AID Rene

This thesis is composed of two independent parts. The first part focuses on the application of the Principal-Agent problem (c.f. Cvitanic & Zhang (2013) and Cvitanic. et al. (2018)) for solving modeling problems in energy markets. The second one deals with the joint laws of a martingale and its current maximum.We first focus on the electricity capacity market, and in particular capacity remuneration mechanisms. Given the increasing share of renewable energies in the electricity production, "classical" power plants (e.g. gas or coal) are less and less used, which makes them unprofitable and not economically viable. However, their closure would expose consumers to the risk of a blackout in the event of a peak in electricity demand, since electricity cannot be stored. Thus, generation capacity must always be maintained above demand, which requires a "capacity payment mechanism" to remunerate seldom used power plants, which can be understood as an insurance to be paid against electricity blackouts.We then address the issue of incentives for decarbonization. The objective is to propose a model of an instrument that can be used by a public agent (the state) to encourage the different sectors to reduce their carbon emissions in a context of moral hazard (where the state does not observe the effort of the actors and therefore cannot know whether a decrease in emissions comes from a decrease in production and consumption or from a management effort. The second part (independent) is motivated by model calibration and arbitrage on a financial market with barrier options. It presents a result on the joint laws of a martingale and its current maximum. We consider a family of probabilities in dimension 2, and we give necessary and sufficient conditions ensuring the existence of a martingale such that its marginal laws coupled with those of its current maximum coincide with the given probabilities.We follow the methodology of Hirsch and Roynette (2012) based on a martingale construction by DHS associated with a well-posed Fokker-Planck PDE verified by the given marginal laws under regularity assumptions, then in a general framework with regularization and boundary crossing.

We study a new kind of nonzero-sum stochastic differential game with mixed impulse/switching controls, motivated by strategic competition in commodity markets. A representative upstream firm produces a commodity that is used by a representative downstream firm to produce a final consumption good. Both firms can influence the price of the commodity. By shutting down or increasing generation capacities, the upstream firm influences the price with impulses. By switching (or not) to a substitute, the downstream firm influences the drift of the commodity price process. We study the resulting impulse-regime switching game between the two firms, focusing on explicit threshold-type equilibria. Remarkably, this class of games naturally gives rise to multiple potential Nash equilibria, which we obtain thanks to a verification-based approach. We exhibit three candidate types of equilibria depending on the ultimate number of switches by the downstream firm (zero, one or an infinite number of switches). We illustrate the diversification effect provided by vertical integration in the specific case of the crude oil market. Our analysis shows that the diversification gains strongly depend on the pass-through from the crude price to the gasoline price.

We develop a model for the industry dynamics in the electricity market, based on mean-field games of optimal stopping. In our model, there are two types of agents: the renewable producers and the conventional producers. The renewable producers choose the optimal moment to build new renewable plants, and the conventional producers choose the optimal moment to exit the market. The agents interact through the market price, determined by matching the aggregate supply of the two types of producers with an exogenous demand function. Using a relaxed formulation of optimal stopping mean-field games, we prove the existence of a Nash equilibrium and the uniqueness of the equilibrium price process. An empirical example, inspired by the UK electricity market is presented. The example shows that while renewable subsidies clearly lead to higher renewable penetration, this may entail a cost to the consumer in terms of higher peakload prices. In order to avoid rising prices, the renewable subsidies must be combined with mechanisms ensuring that sufficient conventional capacity remains in place to meet the energy demand during peak periods.

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We consider a general nonzero-sum impulse game with two players. The main mathematical contribution of this paper is a verification theorem that provides, under some regularity conditions, a suitable system of quasi-variational inequalities for the payoffs and the strategies of the two players at some Nash equilibrium. As an application, we study an impulse game with a one-dimensional state variable, following a real-valued scaled Brownian motion, and two players with linear and symmetric running payoffs. We fully characterize a family of Nash equilibria and provide explicit expressions for the corresponding equilibrium strategies and payoffs. We also prove some asymptotic results with respect to the intervention costs. Finally, we consider two further nonsymmetric examples where a Nash equilibrium is found numerically.

