The life of the scientific program Financial modelling  of the Centre d’Economie de la Sorbonne  is organized around a research seminar from January to June. The seminar takes place online (via the zoom plateform) and/or  on the 6th floor of the MSE building (Maison des Sciences Economiques, 106, boulevard de l’Hôpital, 75013 Paris). During each seminar, original research papers are presented. 

The seminar invites French and International scholars  and  practitioners to present their ongoing research in all the aspects of  financial modelling.

The seminar is organized in partnership with CNRS and Panthéon-Sorbonne University. 

CONTACT: Christophe Chorro, Olivier Guéant , Isabelle Nagot


Monday  October 2 2023 at 15h00 (paris time): Ajouter au calendrier

Ciamac Moallemi (Graduate School of Business, Columbia University)

Title: Automated Market Making and Arbitrage Profits in the Presence of Fees 

Abstract: We consider the impact of trading fees and discrete block generation on the maximal extracted value (MEV) generated by arbitrageurs trading against an automated marker marker (AMM) or, equivalently, on the adverse selection incurred by liquidity providers (LPs) due to DEX-CEX arbitrage. Our starting point is “loss-versus-rebalancing” (LVR) defined by Milionis et al. [2022], which quantifies the loss to arbitrageurs in an idealized, continuous-time setting with no trading fees. Trading fees and discrete blocks, on the other hand, create frictions which impact arbitrage activity.
Under the assumption of Poisson block generation times, we are able to compute arbitrage profits in closed form for general constant function market makers. The formula simplifies when the trading fee is low and blocks are frequent (the “fast block” regime). In this case arbitrage profits are simply LVR scaled down by the probability of an arbitrage trade in any given block. Moreover, though LVR was developed assuming no fees and continuous trading, with fees and discrete blocks LVR is roughly the profit gross of fees of arbing the pool. Introducing fees simply changes how LVR is split and who earns it (arbitrageurs or pool LPs).
Our results have implications for AMM design. First, we show that arbitrage MEV scales with the square root of the interblock time, which suggests that faster blockchains are one mechanism to reduce arbitrage MEV. Further, trading fees create a trade-off between losses to arbitrageurs and larger effective spreads for all traders. Our model quantifies this trade-off and provides a path towards setting revenue optimal fees.

This is joint work with Jason Milionis, Tim Roughgarden, and Anthony Zhang, and is based on these two papers:

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ID de réunion: 938 6042 1093

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Monday  June 26 2023 at 15h30: Ajouter au calendrier

Grigoryeva, Lyudmila (University of St.Gallen)

Title: Reservoir computing for macroeconomic forecasting with mixed frequency data 

Abstract: Macroeconomic forecasting has recently started embracing techniques that can deal with large-scale datasets and series with unequal release periods. MIxed-DAta Sampling (MIDAS) and Dynamic Factor Models (DFM) are the two main state-of-the-art approaches that allow modeling series with non-homogeneous frequencies. We introduce a new framework called the Multi-Frequency Echo State Network (MFESN) based on a relatively novel machine learning paradigm called reservoir computing. Echo State Networks (ESN) are recurrent neural networks formulated as nonlinear state-space systems with random state coefficients where only the observation map is subject to estimation. MFESNs are considerably more efficient than DFMs and allow for incorporating many series, as opposed to MIDAS models, which are prone to the curse of dimensionality. All methods are compared in extensive multistep forecasting exercises targeting US GDP growth. We find that our MFESN models achieve superior or comparable performance over MIDAS and DFMs at a much lower computational cost. 

Joint work with Giovanni Ballarin, Petros Dellaporta,  Marcel Hirt, Sophie van Huellen, Juan-Pablo Ortega 

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ID de réunion : 988 4143 6746

Code secret : 353833


Monday  June 12 2023 at 11h: 


Title: What is Missing in Asset-Pricing Factor Models?

Abstract:  Our objective is to price the cross-section of asset returns. Despite considering hundreds of systematic risk factors (``factor zoo''), factor models still have sizable pricing errors. A limitation of these models is that returns compensate only for systematic risk. We allow compensation also for unsystematic risk while imposing no arbitrage. The resulting stochastic discount factor (SDF) dominates traditional factor models in pricing assets. Empirically, about 70% of this SDF's variation is explained by its unsystematic-risk component, which is correlated with strategies reflecting market frictions and behavioral biases. Our findings provide an avenue for resolving the factor zoo. 

Joint work with Massimo Dello Preite, Raman Uppal, Paolo Zaffaroni

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Monday  June 5 2023 at 17h: Ajouter au calendrier

Thibaut Mastrolia (Berkeley)

Title: Transaction fees for auction markets 

Abstract:  TBA

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ID de réunion : 956 2726 4704

Code secret : 018833

Monday  April 17 2023 at 17h: Ajouter au calendrier

Eduardo Abi Jaber (Ecole Polytechnique)

Title: From the Quintic model that jointly calibrates SPX and VIX to Signature volatility models. 

Abstract:  The quintic Ornstein-Uhlenbeck volatility model is a stochastic volatility model where the volatility process is a polynomial function of degree five of a single Ornstein-Uhlenbeck process with fast mean reversion and large vol-of-vol. The model is able to achieve remarkable joint fits of the SPX-VIX smiles with only 6 effective parameters and an input curve that allows to match certain term structures. Even better, the model remains very simple and tractable for pricing and calibration: the VIX squared is again polynomial in the Ornstein-Uhlenbeck process, leading to efficient VIX derivative pricing by a simple integration against a Gaussian density and simulation of the volatility process is exact.

