Quant of the year: Julien Guyon

Risk Awards 2025: Volatility modeller par excellence (and football fan) achieved breakthrough with joint calibration of S&P and Vix options

Photo: Juno Snowdon

For more than a hundred years, physicists have been searching for a Theory of Everything: a single unifying model that would bridge competing and sometimes conflicting ideas to explain how the universe works.

A similar quest has occupied the minds of quantitative finance experts, albeit on a less grand scale and for a shorter time: how to jointly calibrate options on the S&P 500 and Vix indexes.

It’s a thorny problem that Julien Guyon, professor of finance at the École nationale des ponts et chaussées, describes as the “Holy Grail of volatility modelling”.

Plotting an option’s implied volatility against varying strikes produces a characteristic volatility “smile”. In theory, S&P and Vix options should have the same smile, given that the Vix is a measure of the 30-day volatility of an option on the S&P 500. In reality, the smiles don’t match – S&P options have a smirk to the left, while the Vix smirks to the right.

The mismatch suggests there might be an arbitrage between the two instruments, which are used to hedge portfolios. Guyon previously investigated this notion in a 2020 paper and found there was no arbitrage. This opened the possibility of jointly calibrating the implied volatilities – something quants had tried and failed to achieve.

Path-dependent volatility is the most interesting piece of work that Julien has done. It gives a better description of the behaviour of volatility

Bruno Dupire, Bloomberg

Guyon published his solution in November 2023, using a technique aided by artificial intelligence to establish the relationship between asset prices and volatility. In January 2024, he released another paper showcasing a faster, continuous model for jointly calibrating the two smiles.

“Volatility is a fascinating topic,” Guyon says. “To me, the joint calibration of S&P and Vix options…is going deep into the modelling by capturing the probability distribution not only of an asset price, but also its implied volatility. This is really a challenge mathematically.”

The papers also hinted that volatility is path-dependent: in other words, it has a strong reliance on its past and does not solely depend on current market values. That links to the other area of research Guyon is focused on – and which is expected to have a big impact in the industry: his work on path-dependent volatility models. It’s a neat departure from local volatility and stochastic volatility models, which have been the backbone of derivatives pricing for years.

Guyon’s efforts in producing unique research in relatively unexplored areas of volatility modelling have earned plaudits from senior academics.

“Julien is a very smart and thorough researcher,” says Bruno Dupire, head of the quantitative research team at Bloomberg in New York. “Personally, I think that path-dependent volatility is the most interesting piece of work that Julien has done. It gives a better description of the behaviour of volatility.”

Bridge-builder

Guyon’s first contact with quantitative finance came relatively late in his university career. After studying mathematics at École polytechnique, he moved to École des ponts in 2000 to specialise in applied mathematics and engineering. During this period, and as suggested by the name of the school he was attending, he studied the theory of bridge-building as part of a civil engineering course.

He never built a real bridge, yet that might be where his mindset acquired the instinct of unifying seemingly separate concepts. Only when he went on to take a PhD in probability theory, during which he attended the famous Master’s El Karoui, did he start to learn about financial mathematics.

Guyon took his first steps as a quant in the industry in 2006, when he was hired by Lorenzo Bergomi, now a volatility quant at Squarepoint Capital, to join the equity quant team at Societe Generale. At the time, Societe Generale was a leader in the development of new derivatives products and the best place to learn how to identify and quantify risk, and how to develop the right models to price and hedge complex structures.

His early projects were on Monte Carlo methods for American option pricing and the speeding up of the computation of implied volatilities. This work grew to encompass products from vanilla to exotic and even hybrid products that had anything to do with equities.

We found that instantaneous volatility was very volatile and mean-reverts very fast. In a sense it looks like rough volatility, although it’s not

Julien Guyon

Towards the end of his six years at Societe Generale, Guyon published a paper with colleague Pierre Henry-Labordère showing how to calibrate any local stochastic volatility model with a Monte Carlo method.

Calibrating a model to the surface of implied volatilities is important because the vanilla options on which those volatilities are computed are used to hedge portfolios, including complex ones that contain exotic options. A model that calibrates to the volatility surface of vanilla options correctly enables better hedging.

