Visioconference
WEDNESDAY, 31 AUGUST 2022
8:00 AM–11:30 AM (LOS ANGELES) / 11:00 AM–2:30 PM (NEW YORK) / 5:00–8:30 PM (PARIS)
THURSDAY, 1 SEPTEMBER 2022
9:00–11:00 AM (LOS ANGELES) / 12:00–2:00 PM (NEW YORK) / 6:00–8:00 PM (PARIS)
Sloan conference series
Organized with École Polytechnique and the University of California at Irvine
PROGRAM
WEDNESDAY, 31 AUGUST 2022
8:00 AM–11:30 PM (LA) / 11:00 AM–14:30 PM (NY) / 5:00–8:30 PM (PARIS)
Welcome and Introduction to the Conference
Josselin Garnier (École Polytechnique)
Knut Sølna (University of California at Irvine)
Jean-Philippe Touffut (Fondation Cournot)
Part 1 - 5:00–6:30 PM (Paris) (Chair: David Mordecai, Courant Institute, NYU)
The fractional Brownian motion in volatility modeling
Elisa Alòs (Universitat Pompeu Fabra, Barcelona)
This talk focuses on the properties of volatilities driven by the fractional Brownian motion (fBm) or related processes. In particular, we study the short-end behaviour of the at-the-money level, skew and curvature of fractional volatilities, and we see how this behaviour is more consistent with real market data.
Rough volatility: Fact or artefact?
Rama Cont (University of Oxford)
Based on joint work with Purba Das
We investigate the statistical evidence for the use of "rough" fractional processes with Hurst exponent H<0.5 for the modeling of volatility of financial assets, using a model-free approach. We introduce a non-parametric method for estimating the roughness of a function based on a discrete sample, using the concept of normalized p-th variation along a sequence of partitions. We investigate the finite sample performance of our estimator for measuring the roughness of sample paths of stochastic processes using detailed numerical experiments based on sample paths of fractional Brownian motion and other fractional 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 noise rather than the volatility process itself.
A GMM approach to estimate the roughness of stochastic volatility
Mikko Pakkanen (Imperial College London)
I will present an approach to estimate log normal stochastic volatility models, including rough volatility models, using the generalized method of moments (GMM). In this GMM approach, estimation is done directly using realized measures (e.g., realized variance), avoiding the biases that arise from using a proxy of spot volatility. I will also present asymptotic theory for the GMM estimator, lending itself to inference, and apply the methodology to Oxford–Man realized volatility data. Joint work with Anine Bolko, Kim Christensen and Bezirgen Veliyev.
Part 2 - 7:00–8:30 PM (Paris) (Chair: Knut Sølna, University of California at Irvine)
Quadratic Gaussian models: Analytic expressions for pricing in rough volatility models
Eduardo Abi Jaber (Université Paris I)
Stochastic models based on Gaussian processes, like fractional Brownian motion, are able to reproduce important stylized facts of financial markets, such as rich autocorrelation structures, persistence and roughness of sample paths. This is made possible by virtue of the flexibility introduced in the choice of the covariance function of the Gaussian process. The price to pay is that, in general, such models are no longer Markovian nor semimartingales, which limits their practical use. We derive explicit analytic expressions for Fourier–Laplace transforms of quadratic functionals of Gaussian processes. Such analytic expression can be approximated by closed form matrix expressions stemming from Wishart distributions. We highlight the applicability of such results in the context of rough volatility modeling: (i) fast pricing and calibration in the (rough) fractional Stein–Stein model; (ii) explicit solutions for the Markowitz portfolio allocation problem in a multivariate rough Stein—Stein model.
Efficient inference for large and high-frequency data
Alexandre Brouste (Université du Maine)
Although a lot of attention has been paid to the high-frequency data due to their increasing availability in different applications, several statistical experiments under the high-frequency scheme remain not fully understood. For instance, at high frequency, the variance and the Hurst exponent for the fractional Gaussian noise (fGn) or the scale and the stability index for the stable Lévy process are melting. Weak LAN property with a singular Fisher information matrix was obtained for the fGn and for the stable Lévy process.
Due to this singularity, no minimax theorem can be applied and it has been unclear for a long time whether the maximum likelihood estimator (MLE) possesses any kind of asymptotic optimality property. More recently, we established a non-singular LAN property for the fGn and for the stable Lévy process making it possible to define the asymptotic optimality in these statistical experiments.
The LAN theory does not directly provide the construction of efficient estimators. In regular statistical experiments, the MLE generally achieves optimality. Nevertheless, its computation is time consuming, and alternatives should be found to handle big or high-frequency datasets available in the fields of insurance and finance. In this direction, the Le Cam one-step estimation can be proposed in the aforementioned statistical experiments (and in others).
Time-inhomogeneous Gaussian stochastic volatility models: Large deviations and super roughness
Archil Gulisashvili (Ohio University)
The talk concerns time-inhomogeneous stochastic volatility models in which the volatility is described by a non-negative function of a Volterra type continuous Gaussian process that may have very rough sample paths. We prove small-noise large deviation principles for the log-price process in a Volterra type Gaussian model under very mild restrictions. These results are used to study the small-noise asymptotic behavior of binary barrier options, exit time probability functions, and call options.
Concluding remarks
THURSDAY, 1 SEPTEMBER 2022
9:00–11:00 AM (LA) / 12:00–2:00 PM (NY) / 6:00–8:00 PM (PARIS)
Part 3 - 6:00–7:30 PM (Paris) (Chair: Jean-Philippe Touffut, Fondation Cournot)
Xavier Timbeau (OFCE)
Rough volatility and regime shifts: Views from a macroeconomist
Jean-Bernard Chatelain (Université Paris I)
The presentation will focus on rough volatility on stock market returns, endogenous persistence and regime shifts with feedback from other variables.
Josselin Garnier (École Polytechnique)
Knut Sølna (University of California at Irvine)
Jean-Philippe Touffut (Fondation Cournot)