For pricing SPX products, we show that the Quintic model is part of a larger class of stochastic volatility model where the volatility is driven by a linear function of the signature of a Brownian motion enhanced with the running time. For this larger class of models, we develop pricing and hedging methodologies using Fourier inversion techniques on the characteristic function which is known up to an infinite-dimensional Riccati equation. We illustrate our method on numerical examples for fast pricing, hedging and calibration of vanilla and path-dependent options in several classes of Markovian and Non-Markovian models.

Based on joint works with Louis-Amand Gérard, Camille Illand & Shaun Li.

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ID de réunion : 959 4058 5706

Code secret : 258504

Monday  April 3 2023 at 14h30: Ajouter au calendrier

Giulia Rotundo (Sapienza de Rome)

Title: Network models for financial time series

Abstract:  The seminar is going to focus on two applications

1) Shareholding networks

2) BIS database

1) Shareholding networks

Cross-shareholding occur when a company traded in a Stock market holds shares of another company.

This relationship can be modeled through a direct graph, with cycles. The main questions raise about the ultimate owner, the level of concentration of the market, and scenario analysis.

Open questions concern the reduction of the space of networks for driving the simulations and reducing the total computational time.


R. Cerqueti, G. Rotundo (2022) The weighted cross-shareholding complex network: a copula approach to concentration and control in financial markets, Journal of Economic Interaction and Coordination

R. Cerqueti, G. Rotundo, M. Ausloos (2020) Tsallis Entropy for Cross-Shareholding Network Configurations. Entropy, 22 (6) 676 

R. Cerqueti, G. Rotundo, M. Ausloos (2018) Investigating the Configurations in Cross-Shareholding: A Joint Copula-Entropy Approach. Entropy 20(2) 134 

G. Rotundo, A. M. D’Arcangelis (2014) Network of companies: an analysis of market concentration in the Italian stock market. Quality & Quantity, 48 (4), 1893-1910 

G. Rotundo, A. M. D’Arcangelis (2010) Ownership and control in shareholding networks. Journal of Economic Interaction and Coordination, ISSN 1860-711X, Volume 5, Issue 2, 191-219

2) BIS database

The Bank of International Settlement reports data from 60 countries on exposures also to other countries. Therefore, the unerlying network

is necessarily incomplete. A core-periphery model is implemented to state the relevance of the missing links.


 M. Cinelli, G. Ferraro, A. Iovanella, G. Rotundo (2019) Assessing the impact of incomplete information on the resilience of financial networks. Ann Oper Res. 

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ID de réunion : 985 2011 5287

Code secret : 135955

Monday  March 20 2023 at 14h30: Ajouter au calendrier

Alexey Medvedev (Lombard Odier Asset Management )

Title: Option-like Portfolio Insurance Over a Rolling Window: Introduction and Derivation by Reinforcement Learning 

Abstract: Option-based portfolio insurance is a unique solution that combines a downside protection with

guaranteed minimum upside participation. Its implementation, however, is challenging for public funds,

whose investors have different entry points and uncertain holding periods. In this paper we extend this

methodology to portfolio protection over a rolling window without an explicit investment horizon. Such

an objective reflects the common practice where the performance of funds is regularly monitored over

fixed time intervals. We show how to derive the optimal strategy using methods of reinforcement

learning, and demonstrate that it provides a more effective portfolio protection than the existing

methods both on simulated and real-life data.

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ID de réunion : 983 5322 1207

Code secret : 764383

Monday  March 6 2023 at 14h30: Ajouter au calendrier

Giorgio Ferrari (Bielefeld Universität)

Title: Consumption Decision, Portfolio Choice and Healthcare Irreversible Investment 

Abstract: We propose a tractable dynamic framework for the joint determination of optimal consumption, portfolio choice, and healthcare irreversible investment. Our model is based on a Merton's portfolio and consumption choice problem, where, in addition, the agent can choose the time at which undertaking a costly health investment decision. Health depreciates with age and directly affects the agent's mortality force, so that investment into healthcare reduces the agent's mortality risk. The resulting optimization problem is formulated as a stochastic control-stopping problem with a random time-horizon and state-variables given by the agent's wealth and health capital. We transform this original problem into its dual version, which is now a two-dimensional optimal stopping problem with interconnected dynamics and finite time-horizon. Regularity of the optimal stopping value function is derived and the optimal stopping boundary surface is proved to be Lipschitz continuous and it is characterized as the unique solution to a nonlinear integral equation. In the original coordinates, the agent thus invests into healthcare whenever her wealth exceeds an age- and health-dependent transformed version of the optimal stopping boundary. 

Joint work with Shihao Zhu  

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ID de réunion : 970 8071 0802

Code secret : 352733

Monday  February 13 2023 at 14h30: Ajouter au calendrier

Francesco Violante (ENSAE)

Title: Generalized Autoregressive Conditional Betas: longitudinal feedback in multifactor asset pricing 

Abstract: We propose a new class of observation-driven models describing the joint dynamics of the time-varying slopes in a system of conditionally heteroskedastic simultaneous multiple regressions. The model, dubbed Generalized Autoregressive Conditional Betas (GACB), introduces new and economically meaningful mechanisms of propagation of shocks, namely beta spillovers. The GACB can accommodate large dimensions (both regressors and regressands), testing parametric longitudinal restrictions, introducing exogenous variables influencing betas, as well as the coexistence of constant and time-varying betas.  We derive stationarity and uniform invertibility conditions and present beta and covariance tracking constraints. We propose a variety of computationally convenient quasi-maximum likelihood estimators (parallel and sequential), and we assess their finite sample properties using extensive Monte Carlo experiments. Finally, we use the GACB to illustrate the role of beta spillovers in the Fama-French three factors asset pricing model.

