Bergische Universität Wuppertal Fachbereich Mathematik und Naturwissenschaften Applied and Computational Mathematics (ACM)

Team Research Publications Teaching

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State of the Art in CO₂ Emission Allowances and Renewable Energy Certificates

The project focuses on the mathematical analysis and numerical solution of models related to energy markets, in particular CO₂ Emission Allowances (EA) and Renewable Energy Certificates (RECs). Mathematical models for the pricing of these certificates are formulated using forward backward stochastic differential equations (FBSDEs) and nonlinear partial differential equations (PDEs). The main objective is to establish the existence and uniqueness of solutions to these models and to develop efficient numerical methods for their solution. The project also aims to extend existing models by incorporating additional factors such as jump-diffusion models for electricity price and stochastic fuel price. In addition, the project will investigate the pricing of derivatives on RECs, including European-style options, American-style options, and exotic options.

Scientific Objectives

The scientific objectives of the project include the development of more realistic mathematical models for the pricing of financial products in emissions markets, the proof of the existence and uniqueness of solutions to these models, the development of efficient numerical methods to solve the models, the implementation and validation of the numerical methods, and the application of the models to real cases. The models are formulated in terms of FBSDEs and PDEs, and the numerical methods include Monte-Carlo simulation and semi-Lagrangian techniques. The project aims to provide a rigorous mathematical framework and efficient computational tools for the analysis and pricing of financial products in emissions markets.

German team:

• Matthias Ehrhardt (Leader)
• Michael Günther
• Thomas Kruse
• Long Teng
• Claudia Totzeck
• Erwan Chartier (October 2025 - )

Spanish team:

• Carlos Vázquez Cendón (Leader)
• Inigo Arregui Álvarez
• María del Carmen Calvo-Garrido
• José Germán López-Salas
• María A. Baamonde-Seoane
• Pablo Pérez Picos (April 2025 - )

Associated Portuguese team from ISEG Lisbon:

• João Guerra (Leader)
• Gonçalo Fonseca

Collaborators:

• Jean-Francois Chassagneux (LPSM - Univ. Paris Cité)
• Emmanuel Gobet (CMAP - Ecole Polytechnique)
• Chris Kenyon (Global Head of Quant Innovation at MUFG Securities)
• Rüdiger Kiesel (Univ. Duisburg-Essen)

German institutions:

• Department of Mathematics and Informatics, Faculty of Mathematics and Natural Sciences, University of Wuppertal.

Spanish institutions:

• University of A Coruña, Faculty of Computer Science

Work Packages (Leaders)

• WP1: Pricing Models Statement (Ehrhardt-Vázquez, M01-M12)
• WP2: Analysis of the Models (Kruse-Vázquez, M06-M18)
• WP3: Numerical methods for FBSDE formulations (Kruse-Vázquez, M12-M24)
• WP4: Numerical methods for PDE and PIDE formulations (Ehrhardt-Arregui, M06-M24)
• WP5: Applications to real cases in emission markets (Ehrhardt-Calvo, M25-M36)

Publications related to the Project

Preparatory Work

• M.A. Baamonde-Seoane and C. Vázquez, Pricing renewable energy certificates with a Crank-Nicolson Lagrange-Galerkin numerical method, J. Comput. Appl. Math. 422 (2023), 114891. DOI: 10.1016/j.cam.2022.114891.
• M.A. Baamonde-Seoane, M. Calvo-Garrido, and C. Vázquez, Model and numerical methods for pricing renewable energy certificate derivatives, Commun. Nonlin. Sci. Numer. Simul. 118 (2023), 107066. DOI: 10.1016/j.cnsns.2022.107066.
• M.A. Baamonde-Seoane, M. Calvo-Garrido, M. Coulon, and C. Vázquez, Numerical solution of a nonlinear PDE model for pricing Renewable Energy Certificates (RECs), Appl. Math. Comput. 404 (2021), 126199. DOI: /10.1016/j.amc.2021.126199.
• M.C. Calvo-Garrido, M. Ehrhardt, and C. Vázquez, Jump-diffusion models with two stochastic factors for pricing swing options in electricity markets with partial-integro differential equations, Appl. Numer. Math. 139 (2019), 77-92. DOI:10.1016/j.apnum.2019.01.001.
• M.C. Calvo-Garrido, M. Ehrhardt, and C. Vázquez, Pricing swing options in electricity markets with two stochastic factors using a partial differential equation approach, J. Comput. Finance 20(3) (2017), 81-107. DOI: 10.21314/JCF.2016.317.
• A. Clevenhaus, C. Totzeck and M. Ehrhardt, A gradient based calibration method for the Heston model, Int. J. Comput. Math. (2024), 1-19. DOI: 10.1080/00207160.2024.2353189.
• C. Hendricks, M. Ehrhardt, and M. Günther, Hybrid finite difference / pseudospectral methods for the Heston and Heston-Hull-White PDE, J. Comput. Finance 21(5) (2018), 1-33. DOI: 10.21314/JCF.2018.342.
• M. Hutzenthaler, T. Kruse, and T. Anh Nguyen, On the speed of convergence of Picard iterations of backward stochastic differential equations, Probab., Uncert. Quant. Risk 7(2) (2022), 133-150. DOI: 10.3934/puqr.2022009.
• T. Kossaczká, M. Ehrhardt, and M. Günther, Deep FDM: Enhanced finite difference methods by deep learning, Franklin Open 4 (2023), 100039. DOI: 10.1016/j.fraope.2023.100039.
• A. Poggi, L. Di Persio, and M. Ehrhardt, Electricity Price Forecasting via Statistical and Deep Learning Approaches: the German case, AppliedMath 2023, 3(2), 316-342. DOI: 0.3390/appliedmath3020018.

