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

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

Associated Portuguese team from ISEG Lisbon:

• João Guerra (Leader)
• Gonçalo Fonseca

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

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.

Talks related to the Project

2025