Rising star in quantitative finance: Sigurd Emil Rømer

Risk Awards 2023: Doctoral dissertation outlines more efficient way to simulate rough volatility models

Photo: David Schweiger

Rough volatility was among the hottest topics in quantitative finance when Sigurd Emil Rømer was completing his master’s in mathematics and economics at the University of Copenhagen in 2018. After writing his master’s thesis on it, he decided to make it the subject of his PhD.

Rømer delved into simulation schemes for rough volatility, and developed a more efficient approach that overcomes the drawbacks of two existing methods by combining them. As part of his doctoral dissertation, Rømer published one paper on his hybrid multi-factor simulation method and a second empirical paper on the calibration to option data in which the scheme is used.

His work has impressed one of the founding fathers of the field. “This is not just an empirical analysis. It shows a subtle understanding of the issues, including how to use neural networks to calibrate the model, problems with the model, intuition on how it behaves,” says Jim Gatheral, professor of finance at Baruch College, who (with Mathieu Rosenbaum) jointly won Risk.net’s Quants of the Year award in 2021 for his work on rough volatility.

“It’s a full treatment, and not what you expect a PhD student to generate,” adds Gatheral.

This is not just an empirical analysis. It shows a subtle understanding of the issues, including how to use neural networks to calibrate the model, problems with the model, intuition on how it behaves

Jim Gatheral, Baruch College

Efficient Monte Carlo simulations are essential when using rough volatility to calculate options prices, as Rømer quickly discovered when he began his PhD.

The biggest challenge with rough volatility is calibration. This involves finding the right set of parameters to capture the volatility surface of vanilla options – for instance, on the S&P 500 (SPX). Rømer wanted to compare the quality of the calibration of rough volatility models against their classical counterparts.

“I was able to train neural networks to represent the pricing functions of S&P options for models like rough Bergomi and rough Heston. I then calibrated them to about 15 years of options data, or almost 4,000 volatility surfaces. Many interesting and surprising conclusions resulted from this, and that is what made me want to explore further. But this is where I got stuck, as I needed a more efficient way to simulate rough volatility to perform the Monte Carlo pricing,” Rømer tells Risk.net.

He tried two widely used simulation methods. The first was a hybrid scheme proposed in 2017 by Mikkel Bennedsen et al that achieves a very low approximation error. The other was the multi-factor method proposed in 2019 by Eduardo Abi Jaber and Omar El Euch to simulate rough volatility as a multivariate classical process. Both methods have their advantages, but can be slow when run individually. On closer inspection, Rømer realised the approaches may be complementary.

“I got very intimate with the two original papers, and tried to implement them myself. Looking closely at what happened numerically, and reflecting on the ideas, I realised, essentially, that while the hybrid approach of Bennedsen et al is efficient at short timescales, it is less so at long ones, and vice versa for the multi-factor scheme,” he says.

Photo: David Schweiger

The obvious solution, it seemed to him, was to combine the two approaches in a hybrid multi-factor scheme that reduced the computational burden of the original methods while preserving their accuracy. “Indeed, my numerical results show the [combined] scheme to be much more accurate than the pure multi-factor method, and potentially several hundred times faster than the hybrid method,” says Rømer.

His PhD examiners – Antoine Jacquier, lecturer in financial mathematics at Imperial College London, and Christa Cuchiero, professor of finance at the University of Vienna – were impressed with the empirical evidence for the combined approach.

“The difficult part consists in providing a theoretical proof for the convergence of the proposed scheme,” they say.

“The numerical experiments confirm the validity and quality of the new scheme … The paper develops the new state-of-the-art simulation scheme for rough volatility models, in particular for cases where the diffusion is state-dependent – an issue that was mostly out of reach until now.”

The breakthrough allowed Rømer to press ahead with his original idea of comparing the calibration of rough and classical volatility models to the SPX, and, more intriguingly, jointly to SPX and VIX options – a problem that has captured the attention of many quants in recent years, and for which various solutions have been proposed, although none have been fully accepted.

Rømer concludes that the joint problem can be solved with two-factor volatility models. Surprisingly, though, he found that two-factor classical volatility is sufficient to solve the problem. His work suggests that the difference between classical and rough volatility might be smaller than previously appreciated: “If you take the classic two-factor model that solves the joint calibration problem, and simulate it, you get a behaviour that, visually, is very hard to distinguish from rough volatility.”

As a consequence – and in contrast to many proponents of rough volatility, who are convinced of its advantages – Rømer is cautious: “My point of view is more agnostic. I’m willing to believe in rough volatility, but, in terms of the practical behaviour, it’s hard to see the difference.”

That doesn’t mean rough volatility doesn’t have more to offer. He believes the lessons learned as part of his PhD research could even be applied in his new role as a senior trading analyst at Ørsted, a Danish energy firm that specialises in the development of renewable energy.

“I’m very intrigued by energy markets because of their complexities, like storage constraints, which you don’t find in equity markets,” says Rømer.

“And yet I think there is much we can learn from rough volatility also in energy markets. The same physical constraints, for example, create an interesting decoupling of short- and long-term markets – something which naturally lends itself to modelling at multiple timescales, just like rough volatility.”