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, whilst 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.

This is joint work with Florian Bourgey.

Model calibration under P and under Q are often regarded as two separate branches of finance. P refers to a backward-looking real-world probability measure under which the observed historical price path of a financial asset is viewed as a realization of a stochastic process, while Q denotes a forward-looking risk-neutral measure inferred from the prices of options written on this underlying asset. Accordingly, model estimation based on past prices of the underlying asset is referred to as estimation "under P", while estimation from option prices is known as calibration "under Q". One may question whether such a strict separation is justified or whether it rather reflects the lack of models able to capture the joint dynamics of prices and implied volatilities. Path-dependent volatility models are uniquely positioned to reconcile P- and Q-calibration, since they precisely relate past asset returns to volatility, thus to option prices. In this talk, we introduce the discrete-time 4- (or 3-)factor path-dependent volatility model and we show that combining the path-dependency of volatility that we uncovered in the article Volatility Is (Mostly) Path-Dependent (Guyon and Lekeufack, 2023) with fat-tailed random innovations allows us to reconcile model calibration under P and under Q, which further supports the hypothesis of high endogeneity of volatility. We also propose a new estimation approach that combines P- and Q-information to enhance calibration robustness, and we benchmark its effectiveness against classical methods. This is joint work with Léo Parent.

We propose simple and efficient schemes for Affine Volterra processes, using integrated kernel quantities and the Inverse Gaussian distribution. The schemes preserve positivity, and can be shown to converge weakly by recasting them as stochastic Volterra equations with a measure-valued kernel. Our method applies to two important examples: Volterra square-root/Heston and Hawkes processes. In the first case, when using a fractional kernel, the scheme with large time steps seems to be more performant as the Hurst index H decreases to -1/2. In the second case, our scheme has deterministic complexity, in contrast with exact methods based on sampling jump times that have random complexity, which opens the door to efficient Monte Carlo methods. 

Based on joint works with Elie Attal and Dimitri Sotnikov.

We propose a new theoretical framework that exploits convolution kernels to transform a Volterra-type path-dependent (non-Markovian) stochastic process into a standard (Markovian) diffusion process. Remarkably, it is also possible to go back, i.e., the transformation is reversible. We discuss existence and path-wise regularity of solutions for our class of stochastic differential equations. In the fractional kernel case, when H is in (0,1/2), where H is the Hurst coefficient, we propose a numerical simulation scheme which exhibits a remarkable strong convergence rate of order 1/2, which constitutes a bold improvement when compared with the performance of available Euler schemes, whose strong rate of convergence is H. Joint work with: O. Bonesini, M. Grasselli, G. Pagès. A previous version of the paper was entitled: "From elephant to goldfish (and back): memory in stochastic Volterra processes".

In the setting of stochastic Volterra equations, and in particular rough volatility models, we show that conditional expectations are the unique classical solutions to path-dependent PDEs. The latter arise from the functional Itô formula developed by [Viens, F., & Zhang, J. (2019). A martingale approach for fractional Brownian motions and related path dependent PDEs. Ann. Appl. Probab.]. We then leverage these tools to study weak rates of convergence for discretised stochastic integrals of smooth functions of a Riemann-Liouville fractional Brownian motion with Hurst parameter H∈(0,1/2). These integrals approximate log-stock prices in rough volatility models. We obtain the optimal weak error rates of order 1 if the test function is quadratic and of order (3H+1/2)∧1 if the test function is five times differentiable; in particular these conditions are independent of the value of H. This is a joint work with Antoine Jacquier and Alexandre Pannier.

We consider a microstructure foundation for models driven by Poisson random measures. The model is chosen such that, after suitable rescaling, the joint price-volatility process converges weakly to a rough Bergomi model with Hurst parameter F between 0 and 1/2.

Our main results are twofold. First, we establish weak convergence of the joint log-stock and volatility process in an appropriate Skorohod space. To derive this, we utilize practical C-tightness-criteria, developed by U. Horst, W. Xu and R. Zhang.

Second, we derive weak error rates for moments of the log-stock. We show that a carefully chosen kernel approximation yields a weak rate of order 1/3+3H/4 (for H small). This result is obtained via a moment representation for price processes driven by both Poisson noise and Brownian motion.

This is a joint work with Paul Hager and Ulrich Horst.

We propose a method for option pricing in rough Heston models by expanding the solution to the fractional Riccati equation around H = 1/2. Starting from the closed-form solution of the standard Heston Riccati equation, higher-order terms are obtained recursively by solving linear ODEs. We prove analyticity of the fractional Riccati solution around H = 1/2, implying convergence of the approximation to the characteristic function under usual constraints. Numerically, we obtain fast and accurate implied volatilities down to H = 0.2 (depending on the maturity). 

This is joint work with Dörte Kreher.