We consider a general nonzero-sum impulse game with two players. The main mathematical contribution of this paper is a verification theorem that provides, under some regularity conditions, a suitable system of quasi-variational inequalities for the payoffs and the strategies of the two players at some Nash equilibrium. As an application, we study an impulse game with a one-dimensional state variable, following a real-valued scaled Brownian motion, and two players with linear and symmetric running payoffs. We fully characterize a family of Nash equilibria and provide explicit expressions for the corresponding equilibrium strategies and payoffs. We also prove some asymptotic results with respect to the intervention costs. Finally, we consider two further nonsymmetric examples where a Nash equilibrium is found numerically.

We formulate an equilibrium model of intraday trading in electricity markets. Agents face balancing constraints between their customers consumption plus intraday sales and their production plus intraday purchases. They have continuously updated forecast of their customers consumption at maturity with decreasing volatility error. Forecasts are prone to idiosyncratic noise as well as common noise (weather). Agents production capacities are subject to independent random outages, which are each modelled by a Markov chain. The equilibrium price is defined as the price that minimises trading cost plus imbalance cost of each agent and satisfies the usual market clearing condition. Existence and uniqueness of the equilibrium are proved, and we show that the equilibrium price and the optimal trading strategies are martingales. The main economic insights are the following. (i) When there is no uncertainty on generation, it is shown that the market price is a convex combination of forecasted marginal cost of each agent, with deterministic weights. Furthermore, the equilibrium market price follows Almgren and Chriss's model and we identify the fundamental part as well as the permanent market impact. It turns out that heterogeneity across agents is a necessary condition for the Samuelson's effect to hold. (ii) When there is production uncertainty, the price volatility becomes stochastic but converges to the case without production uncertainty when the number of agents increases to infinity. Further, on a two-agent case, we show that the potential outages of a low marginal cost producer reduces her sales position.

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This work presents a novel policy iteration algorithm to tackle nonzero-sum stochastic impulse games arising naturally in many applications. Despite the obvious impact of solving such problems, there are no suitable numerical methods available, to the best of our knowledge. Our method relies on the recently introduced characterisation of the value functions and Nash equilibrium via a system of quasi-variational inequalities. While our algorithm is heuristic and we do not provide a convergence analysis, numerical tests show that it performs convincingly in a wide range of situations, including the only analytically solvable example available in the literature at the time of writing.

This thesis focuses on electricity spot markets and more particularly on their organization and design. Indeed, the electricity industry is facing new challenges due to the increase of renewable generation capacity but also due to more structural changes related to the behavior of actors. The first chapter of the thesis studies in detail the formation of liquidity in the German continuous market. In particular, it focuses on the evolution of the bid-ask spread during a trading session and the factors that influence it. In a second chapter, I quantitatively evaluate the effect of creating an auction before the start of a continuous market on liquidity, volatility and market concentration. The last part of the thesis theoretically studies the impact of vertical integration in sequential markets as well as the impact of real-time pricing for end consumers on the behavior of market participants.

This thesis aims to analyze the systemic risk on commodity futures markets. Indeed, several research works highlight the importance of these futures in the determination of the physical price of commodities. Their incorporation into traditional finance as a diversifying asset has led to a similar price evolution to that of various financial assets since about 2004. The question that motivated this thesis was therefore to quantify this systemic risk (since it affects commodities, which are directly involved in the real economy), to see precisely the means of transmission (which markets affect which other markets) and finally to allow for an evaluation of the consequences, for example, based on scenarios (stress tests). It therefore makes it possible to develop market surveillance tools and could therefore contribute to the regulation of these markets.

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This paper analyses the interaction between centralised carbon emissive technologies and distributed intermittent non-emissive technologies. In our model, there is a representative consumer who can satisfy her electricity demand by investing in distributed generation (solar panels) and by buying power from a centralised firm at a price the firm sets. Distributed generation is intermittent and induces an externality cost to the consumer. The firm provides non-random electricity generation subject to a carbon tax and to transmission costs. The objective of the consumer is to satisfy her demand while minimising investment costs, payments to the firm and intermittency costs. The objective of the firm is to satisfy the consumer's residual demand while minimising investment costs, demand deviation costs, and maximising the payments from the consumer. We formulate the investment decisions as McKean-Vlasov control problems with stochastic coefficients. We provide explicit, price model-free solutions to the optimal decision problems faced by each player, the solution of the Pareto optimum, and the Stackelberg equilibrium where the firm is the leader. We find that, from the social planner's point of view, the carbon tax or transmission costs are necessary to justify a positive share of distributed capacity in the long-term, whatever the respective investment costs of both technologies are. The Stackelberg equilibrium is far from the Pareto equilibrium and leads to an over-investment in distributed energy and to a much higher price for centralised energy.

