Interest Rate Modelling in the Multi-curve Framework

When the mathematical foundations of the interest rate derivatives market were laid down in the seventies, the first building block to be modelled was the discount factor. In all models, including the simplest ones, in the absence of default this quantity embedded a crucial piece of information: the ‘time-value of money’. No-arbitrage considerations then showed that, in the Q (pricing) measure, the expectation of future lending/borrowing rates could be expressed as suitable ratios of discount factors.

From then on, the concept of the term structure, i.e. the expression of the market agreement on ‘time-value of money’ for all possible maturities, became central to the valuation of all derivative products. Indeed, because of its assumed default-free nature, in these early days the discount factor took on a double role: it was used to discount risk-less pay-offs, and to project the expected values (in the pricing measure) of the forward rates that would index the various interest rate derivatives. So, the same term structure entered both discounting and (via the ‘projection’ of forward rates) payout estimation. People thought that knowing the process for the discount factor was tantamount to having a model for any interest rate derivative.

In the summer of 2007, this long-standing architecture suddenly unravelled. Market quotes showed with clarity that discounting and projection were now performed by market operators using different term structures. The projected rates (i.e. the rates at which risky banks would lend to each other for three or six months in the future) were linked to the now-risky Libor curve. At the same time, as the fear of interbank default risk spread, the practice of bilateral collateralization became the norm. This being the case, it was recognized that collateralized products still needed almost-risk-free discounting, and for this purpose Libor was no longer good. It was therefore replaced by the OIS (Overnight Indexed Swap) rate.

This transformation brought in its wake hitherto unheard of complications. If the Libor curve indeed reflected the default risk inherent in the banking sector, then the different frequencies of payment, which implied a different number of fixings, had to give rise to a different frequency of ‘resetting’ of default risk. As a consequence, a ‘basis’ – a wedge – emerged between term structures associated to different interest rate payment frequency. The old concept of a single ‘the term-structure’ had therefore to be replaced not just by two, but by a multiplicity of term structures.

A detailed explanation is contained for example in Morini (2011), but it is beyond the scope of this review. But let me recall one fact: just a few weeks before this sudden revolution, a paper had appeared on a widespread journal for derivatives practitioners. The title was ‘The Irony of Derivatives Discounting’. This paper claimed that the curve for ‘forwarding’ and the one for discounting need not to coincide, and explored in unprecedented depth the financial meaning of discounting. It was a paper by Henrard (2007).

I do not think Marc possessed any prophetic powers. He had just spotted that one crucial input to derivatives pricing was being treated in a rather superficial way, and set out to improve the situation. (To do so ex ante – i.e. before a crisis forces a change upon the trading community – is, by the way, one of the best and rarest attitudes quants can have.) Then, it was a matter of chance that he made it just a few months before the entire market had to learn, the hard way, that he was right. But by that time, Marc was ahead of most other quants in providing tools to deal with the new reality.

When in Morini (2009) this reviewer came out with the first paper to give a structural explanation to multi-curves, the market had already adapted to this new surprising reality. In fact, a number of papers had already described how bootstrapping, interpolation and pricing could be performed also in a multi-term-structure framework. Among them there was Henrard (2009), the ‘Part 2’ of Henrard’s 2007 paper.

The book under review, aptly titled ‘Interest Rate Modelling in the Multi-Curve Framework’ is the natural continuation and crowning achievement of Marc Henrard’s research. Here, the techniques to manage the multi-curve reality are organized in a consistent way from basic to more and more advanced. With the right balance between clear formulas and verbal explanation, the author takes the reader on a ride from the old single-curve framework towards the new reality – a reality that he describes with all the details needed for a real understanding of OIS discounting and Libor forwarding curve construction. The state-of-the-art approaches, like the use of pseudo-discount factors in forwarding, are described with examples, together with cutting-edge alternatives like direct forward curves.

In keeping with his practical approach, the author devotes specific sections to each crucial technical step like interpolation and bootstrap, and to those practicalities that, if not managed properly, can become the everyday nightmare of a quant. Examples of these ‘devil-is-in-the-detail’ topics are date mismatches, convexity adjustments and the treatment of futures.

If a flaw in this kind of treatment must be found, one can say that the book does not give more details on the ‘economic’ foundations of the new reality. This may be seen as a shortcoming by some readers; but for the large majority this is a plus, because it contributes to keeping the book brief and practical, as a real handbook should be.

The second part of the book deals with more advanced topics, also in terms of mathematics involved: stochastic modelling of the spread, and the formalization of collateral and funding costs leading to different ways to discount. This allows the author to cover also the pricing of cross-currency deals.

The appendices include one crucial issue in any practical implementation of a framework to deal with multi-curves: the computation of sensitivities, also known as ‘greeks’. This is often the computational bottleneck of multi-curve frameworks, since the sensitivities to the different term structures are usually computed with respect to many ‘time-buckets’. In a multi-curve world, the multiplication of time-buckets by the number of different curves generates an explosion of computational complexity when it comes to ‘greeks’. The author presents the most efficient solution to this issue: adjoint differentiation, which allows the user to compute as many sensitivities as desired in the same time as required for just five or six standard sensitivities. This is another topic where the author has made a number of research contributions, besides implementing it directly in his current activity as ‘quant guru’ of a fast-growing software company.

This book is the first practical manual on multi-curve, and is a must-have for an interest rate quant in these days.

Massimo Morini has been a consultant to the World Bank and other supernational institutions. Massimo is Professor at Bocconi University and MSc Director at Milan Polytechnic, and he was Research Fellow at Cass Business School, London. He is a member of the Advisory Board of Numerix. He has published papers in journals and is the author of Understanding and Managing Model Risk: A Practical Guide for Quants, Traders and Validators and other books on credit, funding and interest rate modelling. Massimo holds a PhD in Mathematics and an MSc in Economics.

Massimo Morini
Bocconi University
© 2016, Massimo Morini