Rough volatility models are an increasingly popular class of models in quantitative finance. In contrast to conventional stochastic volatility models, the volatility is driven by a fractional Brownian motion with Hurst index H < 1/2 which is rougher than Brownian motion. This change greatly improves the fit to time series data of underlying asset prices as well as to option prices, see, for instance, [Bayer, Friz, Gatheral, Quantitative Finance 16(6), 887-904, 2016]. Hence, introducing non-Markovian noise improves the predictive power of the model while maintaining parsimoniousness. Unfortunately, the loss of the Markov property poses severe challenges for theoretical and numerical analyses as well as for computational practice.
This minisymposium brings together different approaches for various UQ tasks in the context of rough volatility models and predictive models in finance. The problems addressed range from calibration and statistical analysis of the model parameters to optimal control of rough volatility models. To overcome the considerable practical hurdles posed by the lack of Markovianity, the contributions to the minisymposium use diverse tools such as deep neural networks and large deviation theory, assisted by properly analyzed simulation techniques.
14:00
Weak Error Rates for Option Pricing under the Rough Bergomi Model
Christian Bayer | WIAS Berlin | Germany
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Authors:
Christian Bayer | WIAS Berlin | Germany
Eric Hall | RWTH Aachen University | Germany
Raul F. Tempone | RWTH Aachen University and King Abdullah University of Science and Technology | Germany
In quantitative finance, modeling the volatility structure of underlying assets is a key component in the pricing of options. Rough stochastic volatility models, such as the rough Bergomi model [Bayer, Friz, Gatheral, Quantitative Finance 16(6), 887-904, 2016], seek to fit observed market data based on the observation that the log-realized variance behaves like a fractional Brownian motion with small Hurst parameter, H < 1/2, over reasonable timescales. In fact, both time series data of asset prices and option derived price data indicate that H often takes values close to 0.1 or even smaller, i.e. rougher than Brownian Motion. The non-Markovian nature of the driving fractional Brownian motion in the rough Bergomi model, however, poses a challenge for numerical options pricing. Indeed, while the explicit Euler method is known to converge to the solution of the rough Bergomi model, the strong rate of convergence is only H ([Neuenkirch and Shalaiko, arXiv:1606.03854]). In stark contrast, we prove rate 1 for the weak convergence of the Euler method. Our proof is elementary and relies on Taylor expansions and an affine Markovian representation of the underlying in an extended state space.
14:30
Analysis of rough volatility via rough paths / regularity structures
Paolo Pigato | University of Rome "Tor Vergata" | Italy
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Authors:
Peter Friz | TU Berlin | Germany
Paul Gassiat | Université Paris-Dauphine | France
Christian Bayer | WIAS Berlin | Germany
Paolo Pigato | University of Rome "Tor Vergata" | Italy
Rough paths and regularity structures have emerged as new toolbox to analysis a popular recent class of models from quantitative finance, in which volatility is modelled with fractional noise, in the "rough" regime of Hurst parameter less than 1/2. This talk is based on joint work with P. Gassiat (U Dauphine, Paris), C. Bayer and P. Pigato (WIAS Berlin).
15:00
Moment-based estimation of log-normal volatility models
Mikko Pakkanen | Imperial College London | United Kingdom
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Authors:
Anine Eg Bolko | Aarhus University | Denmark
Kim Christensen | Aarhus University | Denmark
Mikko Pakkanen | Imperial College London | United Kingdom
Bezirgen Veliyev | Aarhus University | Denmark
We introduce a novel method for the estimation of log-normal volatility models, including rough volatility models, based on a generalised method of moments (GMM) approach. Our GMM method is directly applicable to a time series of measures of realised volatility, without relying on estimates or proxies of spot volatility. This talk will present the modelling framework, asymptotic theory, Monte Carlo analysis of finite-sample performance and preliminary empirical results, shedding light on the “volatility is rough” hypothesis.
15:30
Machine Learning for Pricing American Options in High-Dimensional Markovian and non-Markovian models
Ludovic Goudenège | Fédération de Mathématiques de CentraleSupélec | France
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Authors:
Ludovic Goudenège | Fédération de Mathématiques de CentraleSupélec | France
Andrea Molent | Università degli Studi di Udine | Italy
Antonino Zanette | Università degli Studi di Udine | Italy
In this talk we present three efficient methods which allow one to compute the price of American basket options in the multi-dimensional Black-Scholes model. The proposed methods, which are based on Gaussian Process Regression, are termed GPR Monte Carlo, GPR Tree and GPR Exact Integration. Specifically, they are backward dynamic programming algorithms which consider a finite number of uniformly distributed exercise dates. At each time step, the value of the option is computed as the maximum between the exercise value and the continuation value. This is done only for a finite set of points and then Gaussian Process Regression is exploited to approximate the whole value function. The technique employed to compute the continuation value identifies each of the three proposed methods: GPR-Monte Carlo employs Monte Carlo simulation, GPR-Tree a Tree step and GPR-Exact Integration a semi-analytical formula for integration. Numerical tests show that the algorithms are fast and reliable, and they can be used to price American options on very large baskets of assets, overcoming the problem of the curse of dimensionality. Moreover, we also consider the rough Bergomi model, which provides stochastic volatility with memory, and we present how to adapt the GPR-Tree and GPR-Exact Integration methods for pricing American options in this non-Markovian framework.