12-16 Jun 2023, Centre Paul Langevin, Aussois

The aim of the Summer school is to review some aspects of quantitative methods applied to economics and finance. Particular emphasis will be placed on methods that make use of massive or high-dimensional data and on applications to energy issues, in line with the research themes of the FiME laboratory. Three themes will be highlighted, all of which are particularly relevant to today’s world and pose crucial challenges: “High Dimensional Econometrics”, “Differential Privacy”, “Market Microstructure”.

Further information on the Summer school website

by Mohammad Isleimeyyeh

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 →

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 →

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.

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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.

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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.

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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.

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by Charafeddine Mouzouni

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.

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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.

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