Quants of the year: Andrei Lyashenko and Fabio Mercurio

Risk Awards 2020: Quants extend Libor market model to accommodate backward rates

Fabio Mercurio

As Libor’s clock ticks down, there’s consternation about the fate of $370 trillion of financial instruments, which rely on pricing models designed with forward-looking rates in mind. Fallbacks being adopted across derivatives markets would re-hitch contracts to backward-looking versions of overnight risk-free rates on Libor’s demise – expected after 2021. It’s a change that throws the existing interest rate modelling framework into question and presents an existential threat to some products.

How to model backward-looking compounded-in-arrears rates – and price derivatives referencing them – is fundamental to benchmark transition. Two quants set about finding answers and struck gold with an extension to the classic Libor Market Model (LMM). The significance of the new approach, which adapts the existing interest rate modelling framework to alternative overnight benchmarks, has chimed with the wider quant community.

“In terms of topicality, applicability and broadness, it doesn’t get much bigger than this,” says Leif Andersen, global head of the quantitative strategies group at Bank of America Merrill Lynch, himself two-times joint recipient of the Quant of the Year prize “It’s a problem that will affect every bank in the world, and this paper puts a stake in the ground.”

In Libor replacement: a modelling framework for in-arrears rates, published in Risk in June, Andrei Lyashenko, head of market risk and pricing models at Quantitative Risk Management (QRM) in Chicago, and Fabio Mercurio, head of quantitative analytics at Bloomberg, present a new generalised forward market model (FMM) – an extension of the LMM – which allows forward values to be simulated from both forward-looking and backward-looking rates.

“It’s a major enabler,” says Chris Kenyon, head of XVA quant modelling at MUFG Securities EMEA. “It’s something which gives you a reference point for all the other models you can develop in this space; so in that sense it’s truly fundamental.”

The shift from forward-looking to backward-looking rates is a giant leap for derivatives pricing and modelling. Forward-looking rates such as Libor, are known at the beginning of the payment period, while backward-looking rates are fixed at the end of the period. This demands new modelling techniques and could require some contract types to be re-written. It may even spell the end for instruments such as forward rate agreements and Libor-in-arrears swaps.

“We became alarmed early on,” says Lyashenko. “Once Libor disappears, it’s not only that many contracts will need to be redone completely, all the valuation which is used on modelling forward-looking rates would need to be completely changed.”

The widely used LMM prices instruments by decomposing payoffs into a set of forward rates, using volatility and correlation information. Behaving normally, LMM and its variants simulate a forward rate until the beginning of the accrual period. The FMM extension allows modellers to determine the forward value of the associated in-arrears rate by extending the period during which forward values are simulated. Instead of stopping at the beginning of the accrual period, the model extends the simulation of rates until the end of the accrual period.

Photo: Andrei Lyashenko

Andrei Lyashenko

This allows both forward-looking and backward-looking rates to be simulated concurrently – the two sets of rates only diverge once the accrual period is entered. The ability to jointly model forward- and backward-looking rates was seen as a crucial element, as although derivatives markets are heading towards backward-looking compounded rates, market-implied forward-looking versions of RFRs are being flagged as a fallback in some cash markets.

Lyashenko and Mercurio share a long-standing interest in Libor reform. They had been exchanging information on the topic in the months before backward-looking RFRs were chosen as the fallback for Libor-referencing swaps last November, following a consultation led by the International Swaps and Derivatives Association. Once they received confirmation, they got to work quickly.

The lightbulb moment came when the two realised the classic interest rate modelling framework could be naturally extended to cover both forward-looking (Libor-like) and backward-looking (setting-in-arrears) rates “by extending the concepts of zero-coupon bond and forward rate, which are the key concepts in interest rate modelling”, Lyashenko recalls.

It turned out that all the core properties of interest rates used to model and price interest rate derivatives in the existing framework are preserved, allowing the same methodology to be applied to model backward-looking rates and price derivatives referencing them.

In terms of topicality, applicability and broadness, it doesn’t get much bigger than this

Leif Andersen, Bank of America Merrill Lynch

Lyashenko and Mercurio say there are very few obstacles to adopting their proposed method. The expanded technique does not require cutting-edge technology, or even new stores of data. All banks have to do to accomplish the coinciding rate simulation – that is, to convert their in-house implementations of LMM to FMM – is make minor adjustments to the workings of current Libor market models.

