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E-raamat: Forward-Backward SDEs Approach to Pricing in Carbon Markets

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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.
1 A Description of the Carbon Markets and Their Role in Climate Change Mitigation
1(10)
1.1 Why Do We Need Emissions Trading Markets?
1(1)
1.1.1 The Science of Climate Change
1(1)
1.2 Policy Developments and the Paris Agreement
2(1)
1.3 Economic Principles Underlying Emissions Trading as a Policy Tool
3(3)
1.3.1 Tax Versus Market
3(2)
1.3.2 Trading Choices
5(1)
1.4 The European Union Emissions Trading System
6(2)
1.4.1 Phases and Caps
6(1)
1.4.2 EUA Price Evolutions
7(1)
1.5 Carbon Pricing and the Future
8(3)
References
9(2)
2 Introduction to Forward-Backward Stochastic Differential Equations
11(32)
2.1 Backward Stochastic Differential Equations
12(12)
2.1.1 Well-Posedness of BSDEs
13(1)
2.1.1.1 Linear BSDEs
14(1)
2.1.1.2 The Comparison Theorem
15(1)
2.1.2 Application to Non-linear Pricing
16(1)
2.1.2.1 Super-Replication in a Perfect Market
17(1)
2.1.2.2 A Non-linear Market
18(1)
2.1.3 Applications to Stochastic Control
19(2)
2.1.4 Extensions
21(1)
2.1.4.1 Constrained BSDEs
21(1)
2.1.4.2 The Non-lipschitz Setting
22(1)
2.1.4.3 McKean--Vlasov FBSDEs
23(1)
2.2 Markovian BSDEs
24(4)
2.2.1 First Definition and Markov Property
24(1)
2.2.2 The Link with PDEs
25(3)
2.3 Coupled Forward-Backward SDEs
28(15)
2.3.1 The Pontryagin Approach to Stochastic Control Problems
29(2)
2.3.2 Well-Posedness of FBSDEs in Small Time Duration
31(1)
2.3.2.1 Existence and Uniqueness
32(2)
2.3.2.2 The Decoupling Field and a Quasilinear PDE
34(3)
2.3.3 Existence and Uniqueness for Arbitrary Terminal Time
37(1)
2.3.3.1 Non-degenerate Diffusion Coefficient
38(1)
2.3.3.2 FBSDEs with Singular Coefficients
38(2)
References
40(3)
3 A Mathematical Model for Carbon Emissions Markets
43(16)
3.1 Introduction
43(1)
3.2 Market Set-Up
43(2)
3.3 The Bid Stack, Emissions Stack and Emissions Rate
45(6)
3.4 Risk-Neutral Dynamics of Random Factors
51(1)
3.5 The Single-Period Allowance Pricing FBSDE
52(1)
3.6 Extension to a Multi-period Emissions Trading System
53(6)
References
56(3)
4 Numerical Approximation of FBSDEs
59(16)
4.1 Decoupled FBSDEs
59(5)
4.2 A Markovian Iteration Scheme for Fully Coupled FBSDEs
64(2)
4.3 Computation of Conditional Expectations Using Regression
66(3)
4.4 Numerical Examples of the Scheme's Convergence
69(6)
4.4.1 Bender and Zhang Test Model
69(4)
4.4.2 Numerical Investigation of a Simple Singular FBSDE 71 References
73(2)
5 A Case Study of the UK Energy Market
75(28)
5.1 Introduction: An Explicit Model
75(1)
5.2 Specifying the Dynamics of the Market Factors
76(13)
5.2.1 Estimating the Seasonal Component
77(7)
5.2.2 Fitting the Diffusion Processes
84(5)
5.3 Estimating the Bid Stack and Emissions Rate
89(3)
5.4 Simulation Methodology
92(3)
5.5 Results
95(8)
References
100(3)
Index 103
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