by Olivier Davidau, Mireille Bossy, Nadia Maïzi and Odile Pourtallier

The carbon market was launched in the European Union in 2005 as part of the EU's initiative to reduce its greenhouse gas (GHG) emissions. This means that industrial players must now include their GHG emissions in their production costs. We aim to model the behaviour of an industrial agent who faces market uncertainties, and we compare the expected value of an optimal production strategy when buying (or not buying) supplementary emission allowances. This allows us to compute the agent's carbon indifference price which defines its market behaviour (buyer or not). We aim to use the resulting indifference price to study the sensitivity of the carbon market to the shape of the penalty and the emission allowances allocation.

The global warming phenomenon and the related increase in GHG emissions are subjects of great concern in the international community. Developing international cooperation on GHG emissions reduction has led to the ratification of the Kyoto Protocol by most countries. For the European Union, it means an 8% reduction in its GHG emissions in comparison with 1990 levels, to be attained during the period 2008-2012. Such an objective is closely linked to carbon value determination over different time scales.

Determining the value of carbon is therefore an important economic issue and has led to a variety of research activities. It is at the heart of the prospective-based research effort underway at the Center for Applied Mathematics (CMA), at Mines ParisTech. In collaboration with the French Environment and Energy Management Agency (ADEME) and CMA, specialists in stochastic modelling and numerics from the TOSCA and COPRIN teams at INRIA are working on a short-term carbon value derived from the so-called financial 'carbon market', the European Union Emission Trading Scheme (EU ETS).

The EU ETS is a framework for GHG emissions reduction in European industry. It covers specific industrial sectors including energy production, cement, iron and steel, and oil refineries. In February, the beginning of the annual trading period, each production unit receives a certain quantity of emissions allowances. By April, the unit must possess at least the same level of allowances as its emissions during the previous year. Meanwhile, the producer is allowed to sell or buy allowances on the carbon market. If allowances do not cover emissions, the producer must pay a penalty. Such a market is called a compliance or 'cap and trade' market.

We aim to evaluate the consequences of the design of the allowance allocations and penalties on industries' behaviour and GHG emissions reduction. Our approach is based on 'indifference price' evaluation methodology, used to price complex Over The Counter (OTC) financial contracts. We propose to study the EU ETS indifference price according to a given sector's players aggregation.

We model an agent (the producer) taking part in the EU ETS and seeking to optimize its production planning and CO2 allowances trading strategy for one period of the carbon market. The agent faces uncertainties in product sales and in production costs (energy, commodities etc); these are modelled using stochastic processes. The agent then chooses an adequate policy for production, investment and carbon trading in order to maximize the expected utility, a function of its final wealth which represents its preferences. For example, an electricity producer can dynamically switch between coal, gas or hydro power plants, and/or buy/sell emission allowances. To determine whether or not the agent enters the carbon market, we compare the optimal expected utility when buying or selling extra emission allowances at a given price, with the optimal expected utility without a carbon market. The indifference price is the price at which the two utilities become equal. A comparison of the current carbon price with the indifference price shows whether it is more or less financially advantageous for the producer to cover its GHG emissions. That is, when the market (buying) price is higher than the (buying) indifference price, the producer will cover its emissions; otherwise it prefers to pay the penalty. This indifference price gives us information on the investment level the producer will deploy to reduce its GHG emissions.

Mathematically, this optimization problem is a stochastic control problem, where the state variables are accumulated wealth and CO2 emissions, revenue from product sales and production costs. The optimal expected utility is the solution of a Hamilton-Jacobi-Bellman equation. Computing this price and studying its sensitivity to the shape of the allowances allocation and penalty leads us to solve this equation and eventually to invert the resulting value function. We then compute the derivatives of this solution with respect to the level of penalty/allocation. From a mathematical point of view, this leads us to developing efficient numerical schemes. An approach based on backward stochastic differential equations may provide alternative representations, qualitative information on regularity, and derivatives of the solutions and numerical tools.


Please contact:
Olivier Davidau
INRIA and Mines ParisTech, France

Next issue: July 2023
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