My name is Erwan Chartier, and I am a Ph.D. student in the Applied and Computational Mathematics research group at the University of Wuppertal. Under the supervision of Matthias Ehrhardt, my research is conducted within the framework of the bilateral German-Spanish AEI-DFG project “APMANA-EAREC: Advanced Pricing Models and Numerical Approaches for Emission Allowances and Renewable Energy Certificates“
🌍 The Challenge of Carbon Pricing in Modern Energy Systems
Reducing greenhouse gas emissions is one of the major challenges facing modern energy systems. One way to encourage companies to reduce their emissions is to put a price on carbon. Under an emissions trading scheme, regulated companies must surrender emission allowances (EAs) corresponding to their verified emissions. Since these allowances can be traded, their price reflects the economic value of the right to emit greenhouse gases.
The European Union launched its Emissions Trading System (EU ETS) in 2005, and similar systems have since been introduced in countries such as China and South Korea. As climate policies have become more ambitious, carbon prices have gained significant importance in electricity markets:
- 📈 Surging Prices: In the European Union, the price of an allowance rose from around €5 per tonne of CO₂ in 2017 to more than €100 in February 2023.
- ⚡ Market Shifts: Such changes can affect the order in which power plants generate electricity and reduce the competitiveness of carbon-intensive technologies, particularly coal-fired power plants.
🧮 Fully Coupled Forward-Backward Stochastic Differential Equations
Predicting the evolution of allowance prices is important for electricity producers, investors, and regulators. However, these prices cannot always be studied independently of the energy market. Electricity producers choose between different fuels according to their costs, and the price of carbon influences these decisions. The resulting production mix determines the amount of emissions generated, which in turn affects the value of emission allowances. Understanding this feedback requires mathematical models that combine the evolution of market variables with the economic value of carbon.
One approach is to describe these interactions using forward-backward stochastic differential equations (FBSDEs):
- The Forward Equation: Models the evolution of quantities such as electricity prices, fuel prices, and cumulative emissions.
- The Backward Equation: Describes a value process, which can represent the price of an allowance under an appropriate valuation framework. At the end of the compliance horizon, the model must satisfy a terminal condition reflecting assumptions about the emissions cap and non-compliance consequences. Risk-neutral models for emission allowance prices and options were studied by Carmona and Hinz [1], while Carmona and co-authors [2, 3] investigated the mathematical structure of emissions trading models, including singular FBSDEs.
The core difficulty is that the forward and backward equations depend on each other: the allowance price influences producers’ decisions, while those decisions affect cumulative emissions and hence the allowance value. This reciprocal dependence creates a fully coupled system that becomes increasingly challenging to solve numerically as more fuels and market factors are introduced.
💻 Numerical Challenges and Advanced Approaches
A classical way to study coupled FBSDEs is the four-step scheme developed by Ma, Protter, and Yong [4], which links the stochastic system to a quasilinear parabolic partial differential equation (PDE) that can be approximated using finite differences. While well-established in low dimensions, their computational cost grows rapidly due to the curse of dimensionality.
Alternative methodologies help address these high-dimensional hurdles:
- Markovian Iteration: Studied by Bender and Zhang [5], this method iteratively constructs decoupled problems approximated via conditional-expectation estimates and Monte Carlo simulation.
- Neural Networks: In the deep BSDE method proposed by Han, Jentzen, and E [6], neural networks approximate unknown quantities within a stochastic numerical scheme, avoiding full state-space grids. Similarly, Physics-Informed Neural Networks (PINNs) — introduced by Raissi, Perdikaris, and Karniadakis [7] — incorporate governing differential equations directly into the training objective alongside boundary or terminal conditions.
🔬 Current Research & Outlook
Applying these advanced tools to a fully coupled emissions-market model raises a central question: can the discontinuous terminal condition be enforced accurately? This is crucial because errors in the allowance price can distort estimated production decisions and emissions.
In our current work, we investigate a PINN-based approach for the numerical solution of a coupled FBSDE arising in an emissions market with several fuels, enriched with Fourier features to help mitigate spectral bias. The network is trained using the residual of the associated PDE, the terminal condition, and a data loss obtained from Markovian iteration. Our numerical experiments focus on the accuracy of the allowance price and the stability of the approximation with respect to the time horizon and coupling strength.
The longer-term objective is to extend the model to more realistic market settings, potentially incorporating additional fuels, alternative compliance mechanisms, or calibration to observed market data, providing a solid foundation for valuing options on emission allowances in complex energy markets.
References
[1] Carmona, R. and Hinz, J., “Risk-neutral models for emission allowance prices and option valuation”, Manag. Sci. 57(8) (2011), 1453-1468.
[2] Carmona, R., Fehr, M., Hinz, J. and Porchet, A., “Market design for emission trading schemes”, SIAM Rev. 52(3) (2010), 403-452.
[3] Carmona, R., Delarue, F., Espinosa, G.-E. and Touzi, N., “Singular forward-backward stochastic differential equations and emissions derivatives”, Ann. Appl. Probab. 23(3) (2013), 1086-1128.
[4] Ma, J., Protter, P. and Yong, J., “Solving forward-backward stochastic differential equations explicitly: a four step scheme”, Probab. Theory Related Fields 98 (1994), 339-359.
[5] Bender, C. and Zhang, J., “Time discretization and Markovian iteration for coupled FBSDEs”, Ann. Appl. Probab. 18(1) (2008), 143-177.
[6] Han, J., Jentzen, A. and E, W., “Solving high-dimensional partial differential equations using deep learning”, Proc. Natl. Acad. Sci. USA 115(34) (2018), 8505-8510.
[7] Raissi, M., Perdikaris, P. and Karniadakis, G. E., “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations”, J. Comput. Phys. 378 (2019), 686-707.