Bilateral German-Slovakian Project

ENANEFA - Efficient Numerical Approximation of Nonlinear Equations in Financial Applications

financed by DAAD and the Slovakian Ministry of Education

(01/2018 - 12/2019)

Scientific goals

The project is focused on qualitative and numerical analyses of fully nonlinear partial differential equations (PDEs) of parabolic type arising in financial mathematics. The main purpose is to analyze non-linear extensions of the classical Black-Scholes theory for pricing financial instruments, free boundary problems for advection-diffusion equations arising in financial mathematics as well as models of stochastic dynamic portfolio optimization based on the Hamilton-Jacobi-Bellman (HJB) equation.

Main objective:

Analysis of qualitative properties of solutions of nonlinear parabolic partial differential equations arising in financial mathematics, their efficient numerical approximations and interpretation of the results in practice.

Project tasks:

Qualitative and numerical analysis of the HJB equation based on the direct approach and the Riccati transformation approach.
Analysis of PDEs arising in interest rate models and their calibration. Development of new analytic approximations for solution to multi-factor and multi-dimensional models.
Development and analysis of advanced stable numerical schemes for linear and nonlinear derivative pricing models.
Free boundary problems in derivative pricing models and their efficient numerical tracking using the penalty method adjusted to the early exercise free boundary profile.

Specification of objectives and secondary objectives:

The secondary objective is to derive results on existence of solutions and their sensitivity on parameters. Analysis of the properties of the value function of a non-linear programming problem representing the nonlinear diffusion function of the parabolic PDE. Development of new stable numerical solution schemes for quasi-linear parabolic PDEs with non-smooth coefficients and backward diffusion equations as well as multidimensional advection tensor diffusion equations. Calibration of nonlinear models to real data and sensitivity analysis of solutions. Interpretation of the results obtained in practice.

Due to the multidisciplinary composition of the team members and their experience, international cooperation and involvement in international research projects we consider proposed goals as realistic as far as the scientific and time perspectives are concerned.

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:

Christopher Berkhahn
Anna Clevenhaus
Matthias Ehrhardt (Leader)
Ali Ghafoor
Michael Günther
Lorenc Kapllani
Tatiana Kossaczká
Igor Kossaczký
Michelle Muniz
Long Teng
Sarah Treibert

Slovakian team:

Maros Bobulský
Ján Gašper
Zuzana Girová
Soňa Kilianová
Jan Komadel
Radka Litvajová
Igor Melichercik
Daniel Ševčovič (Leader)
Beata Stehliková
Magdaléna Zitnanská

German institutions:

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

Slovakian institutions:

Department of Applied Mathematics and Statistics, Faculty of Mathematics, Physics and Informatics, Comenius University Bratislava

Publications related to the Project

2018

Maros Bobulský, Properties of the value function for the parametric quadratic programming problems, Master Thesis, Comenius University Bratislava, 2018. (Supervisor: Daniel Ševčovič)
Anna Clevenhaus, ADI schemes for option pricing in the Heston model with stochastic correlation, Master Thesis, University of Wuppertal, 2018. (Supervisor: Long Teng)
Michelle Muniz, Untersuchung des Shadow Hamiltonians in der geometrischen Integration mit der Cayley-Abbildung, Master Thesis, University of Wuppertal, 2018. (Supervisor: Michael Günther)
Zuzana Girová, Effect of correlation between factors on bond prices in two-factor short rate models, Master Thesis, Comenius University Bratislava, 2018. (Supervisor: Beata Stehliková)
Christian Hendricks, Matthias Ehrhardt and Michael Günther, Hybrid finite difference / pseudospectral methods for the Heston and Heston-Hull-White PDE, J. Comput. Finance Vol. 21, No. 5 (April 2018), 1-33.
Lorenc Kapllani, On GPU computing for solving BSDEs in multi-step schemes, Master Thesis, University of Wuppertal, 2018. (Supervisor: Long Teng)
Soňa Kilianová, Daniel Ševčovič, Expected Utility Maximization and Conditional Value-at-Risk Deviation-based Sharpe Ratio in Dynamic Stochastic Portfolio Optimization, Kybernetika, accepted for publication, 2018.
Igor Kossaczký, The Tree-Grid Method, Dissertation, University of Wuppertal, September 2018. (Supervisor: Matthias Ehrhardt)
Igor Kossaczký, Matthias Ehrhardt and Michael Günther, A new convergent explicit Tree-Grid Method for HJB equations in one space dimension, Numer. Math. Theor. Meth. Appl. Vol. 11, No. 1, Feb. 2018.
Aleksandr Lapitckii, Multistep schemes for solving BSDEs, Master Thesis, University of Wuppertal, 2018. (Supervisor: Long Teng)
Radka Litvajová, Analysis of relations between time series using methods of network analysis and clustering, Master Thesis, Comenius University Bratislava, 2018. (Supervisor: Beata Stehliková)
Long Teng, Matthias Ehrhardt and Michael Günther, Quanto Pricing in Stochastic Correlation Models, Intl. J. of Theoretical & Applied Finance Vol. 21, No. 05, 1850038 (2018).
Long Teng, Matthias Ehrhardt and Michael Günther, Numerical Simulation of the Heston model under Stochastic Correlation, Int. J. Financial Stud. 2018, 6, 3; (open access)

