Bilateral German-Slovakian Project
BraWu - Bratislava-Wuppertal research group on mathematical modelling in finance
(01/2023 - 12/2024)
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Scientific Gals
The aim of this project is to propose some generalizations/modifications of structural models
and numerical methods previously presented in the literature for the risk-neutral pricing of financial
instruments in emission markets. For the structural models proposed, our goals are to derive a
forward-backward stochastic differential equation (FBSDE)
for the price process of the emissions certificates and solve numerically the associated nonlinear
partial differential equations for the certificate price and for the prices of financial derivatives
in the emissions market.
The basic element of the structural model is an exogenously specified stochastic process modelling the
electricity demand and allowances certificates are considered as derivatives depending on the demand process and cumulative emissions. The generalizations/modifications
of the structural model which we will study are of the following type:
- Consider more general forms for the demand process, allowing for feedback effects between the
price and the demand process or considering that the drift depends on time explicitly;
- Consider different functional forms for the bid and emission stacks.
In order to solve the PDEs for the pricing problem we will propose and study appropriate efficient numerical methods.
Scientific Objectives
I. Structural Models and FBSDEs
Our first goal is to study the standard structural model for the risk neutral pricing of emission
certificates and propose modifications to this model considering the demand process of electricity,
which is the basic underlying stochastic process affecting the emissions certificate
and the functional forms of the bid and emissions stacks
(which model the cumulative emissions and the interaction between the electricity and emission markets).
In particular, we modify the basic structural model considering the following cases:
- Modify the demand process in order to consider feedback effects between the price
and the demand process and/or that the drift depends on time explicitly.
- Consider different concrete stochastic differential equations for the modelling
of the demand process (possibly including different diffusion processes or jump-diffusion processes)
- Consider different functional forms for the bid and the emissions stacks and
study what is the effect on the pricing of emission certificates.
For the modified structural models, we intend to derive the corresponding
forward-backward stochastic differential equation (FBSDE) that represents the price of the
emission certificates and study the problem of existence/uniqueness of solution for this equation
and explore the basic properties of the solution.
II. PDEs for Certificate Pricing and Derivatives Pricing
The risk-neutral price of the allowance certificate can be represented as the solution of a FBSDE.
By using Itô's stochastic calculus and Itô's formula, the price of the certificate can also be
represented as the solution of an appropriate nonlinear partial differential equation (PDE).
In the case of our structural modified models, we want to study two particular problems
related with these PDEs:
- Study the properties of the solution of the nonlinear PDE in some asymptotic cases (e.g. near expiry)
- Propose efficient numerical methods for solving the nonlinear PDE.
Examples of methods that can be tested are alternating direction finite difference schemes or semi-Lagrangian schemes,
among others.
- Study the problem of the pricing of financial derivatives and options written on
the allowance certificate by solving an appropriate PDE.
The industrial partner of this project, the GEFA bank, has
the main office located in Wuppertal and a branch in Bratislava.
This setting will offer the unique opportunity for mutual exchange
between academia and industry, transfer of knowledge and also with
the option for jointly supervised theses.
German team:
Slovakian team:
German institutions:
Slovakian institutions:
Publications related to the Project
2023
- J. Cruz, M. do Rosario Grossinho, D. Ševčovič, and C. Izuchukwu Udeani,
Linear and Nonlinear Partial Integro-Differential Equations arising from Finance,
In: J. Vasundhara Devi, Z. Drici, F. A. McRae (eds.),
Understanding Integro-Differential Equations, Understanding Integro-Differential Equations,
Nova Science Publishers, Inc., Hauppauge, 2023, pp. 191-256,
ISBN: 979-8-89113-040-1, DOI: 10.52305/NRLJ6556, arXiv 2207.11568.
- T. Kossaczká, M. Ehrhardt, and M. Günther,
Deep FDM: Enhanced finite difference methods by deep learning,
Franklin Open, Volume 4, 2023, 100039.
DOI: 10.1016/j.fraope.2023.100039.
- I. Melicherčík and G. Szücs,
Payout phase of defined contribution systems: The case of Slovakia,
Statistika, 103(4), (2023), 445-461.
DOI: 10.54694/stat.2023.12.
- M. Muniz, M. Ehrhardt, M. Günther, and R. Winkler,
Strong stochastic Runge-Kutta-Munthe-Kaas methods for nonlinear Itô SDEs on manifolds,
Applied Numerical Mathematics, Volume 193, November 2023, 196-203.
DOI: 10.1016/j.apnum.2023.07.024.
- D. Ševčovič and C. Izuchukwu Udeani,
Hamilton-Jacobi-Bellman Equation Arising from Optimal Portfolio Selection Problem,
preprint 2023. arXiv:2308.02627
2024
- A. Clevenhaus, C. Totzeck, and M. Ehrhardt,
A gradient based calibration method for the Heston model,
accepted: International Journal of Computer Mathematics, 2024.
- 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: ???
- T. Kossaczká, M. Ehrhardt, and M. Günther,
Deep finite difference method for solving Asian option pricing problems,
to appear in: ECMI 2023 Conference Proceedings,
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: ???
- D. Ševčovič, and C. Izuchukwu Udeani,
Learning the solution operator of HJB equation arising from portfolio management using deep learning,
to appear in: ECMI 2023 Conference Proceedings,
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: ???
