The Quanted Round-up is a curated summary that covers relevant research on key topics in quantitative financial decision-making.

Highlights

This edition looks at recent work testing and evolving the rough volatility framework. Some papers question whether observed roughness in volatility is structural or statistical. Others introduce new models that retain its features without relying on fractional Brownian motion. Several contributions explore how to better approximate or calibrate these models in practice, and where roughness assumptions hold up or break down across different markets and instruments.

Rough differential equations for volatility

Antoine Jacquier, Adriano Oliveri Orioles, Zan Zuric

We introduce a canonical way of performing the joint lift of a Brownian motionW and a low-regularity adapted stochastic rough path X, extending [DOR15]. Applying this construction to the case where X is the canonical lift of a one-dimensional fractional Brownian motion (possibly correlated withW) completes the partial rough path of [FT24]. We use this to model rough volatility with the versatile toolkit of rough differential equations (RDEs), namely by taking the price and volatility processes to be the solution to a single RDE. We argue that our framework is already interesting when W and X are independent, as correlation between the price and volatility can be introduced in the dynamics. The lead-lag scheme of [FHL16] is extended to our fractional setting as an approximation theory for the rough path in the correlated case. Continuity of the solution map transforms this into a numerical scheme for RDEs. We numerically test this framework and use it to calibrate a simple new rough volatility model to market data.

Rough Volatility: Fact or Artefact?

Rama Cont & Purba Das

We investigate the statistical evidence for the use of ‘rough’ fractional processes with Hurst exponent H < 0.5 for modeling the volatility of financial assets, using a model-free approach. We introduce a non-parametric method for estimating the roughness of a function based on discrete sample, using the concept of normalized p-th variation along a sequence of partitions. Detailed numerical experiments based on sample paths of fractional Brownian motion and other fractional processes reveal good finite sample performance of our estimator for measuring the roughness of sample paths of stochastic processes. We then apply this method to estimate the roughness of realized volatility signals based on high-frequency observations. Detailed numerical experiments based on stochastic volatility models show that, even when the instantaneous volatility has diffusive dynamics with the same roughness as Brownian motion, the realized volatility exhibits rough behaviour corresponding to a Hurst exponent significantly smaller than 0.5. Comparison of roughness estimates for realized and instantaneous volatility in fractional volatility models with different values of Hurst exponent shows that, irrespective of the roughness of the spot volatility process, realized volatility always exhibits ‘rough’ behaviour with an apparent Hurst index Ĥ < 0.5. These results suggest that the origin of the roughness observed in realized volatility time series lies in the estimation error rather than the volatility process itself.

Weak Error Estimates For Rough Volatility Models

Peter K. Friz, William Salkeld & Thomas Wagenhofer

We consider a class of stochastic processes with rough stochastic volatility, examples of which include the rough Bergomi and rough Stein–Stein model, that have gained considerable importance in quantitative finance. A basic question for such (non-Markovian) models concerns efficient numerical schemes. While strong rates are well understood (order H), we tackle here the intricate question of weak rates. Our main result asserts that the weak rate, for a reasonably large class of test function, is essentially of order min{3H + ½ , 1} where H ∈ (0, 1/2] is the Hurst parameter of the fractional Brownian motion that underlies the rough volatility process. Interestingly, the phase transition at H = 1/6 is related to the correlation between the two driving factors, and thus gives additional meaning to a quantity already of central importance in stochastic volatility modelling. Our results are complemented by a lower bound which show that the obtained weak rate is indeed optimal.

The rough Hawkes Heston stochastic volatility model

Alessandro Bondi, Sergio Pulido & Simone Scotti

We study an extension of the Heston stochastic volatility model that incorporates rough volatility and jump clustering phenomena. In our model, named the rough Hawkes Heston stochastic volatility model, the spot variance is a rough Hawkes-type process proportional to the intensity process of the jump component appearing in the dynamics of the spot variance itself and the log returns. The model belongs to the class of affine Volterra models. In particular, the Fourier-Laplace transform of the log returns and the square of the volatility index can be computed explicitly in terms of solutions of deterministic Riccati-Volterra equations, which can be efficiently approximated using a multi-factor approximation technique. We calibrate a parsimonious specification of our model characterized by a power kernel and an exponential law for the jumps. We show that our parsimonious setup is able to simultaneously capture, with a high precision, the behavior of the implied volatility smile for both S&P 500 and VIX options. In particular, we observe that in our setting the usual shift in the implied volatility of VIX options is explained by a very low value of the power in the kernel. Our findings demonstrate the relevance, under an affine framework, of rough volatility and self-exciting jumps in order to capture the joint evolution of the S&P 500 and VIX.