We consider competitive capacity investment for a duopoly of two distinct producers. The producers are exposed to stochastically fluctuating costs and interact through aggregate supply. Capacity expansion is irreversible and modeled in terms of timing strategies characterized through threshold rules. Because the impact of changing costs on the producers is asymmetric, we are led to a nonzero-sum timing game describing the transitions among the discrete investment stages. Working in a continuous-time diffusion framework, we characterize and analyze the resulting Nash equilibrium and game values. Our analysis quantifies the dynamic competition effects and yields insight into dynamic preemption and over-investment in a general asymmetric setting. A case-study considering the impact of fluctuating emission costs on power producers investing in nuclear and coal-fired plants is also presented.

We consider the problem of optimal trading for a power producer in the context of intraday electricity markets. The aim is to minimize the imbalance cost induced by the random residual demand in electricity, i.e. the consumption from the clients minus the production from renewable energy. For a simple linear price impact model and a quadratic criterion, we explicitly obtain approximate optimal strategies in the intraday market and thermal power generation, and exhibit some remarkable properties of the trading rate. Furthermore, we study the case when there are jumps on the demand forecast and on the intraday price, typically due to error in the prediction of wind power generation. Finally, we solve the problem when taking into account delay constraints in thermal power production.

We establish explicit socially optimal rules for an irreversible investment decision with time-to-build and uncertainty. Assuming a price sensitive demand function with a random intercept, we provide comparative statics and economic interpretations for three models of demand (arithmetic Brownian, geometric Brownian, and the Cox-Ingersoll-Ross). Committed capacity, that is, the installed capacity plus the investment in the pipeline, must never drop below the best predictor of future demand, minus two biases. The discounting bias takes into account the fact that investment is paid upfront for future use. the precautionary bias multiplies a type of risk aversion index by the local volatility. Relying on the analytical forms, we discuss in detail the economic effects.

Offering a concise but complete survey of the common features of the microstructure of electricity markets, this book describes the state of the art in the different proposed electricity price models for pricing derivatives and in the numerical methods used to price and hedge the most prominent derivatives in electricity markets, namely power plants and swings. The mathematical content of the book has intentionally been made light in order to concentrate on the main subject matter, avoiding fastidious computations. Wherever possible, the models are illustrated by diagrams. The book should allow prospective researchers in the field of electricity derivatives to focus on the actual difficulties associated with the subject. It should also offer a brief but exhaustive overview of the latest techniques used by financial engineers in energy utilities and energy trading desks.

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Many public policies declare electric mobility as a key lever for insuring the carbon emission target and attaining the objectives of oil-dependence reduction. However, the cost of an electric vehicle (ev) is still way too expensive compared to the conventional fuel-powered vehicle (fv) and constitutes a serious barrier against its diffusion. In this note we formulate a tractable model to analyse the dynamics of the adoption of ev’s. The dynamic is driven by increasing marginal production returns and consumer’s willingness to pay. We define the social benefit of replacing an fv by an ev as the fuel-economy it allows to realize, and solve for the optimal subsidy rule. We show that in a context of expensive fuel price, a voluntary policy of subsidy can transform the present fuel-powered fleet into an electric one.

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We consider the problem of optimal trading for a power producer in the context of intraday electricity markets. The aim is to minimize the imbalance cost induced by the random residual demand in electricity, i.e. the consumption from the clients minus the production from renewable energy. For a simple linear price impact model and a quadratic criterion, we explicitly obtain approximate optimal strategies in the intraday market and thermal power generation, and exhibit some remarkable properties of the trading rate. Furthermore, we study the case when there are jumps on the demand forecast and on the intraday price, typically due to error in the prediction of wind power generation. Finally, we solve the problem when taking into account delay constraints in thermal power production.

We consider the problem of optimal trading for a power producer in the context of intraday electricity markets. The aim is to minimize the imbalance cost induced by the random residual demand in electricity, i.e. the consumption from the clients minus the production from renewable energy. For a simple linear price impact model and a quadratic criterion, we explicitly obtain approximate optimal strategies in the intraday market and thermal power generation, and exhibit some remarkable properties of the trading rate. Furthermore, we study the case when there are jumps on the demand forecast and on the intraday price, typically due to error in the prediction of wind power generation. Finally, we solve the problem when taking into account delay constraints in thermal power production.

