and applied unchanged for the remaining life of the existing LIBOR referencing trades. It will be a deterministic spread that would not be revised in the future, whatever what happen in the market after that date. It is not clear to me if the spread would be computed spot between LIBOR and OIS benchmark or on the full term structure of the rates, between LIBOR instrument calibrated curves and OIS calibrated curves. Using a spot spread would change the value of IRS significantly, repricing all of them from a forward spread to a spot spread; the difference can easily be 10s of basis points on some parts of the curve. Using the term structure of spread would require an acceptation of the way to compute the term structure of spread for the curve up to 50 years for each business day. Even leaving aside the question of the reference data (11:00 a.m. London, when LIBOR is fixed but the US market is barely open?), just the question of interpolation (see for example this blog or my multi-curve book) is enough to have also 10s of basis points for some dates. In any case, the question of the manipulation will still exist. If there has been manipulation in the past to gain a one day worth of fixing, what would be the incentive to manipulate if the gain is in one go for 50 years worth of fixing?

Suppose that we have solved all the problem of manipulation and methodology for the fixing of the spread. The spread is chosen to have (at best) the same valuation of swaps with standard collateral rules on the discontinuation date. But does that mean that all the other financial instruments related to LIBOR will have the same value on that date? The answer is clearly no. The spread will simply make the expected value (in the appropriate measure) of the LIBOR-linked coupons equal to the one of the OIS-linked coupons plus spread at that one date. Obviously the prices at other dates will be different (and LIBOR will not be observable anymore), but also the instruments that are not depending only of the expected value of the coupon will be different. This includes the swaptions, cap/floors, STIR futures, swap collateralized in a non-standard way, FRN, CMS, etc.

If we take the case of the CHF market, the current ON benchmark will cease to exist in December 2017, the replacement overnight index exists, it is SARON. To my knowledge no (liquid) OIS market on SARON yet exists. This means that the fallback method based on OIS benchmark would fail. Hopefully the market will start to exist at some stage, will become liquid enough in a couple of years so term OIS benchmarks (up to 12M) can be created. The same question will be relevant in the USD market when Effective Fed Fund Rate (EFFR) is replaced by SOFR. If there is not a liquid enough market by the time LIBOR is discontinued, we will need a fallback to the fallback.

The fallback methodology and data will need to the transparent, for each participant to decide if he want to enter into the standard fallback proposal or enter in a specific agreement with their counterparty. That may requires the full data set used for analysis to be made available without restriction to all interested parties, including end-users, academics, consultants and obviously private citizen. The same will be require for some code that implement the proposed methodologies, including full flexible multi-curve framework and curve calibration. Full term-structure interest rate model coherent with the multi-curve framework will also need to be made available to estimate the non-linear impacts of those spread changes. I’m currently researching on such models, but I don’t know any of them that fulfil all the requirements for such an analysis; the search is still on.

Once a fallback methodology has reached a (relative) consensus, what to do with it? Is it imposed to all users? In that case, I’m expecting multiple years of legal disputes. Is it proposed as a starting point of discussion? Is there also a methodology proposal to estimate the valuation impact of such changes? How is the inherent conflict of interest of market participant being part of the fallback committee and party in trades dealt with? Is the methodology validated by independent external expert? If yes, who is financing those experts? What is the involvement of regulator in choosing or imposing the methodology? The regulators have imposed clearing mandate for derivatives linked to some benchmarks and not to others. Is there a conflict of interest from the regulators regarding the future of those benchmarks? Can the clearing houses impose the fallback proposal to all their members and their clients? From a margining perspective, is the change of a fallback clause to a ISDA master agreement considered as generating new trades for all trades under that agreement? From an economic perspective, it is; from a regulatory perspective, an answer is required. How is the valuation of the trade for which the fallback methodology is not applied affected?

If OIS is a fallback for existing IRS with term reference rate. Do we propose also to have the OIS benchmark as the standard benchmark for IRS? Or do we want to change the market to a overnight only market. The choice of the OIS benchmark as IRS reference benchmark leaves the question of the robust OIS benchmark open. On the other side, the choice of a direct overnight benchmark means that the administrative burden of trading swaps is becoming more important. The reset has to be dealt with on a daily basis. Some IRS with corporates have today a fixing offset of one week, instead of the standard 2 business days, to give enough time to their administrative staff to deal with the reset. Is moving to a daily reset feasible for all of them? Are all the system ready to deal with composition of rates, or even worst, composition of rates with spread — ISDA standard definitions have 4 options for compounding methods.