Joint work with Stefano Grassi (University of Rome 'Tor Vergata') 

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ID de réunion : 931 2760 9443

Code secret : 619245


Wednesday 22 June 2022 at 18h00: Ajouter au calendrier

Jean-Edouard Colliard (HEC)

Title: Algorithmic Pricing and Liquidity in Securities Markets  

Abstract: We run experiments in which machine-learning algorithms play a standard market-making game under adverse selection. We study how the outcome of these experiments differs from standard equilibrium predictions. We find that a monopolist market-maker charges a price lower than the standard monopoly price. In contrast, competing market-makers charge a price at a mark-up above the competitive price. We run comparative statics exercises that deliver new empirical predictions. In particular, the mark-up decreases in the amount of adverse selection. Beyond the case of market-making, our framework is a step towards developing novel predictions on the impact of algorithmic trading in financial markets. 

Joint work with Thierry Foucault, Stefano Lovo 

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ID de réunion : 921 7598 3954

Code secret : 217808

Wednesday 8 June 2022 at 18h00: Ajouter au calendrier

Thierry Foucault (HEC)

Title: Equilibrium Data Mining and Data Abundance 

Abstract: We propose a new theory of information production in financial markets. In this theory, speculators search for new predictors of asset payoffs and optimally decide to trade on predictors whose signal-to-noise ratio exceeds an endogenous threshold. We use the model to derive predictions regarding the effects of progress in information technologies on quantitative asset managers' performance, the similarity of their holdings, and the informativeness of asset prices. We show that data abundance (an expansion of the search space for predictors due to greater data diversity) and greater data processing power do not have the same effects. 

Joint work with Jérôme Dugast.

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ID de réunion : 966 1522 6605

Code secret : 844792

Tuesday 24 May 2022 at 18h00:       Ajouter au calendrier

Christophe Geissler  (Advestis  

Title: Using ESG scores in equity issuer clustering. 

Abstract: We present here an original application of the non-negative matrix factorization (NMF) method, for the case of extra-financial ESG data. These data are subject to high correlations between co-variables, as well as between observations. NMF provides a more easily interpretable clustering of co-variables and observations than a simple principal component analysis (PCA). It gives rise to a natural issuer clustering based on the ESG scores.  

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ID de réunion : 978 6688 3736

Code secret : 989561

Wednesday 11 May 2022 at 18h00:       Ajouter au calendrier

Zhenyu Cui  (Stevens Business School  

Title: Applications of the Dirac Delta Family Method in Implied Volatility, Risk-neutral Density, and High-dimensional Stochastic Control 

Abstract: In this talk, the Dirac Delta family method is introduced, which is based on orthogonal polynomial representations of the Dirac Delta function. We show how this method can lead to new valuation results in three financial applications. First, when combined with the change of variable technique, we obtain an explicit model-free formula for the Black-Scholes implied volatility. The formula is expressed through either a limit or as an infinite series of elementary functions, and we establish that the proposed formula converges to the true implied volatility value. Numerical and empirical examples illustrate the accurateness of the formula. Second, when combined with the celebrated Carr-Madan spanning formula, we derive a novel model-free representation of the risk-neutral density in terms of market-observed options prices. Compared to the widely used method for obtaining the risk-neutral densities via the Breeden–Litzenberger device, our method yields estimate of risk-neutral densities that are model-free, automatically smooth, and in closed-form. Third, when applied to calculating the conditional expectations arising from dynamical programming, we show that it leads to a novel numerical time-stepping approach for solving corresponding HJB system. We demonstrate the accuracy and efficiency of the method through solving some one-dimensional and two-dimensional control problems. 

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ID de réunion : 950 6705 5748

Code secret : 012438

Wednesday 20 April 2022 at 18h00:       Ajouter au calendrier

Serge Tabachnik  (Lombard Odierand Eric Benhamou (AI for alpha)  

Title: Machine Learning Allocation for Volatility Targeting Portfolios

Abstract:  In the context of proactive risk management and volatility targeting portfolio construction, finding robust predictors of future realized volatility is paramount to achieving optimal performance. Up to now, this has motivated the search for a stream of models to forecast volatility based on multiple methodologies. However, because of the complexity and non-stationary behavior of volatility, the appropriate choice of the volatility estimator, and hence the volatility targeting model relying on it, remains an open question. This presentation proposes the implementation of two Machine Learning (ML) based methodologies, to determine the optimal allocation between volatility targeting models. The first methodology is based on Deep Reinforcement Learning (DRL), while the second is based on Gradient Boosting Decision Trees (GBDT). These two innovative methodologies are intrinsically different and distinguished from one another by their learning approach. GBDT uses supervised learning contrary to DRL that falls into the unsupervised learning. Both of these dynamic allocations rely on Adaptive Machine Learning (AML) methods that aim to take the typical regime changes of volatility into account. While this adaptive aspect is inherent to reinforcement learning, in the case of the supervised learning approach it comes from the addition of a filtering step that aims to take the regime changes further into account.

