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.
16:30
- CANCELED - Hierarchical adaptive sparse grids and quasi Monte Carlo for option pricing under the rough Bergomi model
Chiheb Ben Hammouda | King Abdullah University of Science and Technology (KAUST) | Saudi Arabia
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Authors:
Christian Bayer | WIAS Berlin | Germany
Chiheb Ben Hammouda | King Abdullah University of Science and Technology (KAUST) | Saudi Arabia
Raul F. Tempone | RWTH Aachen and KAUST | Germany
The rough Bergomi (rBergomi) model, introduced recently in (Bayer, Friz, Gatheral, 2016), is a promising rough volatility model in quantitative finance. It is a parsimonious model depending on only three parameters, and yet remarkably fits with empirical implied volatility surfaces. In the absence of analytical European option pricing methods for the model, and due to the non-Markovian nature of the fractional driver, the prevalent option is to use the Monte Carlo (MC) simulation for pricing. Despite recent advances in the MC method in this context, pricing under the rBergomi model is still a time-consuming task. To overcome this issue, we have designed a novel, hierarchical approach, based on i) adaptive sparse grids quadrature (ASGQ), and ii) quasi-Monte Carlo (QMC). Both techniques are coupled with a Brownian bridge construction and a Richardson extrapolation on the weak error. By uncovering the available regularity, our hierarchical methods demonstrate substantial computational gains with respect to the standard MC method, when reaching a sufficiently small relative error tolerance in the price estimates across different parameter constellations, even for very small values of the Hurst parameter. Our work opens a new research direction in this field, i.e., to investigate the performance of methods other than Monte Carlo for pricing and calibrating under the rBergomi model.
17:00
Rough Volatility: A Measure-Change Point of View
Aitor Muguruza Gonzalez | Imperial College London | United Kingdom
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Authors:
Aitor Muguruza Gonzalez | Imperial College London | United Kingdom
Antoine Jacquier | Imperial College London | United Kingdom
We consider the family of changes of measure applicable to the Rough Fractional Stochastic Volatility model introduced by Gatheral, Jaisson and Rossembaum and further generalisations. In particular, we aim at understanding the consistent link between the physical (P) and pricing (Q) measures. We discover an unexpected short/long memory relation in rough volatility and discuss the range of change of measures that could produce realistic smiles in VIX options. Finally, we obtain a neat formula relating the risk premium inferred from the market, variance swap quotes and daily realized variance. Being the later two quantities observable, we analyse the empirical behaviour of the risk premium and propose a family of models that resembles such behaviour, yielding a P−Q consistent model.
17:30
Pricing path-dependent Bermudan options: an embarrassingly parallel algorithm
Jérôme Lelong | Université Grenoble Alpes | France
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Author:
Jérôme Lelong | Université Grenoble Alpes | France
In this talk, we propose a new policy iteration algorithm for pricing Bermudan options when the payoff process cannot be written as a function of a lifted Markov process. Our approach is based on a modification of the well-known Longstaff Schwartz algorithm, in which we basically replace the standard least square regression by a Wiener chaos expansion. It allows us to deal with truly path-dependent options but also with non Markovian models such as rough volatility models. The orthogonality property of the Wiener chaos expansion also breaks the bottleneck of the standard least square regression as the coefficients of the chaos expansion are given by scalar products on the L^2(\Omega) space and can therefore be approximated by independent Monte Carlo computations. This key feature enables us to propose an embarrassingly parallel algorithm to efficiently handle non Markovian payoff. We will conclude with some numerical experiments showing the impressive scalability of the parallel implementation and the efficiency of the algorithm for some complex path dependent options.
18:00
On deep calibration of (rough) stochastic volatility models
Blanka Horvath | King’s College London | United Kingdom
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Authors:
Christian Bayer | WIAS Berlin | Germany
Blanka Horvath | King’s College London | United Kingdom
Aitor Muguruza Gonzalez | Imperial College London | United Kingdom
Benjamin Stemper | TU Berlin and WIAS Berlin | Germany
Mehdi Tomas | École Polytechnique | France
We present a consistent neural network based calibration method for a number of volatility models—including the rough volatility family—that performs the calibration task within a few milliseconds for the full implied volatility surface. The aim of neural networks in this work is an off-line approximation of complex pricing functions, which are difficult to represent or time-consuming to evaluate by other means. We highlight how this perspective opens new horizons for quantitative modelling: The calibration bottleneck posed by a slow pricing of derivative contracts is lifted. This brings several model families (such as rough volatility models) within the scope of applicability in industry practice. As customary for machine learning, the form in which information from available data is extracted and stored is crucial for network performance. With this in mind we discuss how our approach addresses the usual challenges of machine learning solutions in a financial context (availability of training data, interpretability of results for regulators, control over generalisation errors). We present specific architectures for price approximation and calibration and optimize these with respect different objectives regarding accuracy, speed and robustness. We also find that including the intermediate step of learning pricing functions of (classical or rough) models before calibration significantly improves network performance compared to direct calibration to data.