This paper is concerned with the mathematical analysis of emissions markets. We review the existing quantitative analyses on the subject and introduce some of the mathematical challenges posed by the implementation of the new phase of the European Union Emissions Trading Scheme as well as the cap-and-trade schemes touted by the U.S., Canada, Australia, and Japan. From a practical point of view, the main thrust of the paper is the design and numerical analysis of new cap-and-trade schemes for the control and reduction of atmospheric pollution. We develop tools intended to help policy makers and regulators understand the pros and cons of the emissions markets. We propose a model for an economy where risk neutral firms produce goods to satisfy an inelastic demand and are endowed with permits in order to offset their pollution at compliance time and avoid having to pay a penalty. Firms that can easily reduce emissions do so, while those for which it is harder buy permits from firms that anticipate they will not need all their permits, creating a financial market for pollution credits. Our equilibrium model elucidates the joint price formation for goods and pollution allowances, capturing most of the features of the first phase of the European Union Emissions Trading Scheme. We show existence of an equilibrium and uniqueness of emissions credit prices. We also characterize the equilibrium prices of goods and the optimal production and trading strategies of the firms. We use the electricity market in Texas to numerically illustrate the qualitative properties of these cap-and-trade schemes. Comparing the numerical implications of cap-and-trade schemes to the business-as-usual benchmark, we show that our numerical results match those observed during the implementation of the first phase of the European Union cap-and-trade CO 2 emissions scheme. In particular, we confirm the presence of windfall profits criticized by the opponents of these markets. We also demonstrate the shortcomings of tax and subsidy alternatives. Finally we introduce a relative allocation scheme which, while easy to implement, leads to smaller windfall profits than the standard scheme.

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The existence of mandatory emission trading schemes in Europe and the United States, and the increased liquidity of trading on futures contracts on CO 2 emissions allowances, led naturally to the next step in the development of these markets: These futures contracts are now used as underliers for a vibrant derivative market. In this paper, we give a rigorous analysis of a simple risk-neutral reduced-form model for allowance futures prices, demonstrate its calibration to historical data, and show how to price European call options written on these contracts. This paper was accepted by Haitao Li, guest editor, finance.emission derivatives, emissions markets, cap-and-trade schemes, environmental finance

We address a method of approximate calculation of optimal control policy applicable to a particular class of stochastic control problems, whose stochastic dynamics exhibit a certain convexity preserving property. Problems of this type appear in many applications and encompass important examples arising in the area of optimal stopping and in the framework of control, based on partial observations. Utilizing this specific structure, we suggest a numerical method which enjoys a number of desirable properties. In particular, we work out a remarkably strong method for calculation of the value function: Within our numerically well-tractable approach, we show a convergence to the value function of the original problem uniformly on compact sets. This issue can be of great advantage, particularly for high-dimensional control problem, where the only competing methods from lest-squares Monte-Carlo family are able to serve merely L p-convergence, under several restrictions. Since the presented algorithm is simple, stable and the procedures are dimensionindependent, the author hopes that it can help solving high dimensional control problems when other methods reach their computational limits.

We address a method for pricing electricity contracts based on the valuation of the ability to produce power, which is considered as the true underlying factor for electricity derivatives. This approach shows that an evaluation of free production capacity provides a framework where a change-of-numeraire transformation converts the electricity forward market into the common settings for money market modelling. Using the toolkit of interest rate theory, we derive explicit option pricing formulas.