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ID de réunion : 917 7934 6457

Code secret : 424113

Wednesday 6 April 2022 at 18h00:       Ajouter au calendrier

Damien Challet  (Centrale Supélec)  

Title: Filtering the covariance matrix of non-stationary systems with constant eigenvalues 

Abstract: We propose a data-driven way to reduce the noise of covariance matrices of nonstationary systems. Our method rests on long-term averaging of the influence of the future on present eigenvalues. This zeroth-order approximation outperforms the latest optimal methods designed for stationary systems as soon as the system is not stationary as seen in large-scale empirical investigations on optimal global minimum variance portfolios and with synthetic data. 


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ID de réunion : 922 6560 8992

Code secret : 502356

Wednesday 9 March 2022 at 18h00:       Ajouter au calendrier

Guillaume Bernis  (Natexis)  

Title: Clustering Effects for Derivatives Pricing: Applications of Hawkes Processes in Finance 

Abstract: We propose an extension of the $\Gamma$-OU Barndorff-Nielsen and Shephard model taking into account jump clustering phenomena.

We assume that the intensity process of the Hawkes driver coincides, up to a constant, with the variance process. By applying the theory of continuous-state branching processes with immigration, we prove existence and uniqueness of strong solutions of the SDE governing the asset price dynamics. We exploit a measure change of self-exciting Esscher type in order to describe the relation between the risk-neutral and the historical dynamics, showing that the $\Gamma$-OU Hawkes framework is stable under this probability change. By exploiting the affine features of the model we provide an explicit form for the Laplace transform of the asset log-return, for its quadratic variation and for the ergodic distribution of the variance process.

We show that the proposed model exhibits a larger flexibility in comparison with the $\Gamma$-OU model, in spite of the same number of parameters required. We calibrate the model on market vanilla option prices via characteristic function inversion techniques, we study its sensitivities and propose an exact simulation scheme. The main financial result is that implied volatility of options written on VIX is upward shaped due to the self-exciting property of Hawkes processes, in contrast with the usual downward slope in the $\Gamma$-OU Barndorff-Nielsen and Shephard model.


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ID de réunion : 926 4884 2718

Code secret : 586575

 Wednesday 9 February 2022 at 18h00:       Ajouter au calendrier

Ryan Donnelly   (King's College)  

Title: Optimal Execution with Exploratory Trading 

Abstract: An agent wishes to liquidate a block of shares subject to price impact effects, but also desires to explore regions of the state and control space that would be avoided according to the optimal strategy of the model. This is accomplished by incorporating unpredictable randomness in the trading strategy at each point in time, and offering a reward in the form of Shannon's differential entropy of the distribution of this random component. We propose a framework in discrete time with captures this objective and solve for the optimal distribution of trades. At each point in time the optimal trade distribution is Gaussian with parameters that are given in terms of the solution to a backwards stochastic difference equation. The solution to this backwards stochastic difference equation can be approximated by a continuous time analogue, which can be solved in closed form. Using this approximation, we demonstrate the relation between this discrete time model and other pieces of literature which work in continuous time. 

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ID de réunion : 984 0419 9876

Code secret : 603818

 Wednesday 26 January 2022 at 15h45:       Ajouter au calendrier

Blanka Horvatz  (TU Munich)  

Title: Data - Driven Market Simulators  & some applications of signature kernel methods in mathematical finance

Abstract: Techniques that address sequential data have been a central theme in machine learning research in the past years. More recently, such considerations have entered the field of finance-related ML applications in several areas where we face inherently path dependent problems: from (deep) pricing and hedging (of path-dependent options) to generative modelling of synthetic market data, which we refer to as market generation.

We revisit Deep Hedging from the perspective of the role of the data streams used for training and highlight how this perspective motivates the use of highly accurate generative models for synthetic data generation. From this, we draw conclusions regarding the implications for risk management and model governance of these applications, in contrast to risk-management in classical quantitative finance approaches.

Indeed, financial ML applications and their risk-management heavily rely on a solid means of measuring (and efficiently computing) similarity-metrics between datasets consisting of sample paths of stochastic processes. Stochastic processes are at their core random variables on path space. However a consistent notion of and efficiently computable similarity-metrics for stochastic processes remained a challenge until recently. We propose such appropriate similarity metrics and contrast them with returns-based similarity metrics. Finally, we discuss the effect of incorporating the information structure (the filtration) of the market into these similarity metrics and the implications of such metrics on options prices.

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ID de réunion : 965 1867 8430

Code secret : 223789

 Wednesday 12 January 2021 at 6pm:       Ajouter au calendrier

Adeline Fermanian (UPMC)

Title: Learning from time-dependent data: signatures, RNN, and neural ODE

Abstract: Time-dependent data arise in many fields of research, such as quantitative finance, medicine, or computer vision. We will be concerned with a novel approach for learning with such data, called the signature transform, and rooted in rough path theory. Its basic principle is to represent multidimensional paths by a graded feature set of their iterated integrals, called the signature. After a general overview of signatures in machine learning, we show its application on one specific problem. Building on the interpretation of a recurrent neural network (RNN) as a continuous- time neural differential equation, we show, under appropriate conditions, that the solution of a RNN can be viewed as a linear function on the signature. This connection allows us to frame a RNN as a kernel method in a suitable reproducing kernel Hilbert space. As a consequence, we obtain theoretical guarantees on generalization and stability for a large class of recurrent networks. 

