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Aïd R., Gruet P., Pham H. (2016), An optimal trading problem in intraday electricity markets, Mathematics and Financial Economics, 10, p. 49-85
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.
Aïd R., Federico S., Pham H., Villeneuve B. (2015), Explicit investment rules with time-to-build and uncertainty, Journal of Economic Dynamics and Control, 51, p. 240â256
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. For example, the impact of volatility on the optimal investment is negligible in some cases. It vanishes in the CIR model for long delays, and in the GBM model for high discount rates
Langrené N., Campi L., Aïd R. (2013), A Structural Risk-Neutral Model for Pricing and Hedging Power Derivatives, Mathematical Finance, 23, 3, p. 387-438
We develop a structural risk-neutral model for energy market modifying along several directions the approach introduced in Aïd et al. In particular, a scarcity function is introduced to allow important deviations of the spot price from the marginal fuel price, producing price spikes. We focus on pricing and hedging electricity derivatives. The hedging instruments are forward contracts on fuels and electricity. The presence of production capacities and electricity demand makes such a market incomplete. We follow a local risk minimization approach to price and hedge energy derivatives. Despite the richness of information included in the spot model, we obtain closed-form formulae for futures prices and semiexplicit formulae for spread options and European options on electricity forward contracts. An analysis of the electricity price risk premium is provided showing the contribution of demand and capacity to the futures prices. We show that when far from delivery, electricity futures behave like a basket of futures on fuels.
Vialas C., Touzi N., Aïd R. (2012), An arbitrage-free interest rate model consistent with economic constraints for Long-Term Asset Liability Management, Bankers, Markets & Investors, 116, p. 4-19
This paper proposes an Heath-Jarrow-Morton model of the yield curve that can fit the particular requirements of long-term asset and liability management (ALM). In particular, the proposed HJM model can reproduce expected long-term statistical properties of any two interest rates, while still satisfying the no-arbitrage constraints. We describe the methodology to calibrate the model in this particular constrainted setting, i.e. to find the model parameters as a function of the expected statistical properties. We precisely give the constraints on these expectations to ensure the existence of a solution.
Chemla G., Porchet A., Aïd R., Touzi N. (2011), Hedging and vertical integration in electricity markets, Management Science, 57, 8, p. 1438-1452
This paper analyzes the interactions between competitive (wholesale) spot, retail, and forward markets and vertical integration in electricity markets. We develop an equilibrium model with producers, retailers, and traders to study and quantify the impact of forward markets and vertical integration on prices, risk premia, and retail market shares. We point out that forward hedging and vertical integration are two separate mechanisms for demand and spot price risk diversification that both reduce the retail price and increase retail market shares. We show that they differ in their impact on prices and firms utility because of the asymmetry between production and retail segments. Vertical integration restores the symmetry between producers and retailers exposure to demand risk, whereas linear forward contracts do not. Vertical integration is superior to forward hedging when retailers are highly risk averse. We illustrate our analysis with data from the French electricity market
Aïd R. (2010), Long-Term Risk Management For Utility Companies: The Next Challenges, International Journal of Theoretical and Applied Finance, 13, 4, p. 517-535
Since the energy markets liberalization at the beginning of the 1990s in Europe, electricity monopolies have gone through a profound evolution process. From an industrial organization point of view, they lost their monopoly on their historical business, but gained the capacity to develop in any sector. Companies went public and had to upgrade their financial risk management process to international standards and implement modern risk management concepts and reporting processes (VaR, EaR...). Even though important evolutions have been accomplished, we argue here that the long-term risk management process of utility companies has not yet reached its full maturity and is still facing two main challenges. The first one concerns the time consistency of long-term and mid-term risk management processes. We show that consistencies issues are coming from the different classical financial parameters carrying information on firms' risk aversion (cost of capital and short-term risk limits) and the concepts inherited from the monopoly period, like the loss of load value, that are still involved in the utility company decision-making process. The second challenge concerns the need for quantitative models to assess their business model. With the deregulation, utilities have to address the question of their boundaries. Although intuition can provide insights on the benefits of some firm structures like vertical integration, only sound and tractable quantitative models can bring answers to the optimality of different possible firm structures.
Campi L., Aïd R., Touzi N., Nguyen Huu A. (2009), A Structural Risk-Neutral Model of Electricity Prices, International Journal of Theoretical and Applied Finance, 12, 7, p. 925-947
The objective of this paper is to present a model for electricity spot prices and the corresponding forward contracts, which relies on the underlying market of fuels, thus avoiding the electricity non- storability restriction. The structural aspect of our model comes from the fact that the electricity spot prices depend on the dynamics of the electricity demand at the maturity T, and on the random available capacity of each production means. Our model explains, in a stylized fact, how the prices of different fuels together with the demand combine to produce electricity prices. This modeling methodology allows one to transfer to electricity prices the risk-neutral probabilities of the market of fuels and under the hypothesis of independence between demand and outages on one hand, and prices of fuels on the other hand, it provides a regression-type relation between electricity forward prices and forward prices of fuels. Moreover, the model produces, by nature, the well-known peaks observed on electricity market data. In our model, spikes occur when the producer has to switch from one technology to the lowest cost available one. Numerical tests performed on a very crude approximation of the French electricity market using only two fuels (gas and oil) provide an illustration of the potential interest of this model.
Ben Tahar I., Aïd R. (2014), A mean Field Game approach to technological transition, PGMO - COPI'14, Paris-Saclay, France
We develop a model to assess the di usion of a new-technology among a population of poten- tial adopters. Our main objective is to analyze the e ect of the strategic interaction of the rms which supply this new technology in a context where production costs decline with cumulated production ( learning by doing e ect ), and where a rm's learning or experience bene ts its rivals ( learning spillover e ect ). To produce a tractable model in a dynamic setting, we adopt a mean- eld-game approach.
Chemla G., Porchet A., Touzi N., Aïd R. (2009), Forward Hedging and Vertical Integration in Electricity Markets, 36th Annual EFA Meeting, Bergen, Norvège
This paper analyzes the interactions between vertical integration and (wholesale) spot, forward and retail markets in risk management. We develop an equilibrium model that fits electricity markets well. We point out that vertical integration and forward hedging are two separate levers for demand and spot price risk diversification. We show that they are imperfect substitutes as to their impact on retail prices and agents' utility because the asymmetry between upstream and downstream segments. While agents always use the forward market, vertical integration may not arise. In addition, in presence of highly risk averse downstream agents, vertical integration may be a better way to diversify risk than spot, forward and retail mar kets. We illustrate our analysis with data from the French electricity market.
Aïd R., Campi L., Lautier D. (2015), A note on the spot-forward no-arbitrage relations in an investment-production model for commodities,, 14
Because of storability constraints, standard no-arbitrage arguments cannot be safely applied in markets of commodities such as energy. In this paper, we propose an alternative approach to justify the convergence of forward towards spot prices as time-to-maturity goes to zero. We show that the classical no-arbitrage relationship between spot and forward prices holds through the well-posedness of an expected profit maximization problem for an agent producing and storing a commodity while trading in forward contracts. A consequence of this is that the forward price of energy can be seen as risk-neutral expectation of the spot price at maturity. Moreover, we obtain an explicit formula for the forward volatility and provide a heuristic analysis of the optimal solution for the production/storage/investment problem in a Markovian setting.