The advent of Vix derivatives added a further tangle to the knotty job of volatility modelling. Vix futures, launched in 2004, and Vix options, in 2006, enabled parties to trade forward volatility and volatility of volatility. Now a model should include the information given by the market on the risk-neutral price of forward volatilities: that is, what the market expects the implied volatility will be in the future.

Photo: Juno Snowdon

These new structures also raised questions on whether there is any arbitrage between the options on the S&P and the options on the Vix. Guyon – who by now had joined Bloomberg working in Dupire’s quant research team – began to tackle the problem of how to jointly calibrate S&P and Vix options.

The contrasting smiles of the two products raised speculation on whether that difference could be exploited. Guyon’s 2020 paper ruled out arbitrage – at least at the time. But he set out to investigate further. He wanted to find a way to model them jointly, so to guarantee the no-arbitrage condition.

His November 2023 paper, jointly authored with Scander Mustapha, then a PhD student at Princeton University, uses a one-factor stochastic local volatility model to jointly fit the two options. The pair created a model on the S&P 500 index that has a process for asset prices, another process linked to volatility, and a dependency structure between them that is captured by a neural network.

“Because of the properties of the neural networks, we hoped that it was rich enough to capture the difficult problem of joint calibration between S&P and Vix options. It was tricky to make it work, but at the end, we made it,” says Mustapha.

Maybe unsurprisingly, they found that the two processes for asset prices and volatility are highly dependent on each other, or – to be more precise – they are inversely correlated. The knowledge of that dependency structure allowed them to model products on the S&P and Vix jointly. And it led to some broader conclusions on the properties of volatility that would link this area of research with Guyon’s work on a path-dependent volatility model.

Julien has done a lot of very interesting work on the harmonisation of Vix and S&P dynamics

Leif Andersen, Bank of America

“That showed us that the volatility process is mostly path-dependent,” Guyon says. “We also found that instantaneous volatility was very volatile and mean-reverts very fast. In a sense it looks like rough volatility, although it’s not.”

The following January came Guyon’s paper presenting a faster model for jointly calibrating S&P and Vix smiles. Co-authored with Florian Bourgey, a quantitative researcher at Bloomberg, the paper featured a continuous time model which is applicable at higher frequency.

Peers praise Guyon’s work, although one practitioner points out that the models may not yet be suitable for live trading environments.

“Julien has done a lot of very interesting work on the harmonisation of Vix and S&P dynamics,” says Leif Andersen, global head of quantitative strategies at Bank of America. “These models may not be used directly on trading floors, where joint modelling of the two smiles will often take a backseat to more targeted models that can achieve the pricing speeds required for active trading. Yet traders and quants keep an eye on the topic, to make sure that they understand the implications and to rule out outright arbitrages.”

While the sense in the quant community is that the matter is solved, Guyon, predictably, begs to differ. As anywhere in science, he says, one should never stop advancing a solution. He thinks improvements in terms of speed or ability to capture the dynamics of volatility are still possible, and he will keep pursuing that objective.

A new path

The idea that implied volatility follows a path, that it isn’t exclusively dependent on the current, instantaneous market, had been gestating in Guyon’s mind for several years. An early paper in 2014 on the topic was followed by a more influential work in 2022, further detailing his theory on volatility’s path dependency.

“When you look at a financial time series of an asset and its implied volatility, for instance the S&P and the Vix, you see that the volatility shocks happen always at the time of market drops, so it makes sense to look at the link between recent returns in the S&P and the Vix,” he says.

Photo: Juno Snowdon

Volatility, he hypothesised, should depend on the average of the past returns and the average of volatility itself. Guyon set out to understand how much of volatility was endogenous, or in other words explainable purely by its own dynamics, and how much of it was exogenous, determined by market shocks. And the result is that the endogenous component is dominant.

“I’m not claiming volatility is purely path-dependent, but I’m claiming that it’s mostly path-dependent,” explains Guyon. The reason volatility is not purely path-dependent is because there needs to be an allowance for exogenous factors, for spikes to happen.

The result is an econometric model to forecast volatility and generate samples of implied vol based on historical data, as outlined in the 2022 paper.

