Course on multi-curve and collateral framework

Since 2007, a new framework has become the standard for interest rate derivative pricing: the multi-curve framework (also called multiple curve). Another market reality has gained more importance: the collateralization of interbank trades.

Even if the frameworks for multi-curves and collateral are nowadays relatively standard, their details and the far-reaching impacts of seemingly small changes are not always fully understood. Over the past few years, I've written several papers on these frameworks and published a book. I have also been one of the architects of a very flexible and efficient open source implementation of those frameworks. It can be found at http://strata.opengamma.io/

Writing technical papers and the code is not an end in itself, as people not only want to know the minutia of the associated mathematics and read the detailed code, but they also want to see the big picture, how it is used in practice, see examples in spreadsheet format, understand what are the impacts and see what is still missing. Reading a collection of unrelated papers with different notations found on internet is not always the most efficient way to get there. To discuss all those details, the best way is often the good old way of listening to a human being in a course or workshop.

Over the last years I presented several of such courses in different formats. It has been presented as a 15-hour course in the master program of University College London where I’m a visiting professor; as a one-day workshop in several WBS conferences; and as in-house training for several banks.
As I have received more questions about this type of course, I have been very systematic at preparing detailed slides, writing a extended agenda, cleaning lecture notes, putting together detailed spreadsheets linked to a production-grade implementation, etc.

I have collected the summary and a typical agenda; they can be found below. Obviously if you are interested in such a course, don’t hesitate to contact me.

Multi-curve framework

  • Definitions and fundamental hypothesis of the framework. The basic instruments. The multi-curve framework is based on relatively simple hypothesis, but those hypothesis are far reaching with subtle impacts.
  • Curve description: Defining flexible curves. Spread curves. What to interpolate? Impact of interpolation on risk.
  • Curve calibration: 
    •  Standard curves or simultaneous calibration. The multi-curve framework is more than a juxtaposition of single curves. The curves interacts and calibrating them simultaneously is often required. The basis swaps have also an impact on how to look at risk. Several markets have idiosyncrasies that need to be taken into account: two-swaps basis swaps in EUR, Fed Funds swaps in USD, change of frequency for AUD IRS, 
    • Curve are never simple. Incorporating turn-of-year, central bank meeting dates, dealing with sparse data, 
    • Risk computation: the growing number of (delta) risk figures. With multiple curves, the number of risk factors is also multiplied. How to look at risks for (linear) products?
    • Jacobian/transition matrices.
    • The market quotes are quite heterogeneous in term of instrument used and tenors. Standardisation of nodes and remapping of risk make it easier to read reports. It can also be used to store/use historical data for VaR, scenarios, statistical analysis. The synthetic curves.
  • Other instruments. The pricing curves have multiplied but the number of liquid instruments has not increased in the same way. The information need to be found where it is, and that includes using different instruments for curve construction and have them in the books for hedging: STIR futures, Fed Funds swaps, Deliverable Swap Futures (CME), Libor coupons with compounding (CAD but also basis swaps), Fed Funds futures, 
  • Modelling stochastic basis spread. The impact of the crisis is not only differentiated curves but also moving spread between them. What is the impact of those stochastic spread on vanilla instruments?
  • Impact of multi-curve framework on interest rate modelling. The standard pre-crisis models have been developed for one (risk-free discounting) curve. How to extend them relatively simply to the multi-curve framework? Black and SABR models in multi-curve. HJM/LMM.
  • Efficient computation of risk (algorithmic differentiation). The increasing number of market quotes used to build curves is not only a challenge for users (risk managers and traders) but also for efficient computation. A single currency vanilla instruments will often have 100 bucketed risk nodes. Algorithmic differentiation is a powerful tool that has been used for a long time in engineering and has made its way to finance in the last 5 years. How efficient is it for curve calibration and risk computation of interest rate books? Impact of multi-curve on quantitative finance library architecture.


  • Cash collateral and generalization. The cash-collateral discounting approach has been around for a couple of years now. The standard results and their exact application. Extension to generalized definitions of collateral. What is hidden behind OIS discounting (and when it can not be used).
  • Assets (bonds) collateral. Not all CSA/collateral agreements are based on cash. Generalization of collateral results for collateral with assets (collateral square).
  • Foreign currency collateral. Impact of foreign currency cash collateral.
  • Multi-curve and collateral. Most of the collateral literature focuses on the ``discounting'' aspect of collateral. Description of a joint multi-curve and collateral framework.
  • Clearing houses (CCP). Cleared swaps and collateral.
  • Collateral adjusted curve calibration. Extending the curve calibration for multiple collateral.
  • Modelling with collateral. Models very similar to the HJM model can be developed with collateral discounting. Even if they are similar to the old HJM, the collateral adds an extra layer of complexity and an extra layer of spreads to deal with. Modelling, even simple instruments like STIR futures, in that set-up is a challenge.
  • Convexity adjustment for change of collateral. There is not yet a consensus on how to compute convexity adjustment for change of collateral (foreign currency in particular). In some special cases, some estimation can be obtained.
  • Risk in multiple collateral environment. Even if all the change of collateral adjustments are not computed, their concentration of risks can be reported.