2025
Talks related to the Project
2023
- Minisymposium "Novel methods in computational finance"
by Daniel Sevcovic (Bratislava) and Matthias Ehrhardt (Wuppertal)
at
ECMI 2023 - 22nd ECMI Conference on Industrial and Applied Mathematics,
Wrocław, Poland, June 26-30, 2023.
- 1st Slovak-German Seminar on Numerical and Applied Mathematics, Bratislava
Tuesday 19.9. at 11:30 room M125
- Michael Günther (Bergische Universität Wuppertal)
"On (stochastic) port-Hamiltonian systems: are there any financial applications?"
Wednesday 20.9. at 13:00, room M125
- Tatiana Kossaczká (Bergische Universität Wuppertal)
"WENO-DS: Deep smoothness WENO scheme for hyperbolic conservation laws"
- Neda Bagheri Renani (Comenius University Bratislava)
"A Comparison Study of ADI and ADE Methods of the Black-Scholes Equation on Option Pricing Models"
- Matthias Ehrhardt (Bergische Universität Wuppertal)
"The idea of Space Mapping in Finance and the first steps towards it"
Break
- Ján Gašper (Comenius University Bratislava)
"Introduction to fractional Laplacians"
- Jakub Hrdina (Comenius University Bratislava)
"Lagrangian duality in convex conic programming"
- Beáta Stehlíková (Comenius University Bratislava)
"Analytical approximations in short rate and shadow rate models"
- 2nd Slovak-German Seminar on Numerical and Applied Mathematics, Wuppertal
Wednesday 29.11. at 10:15, room G.13.18
- Neda Bagheri Renani (Comenius University Bratislava)
"A Comparison Study of ADI and ADE Methods of the Black-Scholes equation on option pricing Models"
- Matúš Padyšák (Comenius University Bratislava)
"Performance and guarantees in Slovak pension funds"
- Igor Melicheršík (Comenius University Bratislava)
"Pension Saving Strategies Based on Samuelson's Lifecycle Theory"
- Tatiana Kossaczká (Bergische Universität Wuppertal)
"Deep finite difference method for solving option pricing problems"
- Cyril Izuchukwu Udeani (Comenius University Bratislava)
"Approximating the solution operator of HJB equation using deep learning"
- Lorenc Kapllani (Bergische Universität Wuppertal)
"Differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations"
2024
- Minisymposium "Novel methods in computational finance"
by Daniel Sevcovic (Bratislava) and Matthias Ehrhardt (Wuppertal) (10 Speakers)
at ALGORITMY 2024 - conference , Podbanske, High Tatra Mountains,
Slovakia, March 20-15, 2024.
- Lorenc Kapllani (BU Wuppertal)
"Differential deep learning-based algorithm for solving high-dimensional
nonlinear backward stochastic differential equations"
- Maria Trnovska (CU Bratislava)
"Path-based data envelopment analysis models with direction defined using the anti-ideal point"
- Jakub Hrdina (CU Bratislava)
"Multiplier form of the path-based data envelopment analysis models and a
single-stage approach for finding an efficient target"
- Igor Melichercík (CU Bratislava)
"Parental bonus in pension systems"
- Matúš Padyšák (CU Bratislava)
"The role of volatility in guaranteed bond funds"
- Neda Bagheri Renani (CU Bratislava)
"A Comparison of Methods ADI and LOD in the Numerical solution of the Black-Scholes Equation in Option Pricing"
- Daniel Sevcovic (CU Bratislava)
"Maximal monotone operator technique for solving Hamilton-Jacobi-Bellman
equations arising in optimal portfolio selection problem"
- Cyril Izuchukwu Udeani (CU Bratislava)
"Learning the solution operator of Hamilton Jacobi Bellman equations using physics-informed DeepONets"
- Minisymposium "Novel methods in computational finance and energy markets"
by Rafal Weron (Wroclaw) and Matthias Ehrhardt (Wuppertal) (20 Speakers)
at
ICCF 2024 - 5th International Conference on Computational Finance,
CWI Amsterdam, The Netherlands, April 2-5, 2024.
- Thomas Kruse (BU Wuppertal) "Multilevel Picard iteration for high-dimensional semilinear parabolic PDEs"
- Long Teng (BU Wuppertal) "A regression-based approach to solve high-dimensional nonlinear pricing BSDEs"
- Anna Clevenhaus (BU Wuppertal) "A gradient-based calibration of the Heston model on real life data"
- Daniel Sevcovic (CU Bratislava) "Multidimensional linear and nonlinear partial
integro-differential equation in Bessel potential spaces with application in option pricing"
- Neda Bagheri (CU Bratislava, Slowakia) "A comparison study of ADI and ADE methods of the Black-Scholes equation on option pricing”
- Cyril Izuchukwu Udeani (CU Bratislava) "Approximating the solution operator of nonlinear parabolic equations arising from portfolio selection using deep learning"
Staff Exchange
2024/2024
Joint Supervision of Theses
2023/2024
Pre-Dissertation Theses
2023
Activities related to the Project
- Minisymposium "Novel methods in computational finance"
by Daniel Sevcovic (Bratislava) and Matthias Ehrhardt (Wuppertal)
at
ECMI 2023 - 22nd ECMI Conference on Industrial and Applied Mathematics,
Wroclaw, Poland, June 26-30, 2023.
- Minisymposium "Novel methods in computational finance and energy markets"
by Rafal Weron (Wroclaw) and Matthias Ehrhardt (Wuppertal)
at
ICCF 2024 - 5th International Conference on Computational Finance,
CWI Amsterdam, The Netherlands, April 2-5, 2024.
Former Projects