Bank Credit

Some users may have used the LIBOR IRS, not as a hedge against level of rate, but as a hedge against banking sector term credit spread. In that case, the fallback to OIS or a repo based benchmark does not help; it would be better for them to have the trades cancelled. Should the fallback provision have some option for cancellation? Should the option for each trade be selected when the fallback changes are introduced or when the discontinuation is active?

Some variable mortgages are linked to LIBOR rates. This is very convenient for banks offering those mortgages as they have their assets and their liabilities linked to the same reference.

In Belgium, by rule, the variable mortgages are linked to Belgium government indices. This has lead in some cases to negative mortgage rates. But it also pushed the risk management issues in the hands of the professional lender, away from the responsibility of the private borrower. The private borrower does not suffer from the lost of credibility (credit spread) of the lender. As the mortgages are usually not marked-to-market in the bank balance sheet, the impact of the LIBOR-Government spread increase are unknown (to me). If LIBOR/EURIBOR was to disappear, this basis risk would appear not only in Belgian banks balance sheet but in many more balance sheets. How will banks deal with that risk when their borrowing costs (IBOR) is not present in vanilla IRS?

Initial Margin

When a new benchmark is created, obviously, no history for it exists. And it will take even more time to obtain an history of derivatives based on those benchmarks. The regulation related to mandatory Initial Margin (IM) call for historical VaR based on a 3-year history and “stressed periods”. Does that mean that the new derivatives have to be IMed with the scheduled based for at least 3-years and will need to wait for a significantly stressed period? Many other models and regulation based on historical estimates will need to be reviewed or adapted to those new situations.


Questions, questions, I have questions. Some call me question Marc! Why?

If you have answers or more questions on the Game of Benchmarks transition, don’t hesitate to contact me — through the comments of this blog or privately.

Maybe we can organize a Game of Benchmarks conference. 

Season 1

Episode 1: The king is dying.

Episode 2: Nobody is running for the throne.

Episode 3: The great pretender.

Episode 4: Transition and transition team.

Episode 5: The court. (forthcoming)



AD in finance - endorsement - Luca Capriotti

I had the honor of having my new book endorsed by experts on Algorithmic Differentiation. The endorsement can also be found on the book's back cover.

(Adjoint) Algorithmic Differentiation has quickly become an indispensable tool in modern financial engineering. Marc Henrard’s book is a lucid and concise introduction to the topic. The subject is introduced simply and clearly, from a mathematician’s prospective, and with the clear intention of demystifying the `black magic’ behind the staggering computational benefits of the technique. A nice read that many practitioners are likely to find useful.

Luca Capriotti
Visiting Professor
Department of Mathematics, University College London


More EUR adjustments?

The ECB has decided to published a new unsecured overnight interest rate benchmark. The press released was posted on its website a couple of days ago. No name has been proposed yet for the new benchmark.

This would be a new benchmark on top of the existing EONIA and STOXX GC Pooling Index.
Those two benchmarks are already the underlying of some derivatives. It is expected that the new index, planned to be published by 2020, will also be used in some derivatives.

That would mean three competing overnight benchmarks in EUR all of them with associated derivatives. That would means twelve (12) possible ON derivatives types. One type for each of the benchmarks used as underlying in combination with each of the benchmarks used for the computation of interest on the mandatory VM amounts, to which you have to add futures, which are equivalent to derivatives with a collateral rate of 0. And you have to add the potential difference in capital treatment for the settle-to-market feature.

The multi-curve framework is definitively not dead, even if LIBOR is dying. Interest rate modelling will continue to be extremely important for all financial markets users. Even if the so-called exotic derivatives have decreased in important since the crisis, the modelling techniques developed for them are becoming more an more important.

I'm working on different research works related to the modelling of the potential impact of the LIBOR disappearance. I will probably take another couple of months to have publishable results, but certainly the changes in the market continue to provide material for academic research and advisory business. Don't hesitate to contact me about any of them.