• Daniel Cezar Salgado, Title, Master Thesis, University of Wuppertal, December 2025.

References related to the Project

Data

• Statistical Review of World Energy (2024|73rd edition), energy institute.
• Data and statistics, International Energy Agency (IEA).

Talks related to the Project

2025

• March 19-21, Wuppertal, Kick-Off Discussions, Carlos Vázquez.
  
• June 23-27, Wuppertal-Lisbon Meeting in Lisbon, Workshop on June 24, 2025 (PRISEMA)
  • Matthias Ehrhardt, Port-Hamiltonian Systems in Quantitative Finance
  • Julia Ackermann, Optimal trade execution with stochastic cross-effects
  • Karel Nana, Detecting asset price bubbles using artificial neural networks
  • Daniel Cezar, Liquidity and counterparty credit risk in energy trading
  • João Guerra, VIX pricing in the rBergomi model under a stochastic change of measure
  • Gonçalo Fonseca, Forward-Backward SDEs with jumps and emissions modelling
  • Miguel Falcão, Jump-diffusion modelling in emissions markets through an integro-differential equation
  • Phuong Nguyen, New approach to minimize probability of Parisian ruin with reinsurance
• September 2-5, Conference Stochastics & Computational Finance 2025 (SCF 25), September 2-5, 2025, ISEG, Lisbon, Portugal.
  • Plenary Talk: Carlos Vázquez, Some recent advances in the pricing of renewable energy certificates and some derivatives
  • Plenary Talk: Matthias Ehrhardt, A novel Space Mapping approach for the calibration of financial models
  • Minisymposium Advanced Pricing Models and Numerical Approaches for Emission Allowances and Renewable Energy Certificates, with Jean-François Chassagneux (Paris) and Carlos Vázquez Cendon (A Coruña)
    • Stéphane Crépey, Sensitivity Analysis of Emissions Markets: A Discrete-Time Radner Equilibrium Approach
    • Liam Welsh, Modelling Equilibrium Behaviour in Solar REC and Offset Credit Markets
    • Yassin El-Hatri, Modelling the Role of Hedgers and Speculators
• October 9-10, Talk: M. Ehrhardt, Mathematical Modeling and Numerical Methods for Pricing Emission and Renewable Energy Certificates in Energy Markets, at 1st TEC2ZERO Symposium for the new Activity-Field Technology Towards Zero-Waste and Zero-Carbon at the University of Wuppertal.

• March 9-13, A Coruña-Lisbon-Wuppertal Meeting in Wuppertal, Workshop on March 11, 2026 (PRISEMA)
  • Pablo Pérez Picos (University of A Coruna), Pricing of Renewable Energy Certificates (RECs) and their derivatives
  • João Guerra (ISEG, University of Lisbon), Optimal investment in Lévy markets
  • Gonçalo Fonseca (ISEG, University of Lisbon), Forward-Backward SDEs with jumps: a numerical approach to the problem
  • Karel Nana (University of Wuppertal), Deep-Learning-Based Methods for calibrating the G-SVJD Model
  
• March 24-27, Climate, Energy and ESG Workshop 2026, University of Duisburg-Essen (Rüdiger Kiesel)

Staff Exchange

2026

• March-May 2026, Pablo Pérez Picos (Research stay in Wuppertal)
  

Outreach

• Emissionshandel: Mathematisch betrachtet (engl. Emissions trading: from a mathematical perspective), Press release DFG Jahresbericht 2024, pages 55-57.
• Press release Meeting March 2025, in Wuppertal (CITIC research center, spanish)
• Press release Meeting March 2025, in Wuppertal (University of A Coruña, Galician)
• Press release German-Spanish research project Innovative Mathematical Models for Emissions Trading, University of Wuppertal, Oct 9, 2024 (German Version).