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Über dieses Buch

In Mathematical Finance, the authors consider a mathematical model for the pricing of emissions permits. The model has particular applicability to the European Union Emissions Trading System (EU ETS) but could also be used to consider the modeling of other cap-and-trade schemes. As a response to the risk of Climate Change, carbon markets are currently being implemented in regions worldwide and already represent more than $30 billion. However, scientific, and particularly mathematical, studies of these carbon markets are needed in order to expose their advantages and shortcomings, as well as allow their most efficient implementation.

This Brief reviews mathematical properties such as the existence and uniqueness of solutions for the pricing problem, stability of solutions and their behavior. These fit into the theory of fully coupled forward-backward stochastic differential equations (FBSDEs) with irregular coefficients. The authors present a numerical algorithm to compute the solution to these non-standard FBSDEs. They also carry out a case study of the UK energy market. This involves estimating the parameters to be used in the model using historical data and then solving a pricing problem using the aforementioned numerical algorithm.

The Brief is of interest to researchers in stochastic processes and their applications, and environmental and energy economics. Most sections are also accessible to practitioners in the energy sector and climate change policy-makers.



Chapter 1. A Description of the Carbon Markets and Their Role in Climate Change Mitigation

The chapter describes the latest evolutions of international agreements on climate change. The “externality” that emissions of greenhouse gases produce, i.e. the fact that their effect on future generations is not included in the price of economic activities that generate them, can be dealt with through different policies. The Chapter therefore exposes the economic theory of carbon markets as an efficient policy to address this problem. The chapter concludes with an explanation of the way that the European Union has implemented a carbon market since 2005 and how the policy has evolved since. This market is the subject of the subsequent chapters.
Jean-François Chassagneux, Hinesh Chotai, Mirabelle Muûls

Chapter 2. Introduction to Forward-Backward Stochastic Differential Equations

This Chapter is a self-contained introduction to Forward-Backward Stochastic Differential Equations (FBSDEs). It reviews first the well-posedness of BSDEs in the Lipschitz setting and their link with a class of non-linear Partial Differential Equation (PDE) in the Markovian case. It presents then recent results on the theory of coupled FBSDEs in the setting of Lipschitz coefficients and non-degenerate noise. The Chapter is concluded by a discussion on the case of equations with irregular coefficients and degenerate noise, which are used to model the carbon allowance price in the subsequent chapters.
Jean-François Chassagneux, Hinesh Chotai, Mirabelle Muûls

Chapter 3. A Mathematical Model for Carbon Emissions Markets

We consider a model in which electricity producers operate in a market that is connected to a carbon emissions market. The link between the allowance market and the electricity market is established through the market bid stack, and other related notions. Then, we derive a forward backward stochastic differential equation (FBSDE) to price an allowance in the case of a single trading period. Finally, we extend these ideas to derive the systems of FBSDEs appropriate for pricing an allowance in the case of multiple trading periods.
Jean-François Chassagneux, Hinesh Chotai, Mirabelle Muûls

Chapter 4. Numerical Approximation of FBSDEs

This chapter gives an overview of some numerical schemes that can be applied to forward-backward stochastic differential equations (FBSDEs). First, a backward version of the Euler-Maruyama discretisation, that is applicable to decoupled FBSDE, is presented. Then, we present a Markovian iteration scheme that allows for the discretisation and numerical resolution of fully coupled FBSDE. We also present a regression based method that allows one to draw samples from the conditional expectation of a random variable, allowing one to turn the aforementioned discretisations into full numerical schemes. Finally, we review on numerical examples the convergence property of the Markovian iteration scheme.
Jean-François Chassagneux, Hinesh Chotai, Mirabelle Muûls

Chapter 5. A Case Study of the UK Energy Market

In this chapter, we carry out a numerical investigation of the model introduced in Chap. 3. The processes and functions appearing in the pricing FBSDE will be chosen so that they model the features of the UK energy market. Their parameters will be estimated using real data. Following this, the pricing FBSDE will be solved numerically, along with a regularized version of the pricing FBSDE. We finally interpret the numerical results.
Jean-François Chassagneux, Hinesh Chotai, Mirabelle Muûls


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