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ID de réunion : 979 1526 3949

Code secret : 683848


 Wednesday 9 June 2021 at 6pm:

Maciej Augustyniak (Université de Montreal)

Title: A Discrete-Time Hedging Framework with Multiple Factors and Fat Tails: On What Matters  

Abstract: This article presents a quadratic hedging framework for a general class of discrete-time affine multi-factor models and investigates the extent to which multi-component volatility factors, fat tails, and a non-monotonic pricing kernel can improve the hedging performance. A semi-explicit hedging formula is derived for our general framework which applies to a myriad of the option pricing models proposed in the discrete-time literature. We conduct an extensive empirical study of the impact of modelling features on the hedging effectiveness of S&P 500 options. Overall, we find that fat tails can be credited for half of the hedging improvement observed, while a second volatility factor and a non-monotonic pricing kernel each contribute to a quarter of this improvement. Moreover, our study indicates that the added value of these features for hedging is different than for pricing. A robustness analysis shows that a similar conclusion can be reached when considering the Dow Jones Industrial Average. Finally, the use of a hedging-based loss function in the estimation process is investigated in an additional robustness test, and this choice has a rather marginal impact on hedging performance.

A draft of the paper can be accessed here for those interested:

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ID de réunion : 945 8470 4249

Code secret : 369060

 Wednesday 26 May 2021 at 6pm:

HAORAN WANG  (Senior Research Data Scientist at Vanguard)

Title: Continuous-time mean–variance portfolio selection: A reinforcement learning framework 

Abstract: We approach the continuous-time mean–variance portfolio selection with reinforcement learning (RL). The problem is to achieve the best trade-off between exploration and exploitation, and is formulated as an entropy-regularized, relaxed stochastic control problem. We prove that the optimal feedback policy for this problem must be Gaussian,with time-decaying variance. We then prove a policy improvement theorem, based on which we devise an implementable RL algorithm.We find that our algorithm and its variant outperform both traditional and deep neural network based algorithms in our simulation and empirical studies. (This is a joint work with Xun Yu Zhou at Columbia University.) 

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ID de réunion : 977 7902 5385

Code secret : 806013

 Wednesday 12 May 2021 at 5pm:

BRUNO DUPIRE (head of quantitative research, Bloomberg )

Title: Functional Itô Calculus


We extend some results of the Itô calculus to functionals of the current path of a process to reflect the fact that often the impact of randomness is cumulative and depends on the history of the process, not merely on its current value. We express the differential of the functional in terms of adequately defined partial derivatives to obtain an Itô formula. We develop an extension of the Feynman-Kac formula to the functional case and an explicit expression of the integrand in the Martingale Representation Theorem. We establish that under certain conditions, even path dependent options prices satisfy a partial differential equation in a local sense. We exploit this fact to find an expression of the price difference between two models and compute variational derivatives with respect to the volatility surface. 

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ID de réunion : 968 5812 1629
Code secret : 585175

 Wednesday 14 April 2021 at 6pm:

MATTHIEU GARCIN (Léonard de Vinci Pôle universitaire, Paris)

Title: Fractional models: estimation, forecast, and market efficiency


The Hurst exponent describes the scaling properties of a time series. One also often links its value to the persistence of the series and consequently to one’s ability to forecast it: if H=1/2 there is no autocorrelation, if H>1/2 the series is persistent, and if H<1/2 the series is anti-persistent. However, the interpretation of the Hurst exponent strongly depends on the model describing the dynamic. Beyond the fractional Brownian motion (fBm), we are interested in various models: non-Gaussian extensions of the fBm, stationary transforms of a fBm, and multifractional motions, which rely on the assumption that the Hurst exponent is time-varying or even is a random process. We expose the specificities of the estimation of the Hurst exponent for all these models as well as the way one can forecast such series, using accuracy metrics that are relevant in the perspective of a portfolio manager. We also present various indicators of market efficiency based on these models.

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ID de réunion : 910 4122 3916

Code secret : 096455

Wednesday 24 MARCH 2021 at 6pm 


Title: The Joint S&P 500/VIX Smile Calibration Puzzle Solved and related topics


The very high liquidity of S&P 500 (SPX) and VIX derivatives requires that financial institutions price, hedge, and risk-manage their SPX and VIX options portfolios using models that perfectly fit market prices of both SPX and VIX futures and options, jointly. This is known to be a very difficult problem. Since VIX options started trading in 2006, many practitioners and researchers have tried to build such a model. So far the best attempts, which used parametric continuous-time jump-diffusion models on the SPX, could only produce approximate fits. In this talk we solve this long standing puzzle for the first time using a completely different approach: a nonparametric discrete-time model. Given a VIX future maturity T1, we build a joint probability measure on the SPX at T1, the VIX at T1, and the SPX at T2 = T1 + 30 days which is perfectly calibrated to the SPX smiles at T1 and T2, and the VIX future and VIX smile at T1. Our model satisfies the martingality constraint on the SPX as well as the requirement that the VIX at T1 is the implied volatility of the 30-day log-contract on the SPX.

The model is cast as the unique solution of what we call a Dispersion-Constrained Martingale Schrodinger Problem which is solved by duality using an extension of the Sinkhorn algorithm, in the spirit of (De March and Henry-Labordere, Building arbitrage-free implied volatility: Sinkhorn's algorithm and variants, 2019). We prove that the existence of such a model means that the SPX and VIX markets are jointly arbitrage-free. The algorithm identifies joint SPX/VIX arbitrages should they arise. Our numerical experiments show that the algorithm performs very well in both low and high volatility environments. Finally, we discuss how our technique extends to continuous-time stochastic volatility models, via what we dub VIX-Constrained Martingale Schrodinger Bridges, inspired by the classical Schrodinger bridge of statistical mechanics. The resulting stochastic volatility model is numerically implemented and is shown to achieve joint calibration with very high accuracy.