Bergomi describes Guyon’s path-dependent volatility model as “the best model for generating a realistic path for the short S&P implied vol”. He adds: “It would be very useful as part of a risk framework that needs to take that variable into account.”

“Path dependency is a fundamental concept in finance,” says Dupire. “Payoffs are path-dependent and so are the dynamics of the underlyings, an important example of which is volatility. I expect that within a couple of years some versions of his model will be implemented in banks.”

In fact, that has already happened – perhaps not in banks, but in investment firms. Quant hedge fund Capital Fund Management has a volatility model that uses the principle of path dependency “that is similar to the one Julien uses in his approach”, reveals Jean-Philippe Bouchaud, chairman of CFM and himself a top researcher in volatility modelling. He explains that the idea of path dependency has its roots in the so-called Zumbach effect, which says that a price trend affects future volatility, irrespective of the direction of the trend. “At the time that effect was first conceptualised, in 2010, it was a feature that none of the models proposed in the literature was capturing,” Bouchaud says.

Simulated data

Guyon’s 10-year spell at Bloomberg ended in 2022, at which point he decided to return to academia full-time. He moved back to Paris, rejoining the École des ponts, this time as a tenured professor of finance. He teaches probability theory and financial mathematics alongside other courses.

Even in the years spent in the industry, the connection with academia has never been interrupted. While at Bloomberg, he was teaching on the masters in quant finance at Columbia University and NYU’s Courant Institute. And now from Paris he regularly goes back to the Big Apple as visiting professor at Tandon School of Engineering.

In Guyon’s immediate future is the expansion of his model. “I really want to unleash the power of path-dependent volatility models,” he says. He plans to deploy his models to a mission that quants have been working on in the past few years: the generation of synthetic data.

Synthetic data is seen as an important new weapon in helping to develop and train a variety of financial models.

Photo: Juno Snowdon

“I really want to unleash the power of path-dependent volatility models”

“It’s super important to have synthetic time series that are indistinguishable from the real ones, because you only have one price path per asset in finance, so you cannot repeat experiments,” Guyon says. “If you’re able to generate realistic scenarios then you can more robustly measure how good your strategy is, you can measure the risk of it and you can run stress tests.”

Guyon is convinced path-dependent volatility models can contribute to purely data-driven solutions to generate synthetic data as well as parametric models for the same purpose.

Meanwhile, he will continue building bridges between industry and academia. His latest appointment is to chair the ‘Futures of quantitative finance’ initiative, a collaboration between École des ponts, Université Paris Cité and BNP Paribas.

Career-wise, he is open-minded. But he will find it impossible to completely sever the link with academia. Of the stellar team at Societe Generale that Guyon started with, Bergomi and Henry-Labordère are now working for hedge funds. Would an opportunity like that tempt Guyon? “If the good offer comes, then I’ll study it. But it’ll have to be a research role that allows me to keep a connection with academia.”

Beautiful game

It is unusual to spot the word ‘football’ in the list of publications of a quant.

Unsatisfied with the draw for the 2014 World Cup Finals which resulted in groups of uneven quality, Guyon, passionate about football as much as probability theory, believed he had a shot at intervening in the debate.

He published articles in the New York Times and Le Monde arguing against the draw methodology that was in use. Fifa, the world football governing body, had designed a system whereby teams were drawn into groups solely based on geographical constraints. Guyon argued that would affect balance and therefore the value of the tournament. So he proposed that the draw of the 2018 World Cup should be driven by the team ranking first, with a secondary rule ensuring geographical constraints were met. Of course, he supported it with a probabilistic analysis of the expected outcomes.

When Fifa adopted such a solution, Guyon decided to continue sharing his ideas with them and gave recommendations on how teams’ rankings should be calculated – which was also adopted – and was vocal against the flawed idea of a 2026 World Cup in which teams were organised in 16 groups of three.

Guyon ranks his contribution to the structure of the world’s biggest sporting event as his crowning feat – above even volatility modelling.

“I’m very happy with what I did with the World Cup and in fact I think that’s the achievement I’m most proud of, more than the things I’ve done in finance!”