Abstract : Volatility is the single most important factor driving the dynamics of financial assets. Volatility modeling has been a very active field of research in the past years. Recent developments include in particular rough volatility, path-dependent volatility, including signature models, as well as designing models that jointly calibrate to S&P 500 and VIX options. This minisymposium aims to present in one place, compare, and bridge different new approaches on volatility modeling. It is our hope that fruitful ideas and collaborations can emerge from it.
Organizer(s) : Julien Guyon
Sponsor : This session is sponsored by the SIAM Activity Group on Financial Mathematics and Engineering.
Jordan Lekeufack Sopze (University of California, Berkeley)
Abstract : Using data, we show that volatility is mostly path-dependent: 90% of the implied volatility of equity indexes is explained endogenously by past index returns thanks to a simple linear model that combined weighted sums of past daily returns and squared returns with different time-shifted power-law weights. It thus suggests a continuous-time Markovian path-dependent volatility model. This model captures key stylized facts of volatility, and fits SPX and VIX smiles well, solving joint SPX/VIX smile calibration problem.
[05626] The 4-Factor Path-Dependent Volatility Model
Author(s) :
Julien Guyon (Ecole des Ponts ParisTech)
Abstract : The natural Markovian continuous-time version of the empirical path-dependent volatility (PDV) uncovered in [Guyon and Lekeufack, Volatility Is (Mostly) Path-Dependent, 2022] is the 4-Factor PDV model. Two factors describe the short and long dependence of volatility on recent returns (trend), while the two other factors describe the short and long dependence of volatility on recent returns squared (historical volatility). We show that this model, which is inferred from the empirical joint behavior of returns and volatility, captures all the important stylized facts of volatility: leverage effect, volatility clustering, large volatility spikes followed by a slower decrease, roughness at the daily scale, very realistic SPX and VIX smiles, joint calibration, Zumbach effect and time-reversal asymmetry. Being Markovian in low dimension, the model is very easy and fast to simulate. It can easily be enhanced with stochastic volatility (PDSV) to account for exogenous shocks. This is joint work with Jordan Lekeufack.
[01960] A theoretical analysis of Guyon's toy volatility model
Format : Talk at Waseda University
Author(s) :
Ofelia Bonesini (Imperial College London)
Antoine Jacquier (Imperial College London)
Chloé Lacombe (Morgan Stanley)
Abstract : We provide a thorough analysis of a path-dependent volatility model introduced by Guyon,
proving existence and uniqueness of a strong solution, characterising its behaviour at boundary points,
providing asymptotic closed-form option prices as well as deriving small-time behaviour estimates.
[05614] Prediction through Path Shadowing Monte-Carlo
Format : Online Talk on Zoom
Author(s) :
Rudy Morel (École Normale Supérieure)
Stéphane Mallat (Collège de France)
Jean-Philippe Bouchaud (CFM)
Abstract : We introduce a Path Shadowing Monte-Carlo method, which provides prediction of future paths, given any generative model. At a given date, it averages future quantities over generated price paths whose past history matches, or “shadows”, the actual (observed) history. We test our approach using paths generated from a maximum entropy model of financial prices, based on a recently proposed multi-scale analogue of the standard skewness and kurtosis called “Scattering Spectra”. This model promotes diversity of generated paths while reproducing the main statistical properties of financial prices, including stylized facts on volatility roughness. Our method yields state-of-the-art predictions for future realized volatility. It also allows one to determine conditional option smiles for the S&P500. These smiles depend only on the distribution of the price process, and are shown to outperform both the current version of the Path Dependent Volatility model and the option market itself.
[04506] Understanding how market impact shapes rough volatility
Author(s) :
Mathieu Rosenbaum (École Polytechnique )
Gregoire Szymanski (Ecole Polytechnique)
Abstract : We explain the subtle connection between the shape of market impact curves and the rough behavior of the volatility. We particularly focus on the celebrated square-root law and on the role of the participation rate in the price and volatility formation process.
[04065] Pricing in affine forward variance models
Format : Talk at Waseda University
Author(s) :
Jim Gatheral ( Baruch College, CUNY)
Abstract : In affine forward variance (AFV) models, the moment generating function may be expressed as the convolution of the forward variance curve and the solution of an associated convolution integral equation. In the case of the rough Heston model, this convolution integral equation may be solved numerically using the Adams scheme and approximately using a rational approximation. We show that in the general case, AFV models may be simulated efficiently using a hybrid version of Andersen's QE scheme. We illustrate convergence of the scheme numerically in the special case of the rough Heston model.
