Bilateral German-Portuguese Project
PRISEMA - Pricing of Financial Instruments in Emission Markets
(05/2024 - 04/2026)
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Scientific goals
The aim of this project is to develop new models and methods for the pricing of carbon emission allowances and renewable energy certificates.
We will focus on stochastic structural models based on the underlying economic factors that determine the price of carbon credits.
Our goal is to derive a forward-backward stochastic differential equation (FBSDE) for the emission allowance pricing process
and numerically solve the associated partial differential equations (PDEs).
We will extend the basic model for risk-neutral pricing of carbon emission certificates by considering more
general processes for the electricity demand process and for the processes modeling cumulative emissions
and the interaction between the electricity and emissions markets.
In particular, we will consider the following extensions:
- In the demand process, we will introduce feedback effects between the price process and demand,
explicit time dependence for the drift function, seasonality, and truncation effects;
- in the stochastic differential equations modeling the demand process, we will introduce appropriate
nonlinear terms and jump diffusion processes;
- we will also consider multiple stochastic fuel prices.
In the generalized models, we will first derive the corresponding FBSDE, which is the risk-neutral price of
carbon emission certificates.
Then, we investigate the problem of well-posedness and existence/uniqueness of solutions for this FBSDE.
In the proposed modified models, we will generalize the PDE approach of Howison and Schwarz.
In particular, we will study the following problems:
- derive properties of the solutions of the pricing PDE in particular asymptotic cases (e.g. near maturity);
- develop efficient numerical methods for solving the pricing PDE, testing the methods based on semi-Lagrangian schemes,
Lagrange-Galerkin methods, and alternating direction (ADI) finite difference schemes, among others;
- obtain pricing PDEs for financial derivatives and options depending on the emission certificates, and
testing appropriate numerical methods for solving these PDEs;
- extend the methodology to the pricing of renewable energy certificates (RECs).
We believe that this project will make significant contributions to the emerging field of environmental (green)
finance. The new models and methods developed will be of interest to researchers/practitioners in the field.
Strategic and Educational Aims
We will further strengthen the academic relations between the two working groups,
but also between the two universities. Right in this context we will establish
a new german-portuguese subgroup in our
ECMI special interest group on computational finance
and an active student and staff exchange in the framework of the
ERASMUS programme.
Here we plan jointly supervised theses and compact courses offered by the staff of the
partner group with the vision of a truly European education and qualification on a high academic level,
also with the emphasis to train fellows to work in multi-cultural research teams.
German team:
Portuguese team:
German institutions:
Portuguese institutions:
Publications related to the Project
2022
2023
- J.F.M. D'Agua,
Modelling electricity and emission markets,
Master Thesis, ISEG - University of Lisbon, October, 2023.
- Jose Cruz, M. R. Grossinho, D. Sevcovic, C. Izuchukwu Udeani,
Linear and Nonlinear Partial Integro-Differential Equations arising from Finance,
In: Understanding Integro-Differential Equations, Eds: J. Vasundhara Devi, Z. Drici, F. A. McRae,
Nova Science Publishers, Inc., Hauppauge, pp. 191-256 (2023).
- H. Guerreiro and J. Guerra,
VIX pricing in the rBergomi model under a regime switching change of measure,
Quantitative Finance 23(5) (2023), 721-738.
DOI: 10.1080/14697688.2023.2178321.
- R. L. Gudmundarson, M. Guerra and A. B. de Moura,
On some effects of dependencies on an insurer's risk exposure, probability of ruin, and optimal premium loading,
European Actuarial Journal, 13(1) (2023), 341-373.
DOI: 10.1007/s13385-022-00326-0.
- R. Sousa, M. Guerra and S. Yakubovich,
Product formulas and convolutions for Laplace-Beltrami operators on product spaces: beyond the trivial case,
Mathematische Annalen, 387:1-2 (2023), 879-926.
DOI: 10.1007/s00208-022-02424-6.
- V. Tavares,
Pricing Renewable Energy Certificates,
Master Thesis, ISEG - University of Lisbon, October, 2023.
2024
- R. Bernardino, J. Janela, C. Martins, C. Mourão, L. S. Pinto and F. G. Rodrigues,
A Multi-Trip Vehicle Routing Problem With Release Dates and Interrelated Periods,
Networks 0, 1-16 (2024).
DOI: 10.1002/net.22258.
- A. Clevenhaus, C. Totzeck and M. Ehrhardt,
A gradient based calibration method for the Heston model,
International Journal of Computer Mathematics Vol. 101, Number 9-10 (2024), 1094-1112.
DOI: 10.1080/00207160.2024.2353189.
- A. Clevenhaus, C. Totzeck and M. Ehrhardt,
A numerical study of the impact of variance boundary conditions for the Heston model,
in K. Burnecki, J. Szwabiński, and M. Teuerle (eds.)