Time permitting, we will also briefly discuss a few related topics:

(i) a remarkable feature of the SPX and VIX markets: the inversion of convex ordering, and how classical stochastic volatility models can reproduce it;

(ii) why, due to this inversion of convex ordering, and contrary to what has often been stated, among the continuous stochastic volatility models calibrated to the market smile, the local volatility model does not maximize the price of VIX futures;

(iii) what are the optimal model-free bounds on the prices of VIX futures given SPX smiles.


ID de réunion : 917 1264 4694

Code secret : 803722

Wednesday 17 MARCH 2021 at 6pm 



Title: American options in the rough Heston model


Abstract: Rough volatility models have emerged as compelling alternatives to classical semimartingale models to capture important stylized features of the implied volatility surface and the time series of realized volatility. The rough Heston model is particularly appealing because its affine structure facilitates the pricing of European options using Fourier techniques. In this work we consider the problem of pricing American options in the rough Heston model. The complexity of the problem stems from the absence of a Markovian-semimartingale structure in the model. To overcome this difficulty, we work with a Markovian multi-factor semimartingale stochastic volatility model, which approximates the rough Heston model. In this approximated model, American options can be priced using a backward approach and simulation-based methods. We prove the convergence of American options prices in the multi-factor model towards the prices in the rough Heston model. The proof relies on the explicit expression of the conditional characteristic function of the joint forward process and the spot price, which is a consequence of the affine structure of the model. We illustrate with some numerical examples the behavior of American option prices with respect to some parameters in the model. This is joint work with Etienne Chevalier and Elizabeth Zuñiga.




Sujet : Financial Modelling seminar : SERGIO PULIDO ENSIIE

Heure : 17 mars 2021 06:00 PM Paris


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ID de réunion : 918 5174 3203

Code secret : 633782


le 18/12  Matthieu Garcin (ESILV) +MIGUEL ANGEL SANCHEZ GRANERO (Alméria) [Self similarities]


 Self similarity in finance.

Abstract: Self similarity is a broad concept popularized by fractals along a great number of topics. Self similar processes are stochastic processes with some kind of self similarity. We will introduce this concept and some algorithms to calculate the self similarity index or Hurst exponent, but with a particular focus on financial applications. The first obvious application is to study self similarity of the (log) price of a stock, but this is just the beginning. In fact, we will see some problems that appears in this context. Second, we will explore the use of the Hurst exponent in Fama-French factor models. Third, the self similarity index is also interesting to measure the degree of comovement of a pair of stocks. In this case, we look for anti-persistent series to choose a pair of stocks which "move together", so we can apply mean-reverting strategies in what is called "Pairs Trading". Four, we will describe how to use the self similarity index to analyze another well-known financial phenomenon: volatility clustering.

6 pm Matthieu Garcin (ESILV) 

 Selection, estimation, and prediction of fractional models: a financial perspective

The Hurst exponent describes the scaling properties of a time series. One also often links its value to the persistence of the series and consequently to one’s ability to forecast it: if H=1/2 there is no autocorrelation, if H&gt;1/2 the series is persistent, and if H&lt;1/2 the series is anti-persistent. However, the interpretation of the Hurst exponent strongly depends on the model describing the dynamic. We are interested in three classes of models: the fractional Brownian motion (fBm), multifractional motions, and transforms of a fBm. The multifractional motions are extensions of the fBm that rely on the assumption that the Hurst exponent is time-varying or even is a random process, whereas it is a constant for the fBm. Transforms of the fBm, such as the fractional Ornstein-Uhlenbeck process or the Lamperti transform of a fBm, are of practical interest, for example in the fixed income world, to model stationary processes. We expose the specificities of the estimation of the Hurst exponent for all these models as well as the way one can forecast such series, using accuracy metrics that are relevant in the perspective of a portfolio manager. We also address the issue of selecting the proper fractional or multifractional model, based on the data.

le 15/01   Jentzen (Université de Munster),  Alvaro Cartea (Oxford)  [Machine Learning] (canceled)

le 12/02   5pm: Roy Cerqueti  (Sapienza University of Rome)

The resilience of a complex network: methods and applications 

Abstract:  The analysis of the resilience of a network is of key relevance in the applied science, for its natural connections with the assessment of the

stability of an overall system and with systemic risk. This paper deals with this topic by advancing theoretical proposals for measuring the resilience

of a weighted network. Empirical experiments on real world data support the theoretical framework.

le 18/03  Alexandru Badescu (Calgary)+Sébastien Laurent (AMSE à confirmer) [Econometrics] (canceled  due to covid 19)

le 22/04    Sergio Pulido (ENSIEE) et Thibaut Mastrolia (Ecole Polytechnique) [Analyse stochastique] (canceled  due to covid 19)

le 27/05  Andrzej Nowak (Univ. Varsovie) + Dehua Shen (Univ. Tianjin)  [Finance comportementale] (canceled  due to covid 19)

le 24/06 TBA [Nowcasting] (canceled  due to covid 19)

Wednesday,  November  27 th 2019 (Room S17)

17h: Bernard de Meyer et Moussa Dabo (CES, Paris 1)

The CMMV Pricing Model in Practice 

Abstract:  Mainstream financial econometrics methods are based on models well tuned to replicate price dynamics, but with little to no economic justification. In particular, the randomness in these models is assumed to result from a combination of exogenous factors. In this paper, we present a model originating from game theory, whose corresponding price dynamics are a direct consequence of the information asymmetry between private and institutional investors. This model, namely the CMMV pricing model, is therefore rooted in market microstructure. The pricing methods derived from it also appear to fit very well historical price data. Indeed, as evidenced in the last section of the paper, the CMMV model does a very good job predicting option prices from readily available data. It also enables to recover the dynamic of the volatility surface.