Edited several times. Last edit: 29 October 2016.



The multi-curve framework book is now the
#1 Amazon bestseller
category Business, Finance & Law > Professional Finance > Interest


Multi-curve and golf

Golf is multi-curve: draw, fade, breaking puts, good and bad bounces, back spin, slice, hook, etc.

As a lot of you know, I'm a keen golfer. When I'm not dreaming the solution to all problems about multi-curve (see page xii of the book for more on that), I'm dreaming about birdies (eagles will be for next year :) ).

I have ordered some swag golf balls with the book title as logo (see picture below). Obviously those balls are for grab. If you are a reader of the multi-curve book, want to play a round of golf, just let me know. I will be happy to offer a multi-curve ball (Pro V1x - 2014) to any reader I meet on the tee.

When I'm not writing or coding about the multi-curve framework, I can be found on different courses around London and Brussels; I will be glad to discover new ones. Don't hesitate to challenge me for a round.
Golf balls with Multi-curve Framework logo resting on a Multi-curve Framework book.

Endorsement: Daminano Brigo and Andrea Pallavicini

This is an important and much needed book looking at multiple interest rate curves, including collateralization. The subject is introduced motivating all developments from a historical perspective and is very pleasant to read. Both a rigorous theoretical approach and detailed practical recipes for bootstrapping and interpolation techniques are provided, in a coordinated fashion, using real market products. Advanced discussion of multiple curve dynamics, with specific modeling choices, is also given in the final part. From one of the originators and protagonists of the recent multiple curves literature, this is an appealing book for a potentially wide audience and is strongly recommended.

Prof Damiano Brigo
Dept of Mathematics, Imperial College London, and Director of the Capco institute, and
Dr Andrea Pallavicini
Imperial College London and Head of Equity, Fx and Commodities models, Banca IMI


Endorsement: Chyng Wen Tee

As you have seen on the book's cover, Professor Chyng Wen Tee (Assistant Professor of Quantitative Finance, Singapore Management University) was kind enough to endorse the book.

The endorsement printed on the back cover was not the full original text of the endorsement; the original endorsement was longer and would not have fit on the back cover.

The full original text:
As a quantitative finance practitioner-turned-academic, I read Dr. Henrard's Interest Rate Modelling in the Multi-curve Framework with great interest and excitement. Seven years after the onset of the infamous financial crisis that started in 2007, credible reference textbooks refurbishing our approach to interest rate modelling remain sparse, leading to a dichotomous gap between the interest rate models taught in a university and the interest rate models applied in practice. In the academic world, all too often the teaching of important concepts about time value of money, discounting and forwarding becomes commingled under the single-curve framework. Students consequently lose context, and graduate with only a loose understanding of the key concepts, without grasping the essence and salient points of interest rate models. On the other hand, from a practitioner's perspective, while numerous leading investment banks have since rolled out a multi-curve interest rate pricing architecture, a good rigorous reference textbook laying out the theoretical foundation and filling in the necessary mathematics with germane rigour is still lacking.

This book provides all the vital missing links for academic and practitioners alike, effectively bridging the gap between theory and practice. Unlike many textbooks which only focus on theory, Dr. Henrard's book takes a unique bottom up approach, guiding the reader all the way from the formulation of fundamental mathematical framework to actual implementation in a production environment, delving along the way into advanced topics including collateralisation, multi-curve calibration, implication on portfolio management as well as exposition on a wealth of interest rate products. This book more than delivers what it promises as far as foundations, evolution and implementation go. Dr. Henrard's elucidating and engaging writing style makes this highly informative book a pleasure to read. On top of that, this a book that also furnishes you with intuition, perspectives and a solid understanding of what forward rates and discount factors really mean. There's little doubt that this will be the de facto textbook on interest rate modelling for many years to come.
Prof. Chyng Wen Tee
Assistant Professor of Quantitative Finance, Singapore Management University


Endorsement: Stéphane Crépey

As you have seen on the book's cover, Professor Stéphane Crépey (Head of Probability and Mathematical Finance, University of Every, France) was kind enough to endorse the book.

The endorsement printed on the back cover was not the full original text of the endorsement; it was a little bit longer and contained some comments to start a dialogue or debate on some philosophical questions about the multi-curve framework.

The full original text:

With his two seminal ''irony'' papers, Marc Henrard is one of the very first to have identified (and in fact, anticipated) the importance of the interest rate multi-curve tsunami that came in the aftermath of the global financial crisis. Quite logically, this is also the focus of his book, one of the very few of its kind. Indeed, "competitors" typically also (and mainly) deal with CVA, FVA and the likes, so that there is usually not much space left for the multi-curve issue per se. By contrast, Marc addresses the question from foundations to curve calibration (a must read in the book). I’m not so sure about the statement that ‘the framework is different from the previous one-curve approach in fundamental aspects’, but in a sense this is a matter of point of view. All together, a very practical and nice read.
Oh, by the way, ''What is the present value of an FRA?''