Progress in Industrial Mathematics at ECMI 2023,
The European Consortium for Mathematics in Industry, Springer, 2024, pp ???.
DOI: ???
- H. Guerreiro and J. Guerra,
Pseudo rough vol-of-vol through Markovian approximation,
REM Working Paper 0310-2024 (2024).
DOI: 10.2139/ssrn.4729788.
- H. Guerreiro,
Option Pricing in Generalized Rough Bergomi Models,
PhD in Applied Mathematics for Economics and Management,
Dissertation, ISEG, University of Lisbon, May 2024. (co-supervised by João Guerra and Maria do Rosário Grossinho)
- D. Jabs,
Mathematische Modellierung von Systemen in der Kreislaufwirtschaft
(Mathematical modeling of systems in the circular economy),
Bachelor Thesis, University of Wuppertal, June 2024.
- T. Kossaczká, M. Ehrhardt and M. Günther,
Deep finite difference method for solving Asian option pricing problems,
in K. Burnecki, J. Szwabiński, and M. Teuerle (eds.)
Progress in Industrial Mathematics at ECMI 2023,
The European Consortium for Mathematics in Industry, Springer, 2024, pp ???.
DOI: ???
- F. Tomé,
Models for spread option pricing in energy markets,
Master Thesis, ISEG, Universidade de Lisboa, October 2024.
2025
- A. Clevenhaus, C. Totzeck and M. Ehrhardt,
A Space Mapping approach for the calibration of financial models with the application to the Heston model,
Preprint 25/02, January 2025.
- A. Clevenhaus,
Hierarchical Approaches in Modeling and Numerical Simulation for Mathematical Finance,
Dissertation, University of Wuppertal, July 2025. (co-supervised by Matthias Ehrhardt and Michael Günther)
- M. Falcão,
Jump-diffusion modeling in emission markets through an integro-differential equation,
Master Thesis, ISEG, Universidade de Lisboa, September 2025.
- R. Henriques,
Monte Carlo VaR Under Reinvestment Risk,
Master Thesis, ISEG, Universidade de Lisboa, June 2025.
- A. B. Santos,
A PDE approach to bilateral counterparty risks and funding costs,
Master Thesis, ISEG, Universidade de Lisboa, June 2025.
2026
- Daniel Cezar Salgado, Risk-neutral pricing of European Emission Allowance derivatives,
Master Thesis, University of Wuppertal, scheduled December 2026.
References related to the Project
Data
Talks related to the Project
2024
- 1st Lisbon-Wuppertal PRISEMA-Workshop, Lisbon, May 22, 2024
- Beatriz Santos, Introduction to the CO2-Equivalent adjustment and some applications
- Carolina Alves, Standard model for market emissions certificates and financial derivatives
- Benjamin Leonhardt, A Multilevel Monte Carlo Method for Uncertainty Quantification in Pricing Financial Products
- Joao Guerra, Least squares Monte Carlo methods in stochastic Volterra rough volatility models
Felix Schlüter, Time Series Analysis and Forecasting of Company Fundamentals- Matthias Ehrhardt The space mapping approach for the Heston model
- Phuong Nguyen, Optimal reinsurance for minimum probability of Parisian ruin
- Tatiana Kossaczká, Deep finite difference method for solving option pricing problems
- Lorenc Kapllani, A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations
- Joint Bratislava-Lisbon-Wuppertal BraWu-PRISEMA-Workshop
on Computational Finance and Energy Markets, Wuppertal
November 12, 2024, Room D.13.06
- Long Teng (Bergische Universität Wuppertal)
A differential deep learning-based method for solving high-dimensional nonlinear BSDEs
- João Guerra (ISEG Lisbon)
Forward Backward Stochastic Differential Equations in Emission markets with demand and cap regime changes
- Neda Bagheri Renani (Comenius University Bratislava)
The Comparison of the Improved Interior-Newton-Smart test method and the Interior-Point Method on Large Scale
Models
- Discussion
Break
- Julia Ackermann (Bergische Universität Wuppertal)
Optimal trade execution with self-exciting price impact
- Phuong Nguyen (ISEG Lisbon)
Optimal reinsurance for minimum probability of Parisian ruin with exponential grace period
- Igor Melichercik (Comenius University Bratislava)
Child - related pension benefits: The case of Slovakia
- Discussion
2025
- 3rd Lisbon-Wuppertal PRISEMA-Workshop, Lisbon, June 24, 2025
- 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
Break
- 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
- Discussion
- September 2-5, 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)
(SCF 25),
September 2-5, 2025, ISEG, Lisbon, Portugal.
- 4th A Coruña-Lisbon-Wuppertal PRISEMA-Workshop, Wuppertal, March 11, 2026
- 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