2019

I. Kossaczký, M. Ehrhardt and M. Günther, The Two-dimensional Tree-Grid Method, J. Comput. Finance, Volume 23, Number 2 (September 2019), 29-57. DOI:10.21314/JCF.2019.373
L. 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, Proceedings of the HPC 2019 Conference, Borovets, Bulgaria.
Tatiana Kossaczká, The Weighted Essentially Non-Oscillatory Method for Problems in Finance, Master Thesis, University of Wuppertal, March 2019. (Supervisor: Matthias Ehrhardt, Barmenia Prize 2019 for outstanding Master Thesis)
- Price for Master Thesis, Barmenia-Mathematik-Preis, Stadthalle Wuppertal, November 9, 2019.
M. Ehrhardt, J. Gašper and S. Kilianová, Mathematical Modeling of an SIR-based infectious disease model with vaccination and waning immunity, Journal of Computational Science Volume 37, October 2019, 101027. DOI:10.1016/j.jocs.2019.101027.
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

Talks related to the Project

2018

Soňa Kilianová, Dynamic portfolio optimization via Hamilton-Jacobi-Bellman equations, Wuppertal, April 11, 2018.
Maros Bobulský, Filter matrices as source of uncertainty in robust portfolio optimization, Wuppertal, April 11, 2018.
Zuzana Girová, Effect of correlation between factors on bond prices in two-factor models, Wuppertal, April 11, 2018.
Anna Clevenhaus, Title, Bratislava, May 29, 2018.
Tatiana Kossaczká, Title, Bratislava, May 29, 2018.
Lorenc Kapllani, Title, Bratislava, May 29, 2018.
Igor Kossaczký, Title, Bratislava, May 29, 2018.
Matthias Ehrhardt, High-Order Methods for Parabolic Equations in Multiple Space Dimensions for Option Pricing Problems, FDM 2018 Conference, Lozenetz, June 14, 2018.
Soňa Kilianová, Dynamic portfolio optimization by means of Riccati transformation of corresponding HJB equation, Workshop on scientific computing 2018, Děčín, Czech Republic, June 14-17, 2018.
Matthias Ehrhardt, High-Order Methods for Parabolic Equations in Multiple Space Dimensions for Option Pricing Problems, ECMI 2018 Conference, Budapest, June 18, 2018.
Long Teng, Title, ECMI 2018 Conference, Budapest, June, 2018.
Sona Kilianová, Dynamic portfolio optimization via Hamilton-Jacobi-Bellman equation, Czech-Japanese Seminar in Applied Mathematics, Japan, Noto-cho, July 13-16, 2018.
Matthias Ehrhardt, High-Order Methods for Parabolic Equations in Multiple Space Dimensions for Option Pricing Problems, Sparse days 2018, CERFACS, Toulouse, September 27, 2018.

2019

Ján Gašper, SIR models with waning imunity, Wuppertal, February 5, 2019.
Soňa Kilianová, Maximizing terminal and intertemporal utility via HJB equation and its Riccati transformation, Wuppertal, February 5, 2019.
Daniel Ševčovič, On solutions to partial integro-differential equations in Bessel potential spaces, Wuppertal, February 5, 2019.
Lorenc Kapllani, Multistep Schemes for Solving Backward Stochastic Differential Equations on GPU, Wuppertal, February 5, 2019.
Anna Clevenhaus, ADI schemes for option pricing in the Heston model with stochastic correlation, Wuppertal, February 5, 2019.
Lorenc Kapllani, A Multistep Scheme to solve Backward Stochastic Differential Equations for Option Pricing on GPUs, Bratislava, June 26, 2019.
Michelle Muniz, Lie Group Methods for Matrix ODEs and SDEs, Bratislava, June 26, 2019.
Ján Gašper, SIR Models with Waning Immunity, Bratislava, June 26, 2019.
Anna Clevenhaus, Pricing American Options with a Non-constant Penalty Parameter, Bratislava, June 26, 2019.
Jakub Hrdina, Copositive Matrices, Bratislava, June 26, 2019.
Terézia Foluvá, Rank-Constrained Semidefinite Problems, Bratislava, June 26, 2019.
ICCF 2019 conference
German-Portuguese-Slovakian FRACTAL-ENANEFA Workshop, Wuppertal, Dec. 12, 2019
- Daniel Sevcovic , Comparison of numerical and analytical approximations of the early exercise boundary for the American put option
- Michelle Muniz, Using Lie group methods to approximate correlation matrices
- Anna Clevenhaus, Pricing American Options with a Penalty function
- Lorenc Kapllani, A Multistep Scheme to solve Backward Stochastic Differential Equations for Option Pricing on GPUs

Staff Exchange

2019

Soňa Kilianová, Lecture Numerics of ODEs and Lecture Advanced Topics Transformation Methods in Computational Finance, University of Wuppertal, Winter term 2019/20.