In this paper, we present a probabilistic numerical algorithm combining dynamic programming, Monte Carlo simulations, and local basis regressions to solve nonstationary optimal multiple switching problems in infinite horizon. We provide the rate of convergence of the method in terms of the time step used to discretize the problem, of the regression basis used to approximate conditional expectations, and of the truncating time horizon. To make the method viable for problems in high dimension and long time horizon, we extend a memory reduction method to the general Euler scheme, so that, when performing the numerical resolution, the storage of the Monte Carlo simulation paths is not needed. Then, we apply this algorithm to a model of optimal investment in power plants in dimension eight, i.e., with two different technologies and six random factors.

We establish explicit socially optimal rules for an irreversible investment decision with time-to-build and uncertainty. Assuming a price sensitive demand function with a random intercept, we provide comparative statics and economic interpretations for three models of demand (arithmetic Brownian, geometric Brownian, and the Cox-Ingersoll-Ross). Committed capacity, that is, the installed capacity plus the investment in the pipeline, must never drop below the best predictor of future demand, minus two biases. The discounting bias takes into account the fact that investment is paid upfront for future use. the precautionary bias multiplies a type of risk aversion index by the local volatility. Relying on the analytical forms, we discuss in detail the economic effects.

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This thesis studies the evolution of natural gas markets in Europe until 2035 using modeling tools. The proposed model, GaMMES, is based on an oligopolistic description of the markets and its main advantages are: a high level of detail of the economic structure of the gas chain and an endogenous consideration of long-term contracts upstream as well as substitution with oil products and coal on the demand side. First, we study the issue of gas supply security in Europe and the conditions for regulating markets vulnerable to the risk of supply disruption, especially from Russia. Three case studies are proposed according to the degree of dependence and the nature of regulation in place: the German market of the 1980s and the current markets of Bulgaria and Spain. In particular, we study the evolution of market characteristics as a function of the risk of disruption and the type of regulation to be put in place to ensure social welfare optimality. Then, we propose a dynamic systems model to take into account the energy substitution between coal, oil and natural gas. Our approach allows us to estimate a new functional form of the demand function for natural gas, which includes both energy substitution and consumption inertia due to end-user investments. In a third step, we use this demand function in a partial equilibrium model of natural gas markets in Europe. The GaMMES model, written as a complementarity problem, represents the main players in the natural gas industry by considering their strategic interactions and market powers. It has been applied to the natural gas market in North-East Europe in order to study the evolution, until 2035, of consumption, spot prices, long-term prices and volumes, production and dependence on foreign imports. Finally, we propose a stochastic extension of the GaMMES model in order to analyze the impact of the strong fluctuation of the Brent price on the gas markets. An econometric study was conducted to calculate the probability law of the oil price, when modeled as a random variable, in order to build and weight the scenario tree. The results allow us to understand how the randomness modifies the strategic behaviors of the actors, in particular at the level of long-term contracts. Finally, the value of the stochastic solution is calculated in order to quantify the importance of taking into account oil price fluctuations for each actor in the chain.

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This work comes from an industrial problem of validation of numerical solutions of ordinary differential equations modeling electrical networks. The chosen approach is the asymptotic estimation of the global error. Four techniques are studied: the Richardson estimator (RS), the Zadunaisky estimator (ZD), the integration of the variational equation (EV) and the calculation of a global correction (SC). The relative order of convergence of these techniques is defined by the speed of convergence of the ratio between the estimator and the error when the discretization is refined. When this ratio is strictly positive, the estimator is said to be valid. We give details on the order of convergence of the SC estimator according to the order of the integration method it uses. To the ZD variants, we add an independent one using the Modified Equation. For variable step integration, the question remained whether ZD and SC retained their order of convergence with respect to the user tolerance. By restricting the type of local error control, we can answer in the affirmative. We further show that some Runge-Kutta methods require weaker assumptions to ensure the validity of this estimator. Numerical tests complete this analysis. They show a negative effect of the arithmetic error on some of these estimation techniques. When the global error reaches its minimum value, contrary to RS, the SC and ZD estimators underestimate it. Finally, an integrator is proposed which avoids the a priori specification of an integration path for complex time equations. Based on a local error control, it allows to automatically bypass isolated singularities.