I develop and test a theoretical model to study the interaction between the commodity and stock markets. The article attempts to clarify the debate between the two conflicting empirical opinions about the effect of the financialization on commodity markets: one that claims there is an effect, and one that denies that effect. The theoretical model determines the futures risk premium by using three factors: the hedging pressure, the stock market returns, and the commodity-equity correlation. I test the futures risk premium in the era of the financialization for three commodities in the energy market: crude oil (WTI), natural gas, and heating oil in Continue reading

## The Role of Financial Investors in Commodity Futures Risk Premium

## The joint dynamics of spot and futures commodity prices

by **Ivar Ekeland, Edouard Jaeck, Delphine Lautier & Bertrand Villeneuve**

We model the dynamic behavior of spot and futures commodity prices with an infinite horizon rational expectations equilibrium model. A new type of proof of existence of the equilibrium is provided. Using simulations with minimal changes between scenarios, we explore the specific effects of market structure, autocorrelation of production, and global risk aversion. The market structure can change a virtually nonstorable commodity into a high-inventory one. A high autocorrelation soften the apparent effects of storage in the short run. Global risk aversion typically decreases when financialization is developed. The effects on the joint price dynamics, risk sharing and physical choices are explored. Continue reading

## Conference on Commodities, Volatility, and Risk Management The Impacts of Trade Restrictions, Market Imperfections, and Green Finance

**13-15 May 2019, Université Paris-Dauphine**

The conference will cover commodity pricing and risk management, viewed through the prisms of market imperfections and environmental concerns. The main focus is on agricultural and energy markets, with specific themes intended to shine light on what the organizers and members of the scientific, industry, and policy advisory committees believe will be prominent issues in the near future.

For further information, call for papers, online submission and registration: https://commodity.sciencesconf.org/

## Deep neural networks algorithms for stochastic control problems on finite horizon, part II: numerical applications

by **Côme Huré, Huyên Pham, Achref Bachouch & Nicolas Langrené**

This paper presents several numerical applications of deep learning-based algorithms that have been analyzed in [Bachouch et al., 2018a]. Numerical and comparative tests using TensorFlow illustrate the performance of our different algorithms, namely control learning by performance iteration (algorithms NNcontPI and ClassifPI), control learning by hybrid iteration (algorithms Hybrid-Now and Hybrid-LaterQ), on the 100-dimensional nonlinear PDEs examples from [Weinan et al. 2017] and on quadratic Backward Stochastic Differential equations as in [Chassagneux et al., 2016]. We also provide numerical results for an option hedging problem in finance, and energy storage problems arising in the valuation of gas storage and in microgrid management.

## Deep neural networks algorithms for stochastic control problems on finite horizon, part I: convergence analysis

by **Côme Huré, Huyên Pham, Achref Bachouch & Nicolas Langrené**

This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming (DP). Differently from the classical approximate DP approach, we first approximate the optimal policy by means of neural networks in the spirit of deep reinforcement learning, and then the value function by Monte Carlo regression. This is achieved in the DP recursion by performance or hybrid iteration, and regress now or later/quantization methods from numerical probabilities. We provide a theoretical justification of these algorithms. Consistency and rate of convergence for the control and value function estimates are analyzed and expressed in terms of the universal approximation error of the neural networks. Numerical results on various applications are presented in a companion paper [Bachouch et al., 2018b] and illustrate the performance of our algorithms.

## Day-ahead probabilistic forecast of solar irradiance: a Stochastic Differential Equation approach

by **Jordi Badosa, Emmanuel Gobet, Maxime Grangereau & Daeyoung Kim**

In this work, we derive a probabilistic forecast of the solar irradiance during a day at a given location, using a stochastic differential equation (SDE for short) model. We propose a procedure that transforms a deterministic forecast into a probabilistic forecast: the input parameters of the SDE model are the AROME numerical weather predictions computed at day D-1 for the day D. The model also accounts for the maximal irradiance from the clear sky model. The SDE model is mean-reverting towards the deterministic forecast and the instantaneous amplitude of the noise depends on the clear sky index, so that the fluctuations vanish as the index is close to 0 (cloudy) or 1 (sunny), as observed in practice. Our tests show a good adequacy of the confidence intervals of the model with the measurement.

