Bilateral German-Portuguese Project
FRACTAL - FRActional models and CompuTationAL Finance
(01/2019 - 12/2020)
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Scientific goals
The aim of this project is to investigate analytically and numerically boundary value problems
of fractional partial differential equations (fPDEs).
Our focus is on possible real world applications in close cooperation with local industry partners.
Suggested by the recent literature for the application of fPDEs in finance
(e.g. in option pricing models based on Lévy processes or fractional Brownian motion),
we are concerned with:
- analytical study of fPDEs (existence and uniqueness results; properties of the solution
- numerical methods for fPDEs (investigate algorithmic realisations and numerical results)
- application to financial option pricing (including the rigorous derivation of the fPDE associated to
the fractional Black-Scholes model, Lévy processes, and clarification of absence of arbitrage in
the resulting model.)
The project will reinforce the academic relations between both teams,
exploring significant complementarities in cooperative research work,
with special emphasis on qualification
of young academics in an international environment.
Main objective:
Analysis of qualitative properties of solutions of fractional
parabolic partial differential equations arising in financial mathematics,
their efficient numerical approximations and interpretation of the results in practice.
Project tasks:
- ANALYTICAL STUDY OF fPDEs
- NUMERICAL METHODS FOR fPDEs
- APPLICATIONS TO FINANCIAL OPTION PRICING
Due to the multidisciplinary composition of the team members
(statistics, numerics and economics) and their interdisciplinary experience,
international cooperation and involvement in international research projects,
we consider the proposed goals as realistic as far as the scientific and time perspectives are concerned.
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
2019
- Igor Kossaczký, Matthias Ehrhardt and Michael Günther,
The Two-dimensional Tree-Grid Method,
J. Comput. Finance Volume 23, Number 2 (September 2019), 29-57. DOI:10.21314/JCF.2019.373
- Lorenc Kapllani,
Multistep Schemes for Solving Backward Stochastic Differential Equations on GPU,
Master Thesis, University of Wuppertal, March 2019. (Supervisors: Long Teng, Matthias Ehrhardt)
- L. Kapllani, L. Teng and M. Ehrhardt,
A Multistep Scheme to solve Backward Stochastic Differential Equations for Option
Pricing on GPUs,
in: I. Dimov and S. Fidanova (eds.),
Advances in High Performance Computing,
Results of the International Conference on "High Performance Computing" Borovets, Bulgaria, 2019,
Studies in Computational Intelligence 902, Springer, 2021, pp. 196-208.
DOI: 10.1007/978-3-030-55347-0
- Tatiana Kossaczká,
The Weighted Essentially Non-Oscillatory Method for Problems in Finance,
Master Thesis, University of Wuppertal, March 2019. (Supervisor: Matthias Ehrhardt)
- Price for Master Thesis, Barmenia-Mathematik-Preis, Stadthalle Wuppertal, November 9, 2019.
2020
- L. Teng, X. Wu, M. Günther and M. Ehrhardt,
A new methodology to create valid time-dependent correlation matrices via isospectral flows,
ESAIM: Mathematical Modelling and Numerical Analysis (M2AN), Volume 54, Number 2,
(March-April 2020), 361-371.
DOI:10.1051/m2an/2019064
- M. Muniz, M. Ehrhardt and M. Günther,
Approximating correlation matrices using stochastic Lie group methods,
Mathematics 2021, 9(1), 94.
DOI: 10.3390/math9010094
- A. Clevenhaus, M. Ehrhardt, M. Günther and D. Sevcovic,
Pricing American Options with a Non-constant Penalty Parameter,
J. Risk Financial Manag. 2020, 13(6), 124. DOI 10.3390/jrfm13060124
- A. Clevenhaus, M. Ehrhardt and M. Günther,
An ADI Sparse Grid method for pricing efficiently American Options under the Heston model,
Preprint 20/45, October 2020.
- M. Muniz, M. Ehrhardt, M. Günther and R. Winkler,
Higher Strong Order Methods for Itô SDEs on Matrix Lie Groups,
Preprint 21/03, January 2021.
- Beatriz Leal, cotutelle thesis.
Talks related to the Project
2019
- 1st Lisbon-Wuppertal FRACTAL Workshop, Lisbon, Oct. 9, 2019
- Michelle Muniz,
Using Lie group methods to approximate correlation matrices
- Anna Clevenhaus,
Pricing American Options with a Penalty function
- João Guerra,
Some problems and research topics in fractional processes, fractional equations and finance
- Francisco Fonseca,
Fractional Diffusion Models for Option Pricing
- German-Portuguese-Slovakian FRACTAL-ENANEFA Workshop, Wuppertal, Dec. 12, 2019
- Michelle Muniz,
Using Lie group methods to approximate correlation matrices
- Anna Clevenhaus,
Pricing American Options with a Penalty function
- João Guerra,
Some problems and research topics in fractional processes, fractional equations and finance
- Francisco Fonseca,
Fractional Diffusion Models for Option Pricing
2020
Minisymposium "Novel methods in computational finance" by Matthias Ehrhardt and Jan ter Maten
at ECMI 2020 conference, Limerick, Ireland, June 21-26, 2020.
(ECMI 2020 has been postponed due to Covid-19. Provisional rescheduled dates are June 21-25, 2021).- Minisymposium "Novel methods in computational finance" by Daniel Sevcovic (Bratislava) and Matthias Ehrhardt (Wuppertal)
at ALGORITMY 2020 - Conference on Scientific Computing, Vysoke Tatry, Podbanske, Slovakia,
September 10-15, 2020.
German-Portuguese FRACTAL Workshop, Lisbon, September 23, 2020
(postponed to 2021)
2021
2020