[02829] The rough Hawkes Heston model
Format : Talk at Waseda University
Author(s) :
Sergio Andres Pulido Nino (ENSIIE-LaMME, Evry)
Alessandro Bondi (Scuola Normale Superiore di Pisa)
Simone Scotti (Universita di Pisa)
Abstract : We introduce an extension of the Heston stochastic volatility model that incorporates rough volatility and jump clustering phenomena. In our 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. Prices of options on the underlying and its volatility index can then be obtained using Fourier-inversion techniques. We show that a parsimonious setup, characterized by a power kernel and an exponential law for the jumps, is able to simultaneously capture the behavior of the implied volatility smile for both S&P 500 and VIX options. Our findings demonstrate the relevance, under an affine framework, of rough volatility and self-exciting jumps in order to jointly calibrate S&P 500 and VIX smiles.
[05086] Recent advances on rough volatility
Format : Talk at Waseda University
Author(s) :
Antoine Jacquier (Imperial College London)
Abstract : Empirical evidence has recently highlighted that volatility of financial markets was not Markovian, giving rise to a new paradigm called "rough volatility". In this talk, we consider several recent advances in the topic, from a modelling point of view (suggesting interesting extensions of fractional Brownian motion) and from numerical aspects.
[04764] Does the Term-Structure of Equity At-the-Money Skew Really Follow a Power Law?
Format : Talk at Waseda University
Author(s) :
Mehdi El Amrani-Zirifi (Bloomberg LP)
Julien Guyon (Ecole des Ponts ParisTech)
Abstract : Using two years of S&P 500, Eurostoxx 50, and DAX data, we empirically investigate the term-structure of the at-the-money-forward (ATM) skew of equity indexes. While a power law (2 parameters) captures the term-structure well away from short maturities, the power law fit deteriorates considerably when short maturities are included. By contrast, 3-parameter shapes such as time-shifted or capped power laws, are shown to fit well regardless of whether short maturities are included or not.
[04899] Fast exact joint S&P 500/VIX smile calibration in discrete and continuous time
Format : Talk at Waseda University
Author(s) :
Florian Bourgey (Bloomberg L.P.)
Julien Guyon (Ecole des Ponts ParisTech)
Abstract : We introduce the Newton--Sinkhorn and implied Newton algorithms which significantly speed up the Sinkhorn algorithm that (Guyon, Risk, April 2020) used to build the first arbitrage-free model exactly consistent with S&P 500 and VIX market data. Using a purely forward Markov functional model, we show how to build a continuous-time extension of the previous discrete-time model. We also compute model-free bounds on S&P 500 options that show the importance of taking VIX smile information into account. Extensive numerical tests are conducted.
[03758] Joint calibration to SPX and VIX options with signature-based models
Format : Talk at Waseda University
Author(s) :
Christa Cuchiero (University of Vienna)
Guido Gazzani (University of Vienna)
Janka Möller (University of Vienna)
Sara Svaluto-Ferro (University of Verona)
Abstract : We consider a stochastic volatility model where the dynamics of the volatility are described by linear functions of the signature of a primary process. Under the assumption that this process is polynomial, we can express the log-price and the VIX squared as linear functions of the signature of an (augmented) primary process. This feature can be efficiently used for calibration purposes since the signature samples can be easily precomputed. We also propose a Fourier approach for VIX and SPX options exploiting that the signature of the augmented primary process is an infinite dimensional affine process.
[03638] Neural Joint SPX/VIX Smile Calibration
Format : Talk at Waseda University
Author(s) :
Scander Mustapha (Princeton University)
Julien Guyon (CERMICS, Ecole des Ponts ParisTech)
Abstract : We calibrate neural stochastic differential equations jointly to S&P 500 smiles, VIX futures, and VIX smiles. Drifts and volatilities are modeled as neural networks. Minimizing a suitable loss allows us to fit market data for multiple S&P 500 and VIX maturities. A one-factor Markovian stochastic local volatility model is shown to fit both smiles and VIX futures within bid-ask spreads. The joint calibration actually makes it a pure path-dependent volatility model, confirming the findings in \[Guyon, 2022, The VIX Future in Bergomi Models: Fast Approximation Formulas and Joint Calibration with S&P 500 Skew\].