## Optimal electricity demand response contracting with responsiveness incentives

by **René Aïd, Dylan Possamai & Nizar Touzi**

Despite the success of demand response programs in retail electricity markets in reducing average consumption, the literature shows failure to reduce the variance of consumers’ responses. This paper aims at designing demand response contracts which allow to act on both the average consumption and its variance. The interaction between the producer and the consumer is modeled as a Principal-Agent problem, thus accounting for the moral hazard underlying demand response programs. The producer, facing the limited flexibility of production, pays an appropriate incentive compensation in order to encourages the consumer to reduce his average consumption and to enhance his responsiveness. We provide closed-form solution for the optimal contract in the case of linear energy valuation. Without responsiveness incentive, this solution decomposes into a fixed premium for enrolment and a proportional price for the energy consumed, in agreement with previously observed demand response contracts. The responsiveness incentive induces a new component in the contract with payment rate on the consumption quadratic variation. Furthermore, in both cases, the components of the premium exhibit a dependence on the duration of the demand response event. In particular, the fixed component is negative for sufficiently long events. Finally, under the optimal contract with optimal consumer behaviour, the resulting consumption volatility may decrease as required, but it may also increase depending on the risk aversion parameters of both actors. This agrees with standard risk sharing effects. The calibration of our model to publicly available data of a large scale demand response experiment predicts a significant increase of responsiveness under our optimal contract, a significant increase of the producer satisfaction, and a significant decrease of the consumption volatility. The stability of our explicit optimal contract is justified by appropriate sensitivity analysis.

## Conference “Advances in Modelling and Control for Power Systems of the Future”

**Conference CAESARS 2018**

September 5-7 2018 – EDF Lab, Palaiseau

In electrical system, strong evolutions are under way that will change deeply the organisation of the whole sector in the short and long term horizon: quick development of renewable technology, volatile and unpredictable production, costly investments in a difficult economic context, competing environment, emission market, new usages of electric vehicles…

The simulation and analysis of the evolution of the electrical system is fundamental for better sustaining the energetic transition to a Clean Energy World but it is challenging Continue reading

## “Des Mathématiques de la décision aux jeux à champ moyen”

**Conférence en l’honneur de Jean-Michel Lasry **

27 JUIN 2018 – Université Paris Dauphine

#JML70

Le programme de la conférence en l’honneur de Jean-Michel Lasry pour son 70e anniversaire fait écho à la riche carrière scientifique de celui-ci, mêlant divers champs des mathématiques appliquées (analyse convexe, contrôle optimal, équations aux dérivés partielles, calcul stochastique, théorie des jeux, etc.) et faisant la part belle aux applications, notamment à l’économie et à la finance. Continue reading

## Thematic Semester “Statistics for Energy Markets”

The large-scale development of renewable energy production prompts us to rethink the structure of the modelling of pricing processes and how we conceive financial risks in the energy markets. Because this development accentuates the impact on market prices of global warming, better consideration of this link should enable portfolio managers exposed to the energy markets to more clearly understand the risks and to determine optimal strategies for hedging them. In this context, the current challenges concern: (1) the definition of models suited to risk assessment, measurement and hedging applications; (2) development of forecasting tools, both of climate variables and market prices; and (3) effective statistical estimation procedures.

This thematic semester, titled “Statistics for the energy markets”, aims to present the recent thinking and developments on this topic, and to identify areas for research and collaboration between practitioners in the industry and academic researchers. It is supported by the “Finance and sustainable growth” Labex of the Institut Louis Bachelier. Continue reading

## HJB Equations and Extensions of Classical Stochastic Control Theory

These notes are a transcription of the course ”Equations de HJB et extensions de la théorie classique du contrôle stochastique”, given by P.-L. Lions at the Collège de France in 2016/2017. The course contains a few developments on Bayesian learning, on optimal control of conditioned processes, on interfaces and junction problems, and a seminar on scalar conservation laws which was presented by P.-L. Lions as a part of his course.

## Mean Field Games and Related Topics #4

**Chiostro di S. Pietro in Vincoli, Rome**

**June 14-16, 2017**

Mean field games theory describes the equilibria in strategic interactions of a large number of rational agents. In the recent years, this research area has been rapidly spreading in several different directions which are concerned with significant probabilistic and analytic issues. The goal of this conference is to discuss recent advances in the modeling, analysis and numerical approximation of mean field games systems as well as some related, even if different, approaches to the description of macroscopic behaviors in collective dynamics.

## PDE and Probability Methods for Interactions

**Sophia Antipolis (France) – March 30-31, 2017**

The conference PDE and Probability Methods for Interactions will be held at Inria, on the French Riviera, March 30-31, 2017. The goal of this conference is to exchange ideas on new trends in PDE analysis and stochastic analysis which are motivated by interactions between themselves, or by interactions between them and applications in various fields.