6pm: Ramy Sukarieh (United Nations  JSPF  )

Stock selection and Portfolio construction using VAR models and Genetic Algorithm

We seek to construct portfolios with the following intuitive characteristics: ordered, segregated, integrated, conditional, and concentrated. To this end, we use vector auto-regressive VAR models and Genetic Algorithm GA. The VAR models were estimated using normal equations and MLE. The GA randomly permutes from the uniform discrete distribution without replacement, compute rank based fitness scaling function, and selects via stochastic universal sampling function (SUS). We use the Akaike Information Criterion (AIC) as the fitness function to the GA algorithm. This research shows that combining linear models with GA lead to well behaved investable portfolios with reasonable performance when compared to the market. In addition, it provides a simple yet rich framework from which one can bifurcate into more sophisticated model such as, Bayesian in time-varying VAR TVP-VAR with stochastics volatility, where inputs can be obtained from nonlinear models such as Poly-models. 

Wednesday,  October 16 th 2019

17h:  Eduardo Abi Jaber (Paris 1)

Invariance and viability theory: from finite dimensional ODEs to infinite dimensional SDEs 

Abstract:  We will give a general overview of viability theory for arbitrary closed sets with respect to deterministic and stochastic systems. A closed set D is said to be viable for a dynamical system, if there exists a D-valued solution to this system. We will provide a new geometric characterization for finite-dimensional stochastic differential equations (with jumps), which extends the well-known inward pointing Stratonovich drift condition to the case where the diffusion matrix may fail to be differentiable. Finally, we will discuss the extension to infinite dimensional S(P)DEs.

18h:  Luciano Campi (London School of economics)

N-player games and mean-field games with smooth dependence on past absorptions

Abstract:  N-player games and mean-field games with smooth dependence on past absorptions Mean-field games with absorption is a class of games, that have been introduced in Campi and Fischer (2018) and that can be viewed as natural limits of symmetric stochastic differential games with a large number of players who, interacting through a mean-field, leave the game as soon as their private states hit some given boundary. In this talk, we push the study of such games further, extending their scope along two main directions. First, a direct dependence on past absorptions has been introduced in the drift of players' state dynamics. Second, the boundedness of coefficients and costs has been considerably relaxed including drift and costs with linear growth. Therefore, the mean-field interaction among the players takes place in two ways: via the empirical sub-probability measure of the surviving players and through a process representing the fraction of past absorptions over time. Moreover, relaxing the boundedness of the coefficients allows for more realistic dynamics for players' private states. We prove existence of solutions of the mean-field game in strict as well as relaxed feedback form. Finally, we show that such solutions induce approximate Nash equilibria for the N-player game with vanishing error in the mean-field limit as N goes to infinity. This talk is based on a joint work with Maddalena Ghio and Giulia Livieri (SNS Pisa).


Tuesday 18 June 2019 

12:30: Marcos Costa Santos Carreira (Ecole Polytechnique - CMAP)

Learning Interest Rate Interpolation

Abstract:  The usual methods for interest rate interpolation consider only the values and time to maturity of spot rates as the inputs, and differ mainly on the continuity of the implied forward rates. We treat the interpolation problem as a replication problem, where a bond (or interest rate future/swap) is priced as a function of the minimum variance replicating portfolio of the traded bonds (or derivatives). In this view, the hedging ratios determined by the interpolation are as important (if not more) than getting the "right" interpolated rate; this is similar to the adjustments to the Black and Scholes delta as a consequence of the joint dynamics of the asset price and volatility in the different volatility models. We show how to learn the parameters of the weight functions and apply this method to the overnight rate indexed interest rates derivatives in Brazil. We then extend the concept from interpolating broken dates to the market references, in order to determine which points are key to the shape and dynamics of the curve and which points can be replicated by these real anchors. 

1:30: Stéphane Crépey (Université Evry Val d’Essonne)

XVA analysis from the balance sheet 

Abstract:  Since the 2008-09 financial crisis, derivative dealers charge to their clients various add-ons, dubbed XVAs, meant to account for counterparty risk and its capital and funding implications.  As banks cannot replicate jump-to-default related cash-flows, deals trigger wealth transfers from bank shareholders to bondholders and shareholders need to set capital at risk. On this basis, we devise a theory of XVAs, whereby the so-called contra-liabilities and cost of capital are sourced from bank clients at trade inceptions, on top of the fair valuation of counterparty risk, in order to compensate shareholders for wealth transfer and risk on their capital. The resulting all-inclusive XVA formula, meant incrementally at every new deal, reads (CVA+FVA+KVA), where C sits for credit, F for funding, and where the KVA is a cost of capital risk premium. All these XVAs are nonnegative and, even though we do crucially include the default of the bank itself in our modeling, unilateral in a certain sense. The corresponding XVA policy ensures to bank shareholders a submartingale wealth process corresponding to a target hurdle rate h on their capital at risk, consistently between and throughout deals.