Professor Stéphane Crépey
Head of Probability and Mathematical Finance, University of Every, France


The picture on the book's cover: the story behind it.

The picture used as background for this blog is also the one used on the book's cover. It may look like a set of greenish color bands, but it is more than that. The picture used on the cover is not the most important part of the book (if it is for you, maybe I have missed my goal writing the book). Nevertheless there is a small story behind it and I though it was worth sharing it.

The initial book jacket proposal from the editor was not really to my taste. At the time the editor didn't want to change the collection cover. I made a couple of suggestions to change the small color square on the original design using some of the pictures in my personal library. A couple of weeks before the publication date for the book, to my surprise (and joy), the editor decided to change completely the design for the entire collection. For my book, which is the first with the new design, he used one of the pictures I had suggested, the one you can see on the blog background. The picture is simply the close-up of a palm-tree leave. It is a "natural" picture. The other books in the collection will use computer generated pictures with geometrical figures.

When my Mum saw the book cover for the first time, she said that it looked like a "botanical book". As you know from the story of the cover picture, this is a true statement, in some sense of the term "botanical".
La mathématique est l’art de donner le même nom à des choses différentes.

Henri Poincaré.

Personal translation: Mathematics is the art of giving the same name to different items.

My Belgium house, the place from where I write most of my articles, is located in the region of Brussels (technically it is not located in the municipality of Brussels, which is only one of the 19 municipalities composing Brussels region, but I don't want to bore you with Belgian political system). The precise location is unimportant, except that it is located just next to the "Jardin Botanique", i.e. the botanical garden of Brussels. So even if the picture on the cover was not taken in that botanical garden, my links to botanical science extend not only to the cover of my book but also to the location where it was mainly written.

I have to add a third botanical connection. My grand fathers were both agronomist. They worked most of their careers in what was then the "Belgian Congo". Looking at books in the familly library, I can find books like "L'Arboretum de Stanleyville" and "Reforestation sur grande échelle au Kivu", written by my grand father Paul Liégeois in the 50s. So certainly a family connection to trees (not of the binomial/trinomial type but of the wood type) and leaves.

Book's cover

Sample chapter

A sample chapter, the table of content and the index of the book are now available on the Palgrave website.

The sample chapter is the long introduction (10 pages); it is an introduction to the book but also to the history of the multi-curve framework.

The book page on Palgrave site is: http://www.palgrave.com/products/title.aspx?pid=707837

The sample chapter can be found at: http://www.palgrave.com/resources/sample-chapters/9781137374653_sample.pdf


How it came to life.

The first lines of the book were written in 2006. At the time the term multi-curve framework, which is used for the book's title, had not been coined and my idea was only to write a couple of pages for a note. In the mean time, August 2007 changed the course of writing on interest rate curve modelling for ever.
Festina lente.
Latin saying
Personal translation: Haste slowly.

The starting point of the reflection was my quest to answer the question ``What is the present value of a FRA?'' in a way that convinces me. The initial intend was a personal quest to understand the foundation of a fundamental, and in appearance simple, quantitative finance question. I could not find an answer in the literature satisfactory to me. The answers I could find were either ``it is trivial'', i.e. ``don't ask silly questions'', either a description of a replication argument for which it was not acceptable to discuss the numerous hidden hypothesis. Discussing the hypothesis was not politically correct as a scientist and as a business manager. For the former, it questioned a foundation of quantitative finance, and thus the developments build on those foundations. For the latter, it was not seen positively by executive committee members, board members and supervisors as it casted doubts on official accounting figures - maybe rightly so - and had legal implications; it was not acceptable to say that official figures were "fishy" or even simply uncertain.

Ironically, the first article to come out of those reflections, titled The Irony in derivatives discounting, was published just one month before the now famous August 2007. The publication was not a prediction of what would happen in the derivative market just after and should not be seen as a premonition. Neither should it be seen as a cause of the crisis. It was nevertheless an indication of inconsistencies in the practice of derivative pricing that were not answered by a coherent theoretical framework.

The book, which started, unbeknown to its author, seven years ago, is intended to be a description of the current status of the subject, which is now called the multi-curve framework. It borrows from the developments of numerous practitioners and academics working on the subject. The length of the bibliography, with most of the references dating from 2009 and after, is a witness of the activity on the subject over the last years.

Hopefully the reader will find as much interest in the subject as I have over the last eight years. Hopefully he will also be intrigued, surprised, amused and maybe amazed at some of the subject facets.

Oh, by the way, my quest is still on! I'm still asking myself ``What is the present value of a FRA?'', even if the question has changed to ``What is the collateral quote of a FRA?'' (read the book for the meaning behind the new question ;) ). A question source of insomnia ... and maybe the starting point of another book in a couple of years.


P.S. This post is largely inspired by the book preface.