## Volatility in electricity derivative markets: the Samuelson effect revisited

by **Edouard Jaeck & Delphine Lautier**

This article proposes an empirical study of the Samuelson effect in electricity markets. Our motivations are twofold. First, although the literature largely assesses the decreasing pattern in the volatilities along the price curve in commodity markets, it has not extensively tested the presence of such a dynamic feature in electricity prices. Second, the analysis of a non-storable commodity enriches the literature on the behavior of commodity prices. Indeed, it has been sometimes asserted that the Samuelson effect results from the presence of inventories. We examine the four most important electricity futures markets worldwide for the period from 2008 to 2014: the German, Nordic, Australian, and US markets. We also use the American crude oil market as a benchmark for a storable commodity negotiated on a mature futures market. Our analysis has two steps: i) in addition to the traditional tests, we propose and test a new empirical implication of the Samuelson effect: price shocks should spread from the physical market to the paper market, and not the reverse; ii) based on the concept of “indirect storability”, we investigate the link between the Samuelson effect and the storability of the commodity. We find evidence of a Samuelson effect in all of the electricity markets and show that storage is not a necessary condition for such an effect to appear. These results should be taken into account for the understanding of the dynamic behavior of commodity prices, for the valuation of electricity assets, and for hedging operations.

## A Long-Term Mathematical Model for Mining Industries

by **Yves Achdou, Pierre-Noel Giraud, Jean-Michel Lasry & Pierre-Louis Lions**

A parcimonious long term model is proposed for a mining industry. Knowing the dynamics of the global reserve, the strategy of each production unit consists of an optimal control problems with two controls, first the ux invested into prospection and the building of new extraction facilities, second the production rate. In turn, the dynamics of the global reserve depends on the individual strategies of the producers, so the models leads to an equilibrium, which is described by low dimensional systems of partial differential equations. The dimensionality depends on the number of technologies that a mining producer can choose. In some cases, the systems may be reduced to a Hamilton-Jacobi equation which is degenerate at the boundary and whose right hand side may blow up at the boundary. A mathematical analysis is supplied. Then numerical simulations for models with one or two technologies are described. In particular, a numerical calibration of the model in order to t the historical data is carried out.

## Negotiated Red Zones Around Hazardous Plants

by **Céline Grislain-Letrémy & Bertrand Villeneuve**

The industrialists are liable for any damage they cause to neighboring households. Consequently, households do not have to pay for the risk they create by locating in exposed areas. To contain their liabilities, the firm can purchase or rent land, establishing an exclusion zone, also called a red zone. This paper studies the negotiations of red zones in urban areas exposed to industrial disasters. We compare typical scenarios regarding the distribution of bargaining power between the firm and the city. Using a microeconomic model, we explain how red zones are revised as technology, climate, or demography change, and we show how and why responses are neatly distinct across scenarios. Further, we give and explain the conditions for industrial sanctuaries (as the population grows) and city cores (as the risk grows).

## Natural Disasters, Land-Use, and Insurance

by **Céline Grislain-Letrémy & Bertrand Villeneuve**

This paper addresses the urbanization of areas exposed to natural disasters and studies its dependency on land-use and insurance policies. The risk-map paradox that we describe explains why an insurance system with simplistic maps and tariffs is the rule. Indeed, in practice we observe simple policies, consisting of a prohibited red zone and a zone without insurance tariff differentiation. We show that they implement the optimal land-use in specific cases. Even if there are fixed damages per dwelling, the red-zone policy is relatively efficient. In a central proposition, we detail the effects redefining the optimal red zone as the climate or the population changes. We use this analysis to expose and comment plausible cases in which, as the population grows, the red zone shrinks, the red zone grows, and the red zone shrinks and then grows.

## StOpt: A New Stochastic Optimization Library

The STochastic OPTimization library (StOpt) was developped at **OSIRIS** department at **EDF R&D** and used in some operational projects as an open source project.

The StOpt library aims at providing tools in C++ for solving some stochastic optimization problems encountered in finance or in the industry. A python binding is available for some C++ objects provided permitting to easily solve an optimization problem by regression.

Different methods are available :

– dynamic programming methods based on Monte Carlo with regressions (global, local and sparse regressors), for underlying states following some uncontrolled Stochastic Differential Equations (python binding provided). Continue reading

## Jeux à champ moyen – Bilan et perspectives

par **Pierre-Louis Lions**

**16 juin 2016, ****Abbaye des Vaux de Cernay**

Présentation donnée dans le cadre des journées ateliers FiME à l’Abbaye des Vaux de Cernay.

## Journées ateliers FiME

**Abbaye des Vaux de Cernay**

**16-17 juin 2016**

L’objectif de ces deux journées était de faire le point sur les travaux conduits au sein de l’IdR FiME, les résultats déjà obtenus et les perspectives de développement de ces travaux. Les travaux étaient présentés au travers d’ateliers co-animés par les ingénieurs chercheurs d’EDF R&D et les chercheurs académiques, complétés de quelques exposés magistraux (G. Stolz, PL Lions, B Villeneuve, I Ekeland).