The inputs remain largely the same, they add. Curves and volatilities for each rate can be derived from existing market information in familiar instruments like swaps, bonds, caps and swaptions. Further, the FMM uses the same stochastic methods as the LMM.

The applicability of the proposed solution and the simplicity of its implementation received praise from top quants who spoke to Risk.net as part of the awards process.

“[The paper] is so simple – you have simple expressions for everything, and to me that’s really a mark of genius,” says MUFG’s Kenyon. “It’s also a thorough paper. You’ve got backward-looking in-arrears term rates – which is linguistically tortuous – done very nicely.”

Vladimir Piterbarg, head of quantitative analytics and quantitative development at NatWest Markets – twice named Risk.net’s Quant of the Year himself, and a major contributor to the literature on interest rate modelling – says the paper is straightforward and “doesn’t have any unnecessary complications”.

“They wrote down how the LMM-like model would and should look in the world where we only have overnight rates,” he adds.

Completing the framework

Lyashenko and Mercurio argue the development ensures LMM not only survives in the compounded-in-arrears rate environment, but thrives as a stronger, more complete tool and should be used even without Libor reform.

A version of the FMM has already been implemented at Lyashenko’s organisation, Quantitative Risk Management, despite the fact that Libor is still very much alive. It is applicable to Libor derivatives and calibrated to Libor options, and the authors say early adoption is worth the effort; the benefits, they explain, include gathering additional information that the standard LMM doesn’t offer, like forward rates dynamics.

Mercurio explains the rates are simulated in a single-curve environment for the sake of simplicity; he adds that modelling multiple curves and currencies is relatively easy.

BofA’s Andersen says Lyashenko and Mercurio’s work will be of particular value for smaller firms with fewer quants at their disposal. Large banks, he says, will be familiar with the problem and, while those firms will likely extend their LMMs in line with the quants’ proposals, they are more likely to make specific tweaks and refinements to the methodology to suit their businesses. For less-sophisticated firms – a “good chunk of the market” – the paper will be highly valuable.

Mercurio and Lyashenko are confident on the issue of the FMM’s breadth of appeal. “Bigger banks have quant teams that may be working on similar solutions,” Mercurio says. “So in a sense, smaller banks may be more in need of this information to guide them through development.”

But he adds that the potential of the FMM and the logic behind it is such that the paper should be of significant interest to the whole market, including well-resourced majors.

From now on, people are going to think about interest rate modelling differently, thanks to what we propose in the paper

Andrei Lyashenko, Quantitative Risk Management

“The approach we are presenting is really new,” he says. “It provides a consistent framework, and we will be submitting a follow-up paper that provides more evidence.”

Front-office quants, Mercurio speculates, who are involved in valuing new deals and pricing fallbacks, will already be looking into the topic: “They want to assess, themselves, the price impact. And they also want to assess the valuation challenges for new deals based on in-arrears rates.”

“There is no other way,” adds Lyashenko. “It’s the only logical way to do it. From now on, people are going to think about interest rate modelling – especially term structure modelling – differently, thanks to what we propose in the paper. Whether they use the FMM [itself] or some other model, is a different story, but the methodology is definitely going to be used.”

Mercurio and Lyashenko have much in common: more than 20 years of experience in the industry, management responsibility for large teams of quants for their respective organisations and teaching duties at renowned quantitative finance courses – Mercurio at New York University, Lyashenko at the Illinois Institute of Technology.

A striking difference is the public profile of the two. Mercurio regularly publishes research, predominantly on interest rate derivatives, and speaks at hundreds of conferences. In contrast, Lyashenko has a rather private profile, at least until now, liaising predominantly with clients rather than the wider quant community.

He has worked with some of the best known names in the field, including Alex Lipton, now a MIT Connection Science fellow among other roles. He considers Riccardo Rebonato, Edhec professor of finance, to be his mentor.

Looking forward, Mercurio explains that their method allows for the eventual construction of general Heath-Jarrow-Morton frameworks (HJMs), which model the evolution of forward rate curves. The pair will expand on the HJM topic in an upcoming paper, Mercurio says, describing the proposal as the bridge between the LMM and HJM models.

Lyashenko adds that the hybrid FMM model, on top of the valuation tools promised by the extension, “will have the fine resolution from an HJM at the same time, at practically no additional cost”.