Wednesday,  May 22 th 2019, 6th Floor MSE

17h-18h:  Maxence Soumare, Ekimetrics

Machine learning applied to Healtcare – How Ekimetrics Datascience expertise translates to the medical sector

Machine learning (ML) is a current hot topic in computer science research. This new promotion alongside an ever increasing ease to access and harness computing power has seen an explosion of ML applications to various fields. Although the medical domain is not Ekimetric’s main activity, the proposed use case demonstrates the capacity of the company to use data science to approach a wide range of questions and sectors. The presentation will introduce Ekimetric’s core activities and will then list some current Machine learning endeavors in Healthcare. We will then reintroduce some basics about neural networks, first for the biological ones followed by the same for their artificial counterparts. Lastly we will develop (methodology and technical choices) a use case that aims at answering a subject from Institut Gustave Roussy to detect renal cortex on radiologies. 

18h-19h: Srinivas Raghavendra, department of economics, Galway University, Ireland

Dynamics of conflict between shareholders and managers: Revisiting the theory of firm under financialization

 The period since 1980s, referred to as the “financialization” era, has witnessed profound changes in the way in which the financial markets and institutions interacted with the real economy. Although there is no one commonly agreed definition of the phenomenon “financialization”, the term is often used to refer the growing ascendency of ‘shareholder value’ as a mode of corporate governance. The shareholder power over manager is one of the channels through which financialization is argued to impact on the accumulation and growth decisions of firms. Here, we revisit the conventional theory of firm and introduce the shareholders-managers conflict to study the implications of financialization on the real economy in terms of some of the macroeconomic variables such as capacity utilization and growth.

Wednesday,  March 20 th 2019, 6th Floor MSE

17h-18h: Eric Jondeau, HEC Lausanne

Long-Term Financial Returns: VAR vs. DSGE Model – A Horse-Race 

This paper considers an institutional investor who is implementing a long-term portfolio allocation using forecasts of financial returns. We compare the predictive performance of two competing macro-finance models —an unrestricted Vector AutoRegression (VAR) and a fully structural Dynamic Stochastic General Equilibrium (DSGE) model— for investment horizons from 2 to 15 years. We show that, although the performances are similar for short horizons, the DSGE model outperforms the unrestricted VAR at forecasting financial returns in the long term. This model also generates substantially higher Sharpe ratios for mean-variance allocations. Even if it contains fewer unknown parameters, the DSGE model benefits from economically grounded restrictions that help anchor financial returns in the long term.

18h-19h: Florian Ielpo, Unigestion SA, Paris 1 Sorbonne University, Labex ReFi, and IPAG

 Factor Timing Revisited: Alternative Risk Premia Allocation Based on Nowcasting and Valuation Signals 

Alternative risk premia are encountering growing interest from investors. The vast majority of the academic literature has been focusing on describing the alternative risk premia (typically, momentum, carry and value strategies) individually. In this article, we investigate the question of allocation across a diversified range of cross-asset alternative risk premia over the period 1990-2018. For this, we design an active (macro risk-based) allocation framework that notably aims to exploit alternative risk premia’s varying behaviour in different macro regimes and their valuations over time. We perform backtests of the allocation strategy in an out-of-sample setting, shedding light on the significance of both sources of information.

Wednesday,  February 20 th 2019, 6th Floor MSE

17h-18h: Othmane Mounjid, Ecole Polytechnique

Optimal liquidity-based trading tactics (with Charles-Albert Lehalle and Mathieu Rosenbaum)  

We consider an agent who needs to buy (or sell) a relatively small amount of asset over some fixed short time interval. We work at the highest frequency meaning that we wish to find the optimal tactic to execute our quantity using limit orders, market orders and cancellations. To solve the agent’s control problem, we build an order book model and optimize an expected utility function based on our price impact. We derive the equations satisfied by the optimal strategy and solve them numerically. Moreover, we show that our optimal tactic enables us to outperform significantly naive execution strategies.

Wednesday,  January 23 th 2019, 6th Floor MSE

17h-19h: Giulia Rotundo, Sapienza University of Rome

Complex networks modeling for financial data

The seminar aims to show some usage of complex networks analysis for modeling financial data. The main focus is going to be on three applications

- A copula approach to cross-ownership of companies,

- Herding in mutual funds,

- Assessing the impact of incomplete information on the resilience of financial networks.

In details:

- The cross-ownership of companies creates no trivial links among them. Within this respect, the diversification is intended to describe the holdings;  and the integration represents the number of other companies that hold the shares. A high integration allows to spread fluctuations on the other companies, but it reduces the amount of profit to be kept in the company; a high diversification allows to spread the sources of risk, but at the same time it increases the probability to be exposed to fluctuations of other companies.

A copula approach for the detection of the most instable network topologies gives the results for the interaction among integration and diversification. A case study is used to outline the method.

- Herding in mutual funds

Some previous studies were emphasizing that correlations among stocks were loose during expansion periods of the market; and that they were stronger during recessions. Does it hold for mutual funds? We are going to build the network of mutual funds through the correlation network and to insert in the regression of the herding the centrality measures as explanatory variables. The results show that the behavior of mutual funds is just the opposite of the stocks.

- Assessing the impact of incomplete information on the resilience of financial networks

The paper faces the problem of the robustness of the network of interbank exposures against the propagation of failure cascades. The available data are retrieved through the BIS database and report only incomplete information, and show a core-periphery structure. A model of financial contagion is set up to estimate the width and length of the cascades, both in real and simulated data, that insert some percentages of the missing links. Simulations show that the network is far from the worst scenario for the propagation of contagions, and that the detection of the missing links is not trivial in the overall dynamic.