The SSR under Quadratic Rough Heston

Florian Bourgey & Jim Gatheral

We extend the hybrid scheme of Gatheral (2022) and apply the finite difference methodology of Bourgey et al. (2024) to compute the skew-stickiness ratio (SSR) under quadratic rough Heston. We find that the quadratic rough Heston model not only provides good joint fits to both SPX and VIX volatility smiles but also produces credible SSR values, while remaining extremely parsimonious. By examining the historical evolution of the quadratic rough Heston model, and relating it to well-known classical stochastic volatility models, we can begin to understand the underlying reasons for its seemingly unreasonable effectiveness.

Rough Bergomi turns grey

Antoine Jacquier, Adriano Oliveri Orioles, Zan Zuric

We propose a tractable extension of the rough Bergomi model, replacing the fractional Brownian motion with a generalised grey Brownian motion, which we show to be reminiscent of models with stochastic volatility of volatility. This extension breaks away from the log-Normal assumption of rough Bergomi, thereby making it a viable suggestion for the Equity Holy Grail -- the joint SPX/VIX options calibration. For this new (class of) model(s), we provide semi-closed and asymptotic formulae for SPX and VIX options and show numerically its potential advantages as well as calibration results.

Volatility Models in Practice: Rough, Path-Dependent, or Markovian?

Eduardo Abi Jaber & Shaun (Xiaoyuan) Li

We present an empirical study examining several claims related to option prices in rough volatility literature using SPX options data. Our results show that rough volatility models with the parameter H ∈ (0, 1/2) are inconsistent with the global shape of SPX smiles. In particular, the at- the-money SPX skew is incompatible with the power-law shape generated by these models, which increases too fast for short maturities and decays too slowly for longer maturities. For maturities between 1 week and 3 months, rough volatility models underperform one-factor Markovian models with the same number of parameters. When extended to longer maturities, rough volatility models do not consistently outperform one-factor Markovian models. Our study identifies a non-rough path-dependent model and a two-factor Markovian model that outperform their rough counterparts in capturing SPX smiles between 1 week and 3 years, with only three to four parameters.

References

1. Rough Bergomi turns grey. May 2025. Jacquier, A.; Orioles, A.O. and Zuric, Z. Available at arXiv: https://doi.org/10.48550/arXiv.2505.08623

2. Rough differential equations for volatility. December 2024. Bonesini, B.; Ferrucci, E.; Gasteratos, I. and Jacquier, J. Available at arXiv: https://doi.org/10.48550/arXiv.2412.21192

3. Rough Volatility: Fact or Artefact?. February 2024. Cont, R. and Das, P. Sankhya B 86, 191–223 (2024). Available at Springer Nature Link: https://doi.org/10.1007/s13571-024-00322-2

4. The rough Hawkes Heston stochastic volatility model. September 2024. Bondi, A.; Pulido, S. and Scotti, S. Mathematical Finance, 34, 1197–1241. Available at Wiley: https://doi.org/10.1111/mafi.12432

5. The SSR under Quadratic Rough Heston. May 2025. Bourgey, F. and Gatheral, J. Available at SSRN: http://dx.doi.org/10.2139/ssrn.5239929

6. Volatility Models in Practice: Rough, Path-Dependent, or Markovian? May 2025. Abi J.E. and Li. S. Mathematical Finance. Available at Wiley: https://doi.org/10.1111/mafi.12463

7. Weak error estimates for rough volatility models. February 2025. Friz, P.K.; Salkeld, W. and Wagenhofer, T. Ann. Appl. Probab. 35 (1) 64 - 98. Available at Project Euclid: https://doi.org/10.1214/24-AAP2109