Finance

The famous Black-Scholes equation is an effective model for option pricing. It was named after the pioneers Black, Scholes and Merton who suggested it 1973.
In this research field our aim is the development of effective numerical schemes for solving linear and nonlinear problems arising in the mathematical theory of derivative pricing models.
An option is the right (not the duty) to buy (`call option') or to sell (`put option') an asset (typically a stock or a parcel of shares of a company) for a price E by the expiry date T. European options can only be exercised at the expiration date T. For American options exercise is permitted at any time until the expiry date. The standard approach for the scalar Black-Scholes equation for European (American) options results after a standard transformation in a diffusion equation posed on an bounded (unbounded) domain.
Another problem arises when considering American options (most of the options on stocks are American style). Then one has to compute numerically the solution on a semi-unbounded domain with a free boundary. Usually finite differences or finite elements are used to discretize the equation and artificial boundary conditions are introduced in order to confine the computational domain.
In this research field we want to design and analyze new efficient and robust numerical methods for the solution of highly nonlinear option pricing problems. Doing so, we have to solve adequately the problem of unbounded spatial domains by introducing artificial boundary conditions and show how to incorporate them in a high-order time splitting method.
Nonlinear Black-Scholes equations have been increasingly attracting interest over the last two decades, since they provide more accurate values than the classical linear model by taking into account more realistic assumptions, such as transaction costs, risks from an unprotected portfolio, large investor's preferences or illiquid markets, which may have an impact on the stock price, the volatility, the drift and the option price itself.
Special Interests
Publications
- 2024
5250.
Vinod, Vivin; Zaspel, Peter
Assessing Non-Nested Configurations of Multifidelity Machine Learning for Quantum-Chemical Properties
Machine Learning: Science and Technology, 5 (4) :045005
20245249.
B. Jacob, C. Totzeck
Port-Hamiltonian Structure of Interacting Particle Systems and Its Mean-Field Limit
SIAM Multiscale Modelling & Simulation, 22
20245248.
Günther, M.; Jacob, Birgit; Totzeck, Claudia
Data-driven adjoint-based calibration of port-Hamiltonian systems in time domain
Math. Control Signals Syst.
20245247.
Schaefers, Kevin; Peardon, Michael
A modified Cayley transform for SU(3)
20245246.
Petrov, Pavel S.; Ehrhardt, Matthias; Kozitskiy, Sergey B.
A generalization of the split-step Padé method to the case of coupled acoustic modes equation in a 3D waveguide
Journal of Sound and Vibration, 577 :118304
2024
Herausgeber: Academic Press5245.
Petrov, Pavel S.; Ehrhardt, Matthias; Kozitskiy, Sergey B.
A generalization of the split-step Padé method to the case of coupled acoustic modes equation in a 3D waveguide
Journal of Sound and Vibration, 577 :118304
2024
Herausgeber: Academic Press5244.
Clevenhaus, Anna; Totzeck, Claudia; Ehrhardt, Matthias
A gradient-based calibration method for the Heston model
International Journal of Computer Mathematics, 101 (9-10) :1094–1112
2024
Herausgeber: Taylor & Francis5243.
Clevenhaus, Anna; Totzeck, Claudia; Ehrhardt, Matthias
A gradient-based calibration method for the Heston model
International Journal of Computer Mathematics, 101 (9-10) :1094–1112
2024
Herausgeber: Taylor & Francis5242.
Clevenhaus, Anna; Totzeck, Claudia; Ehrhardt, Matthias
A gradient-based calibration method for the Heston model
International Journal of Computer Mathematics
20245241.
Schäfers, Kevin; Peardon, Michael; Günther, Michael
A modified Cayley transform for SU (3)
Preprint
20245240.
Schäfers, Kevin; Peardon, Michael; Günther, Michael
A modified Cayley transform for SU (3)
Preprint
20245239.
Ehrhardt, Matthias
A nonstandard finite difference scheme for a time-fractional model of Zika virus transmission
Mathematical Biosciences and Engineering, 21 (1) :924–962
2024
Herausgeber: AIMS Press5238.
Gaul, Daniela; Klamroth, Kathrin; Pfeiffer, Christian; Stiglmayr, Michael; Schulz, Arne
A Tight Formulation for the Dial-a-Ride Problem
European Journal of Operational Research
September 2024
Herausgeber: Elsevier BV
ISSN: 0377-22175237.
Ehrhardt, Matthias
A nonstandard finite difference scheme for a time-fractional model of Zika virus transmission
Mathematical Biosciences and Engineering, 21 (1) :924–962
2024
Herausgeber: AIMS Press5236.
5235.
Clevenhaus, Anna; Totzeck, Claudia; Ehrhardt, Matthias
A numerical study of the impact of variance boundary conditions for the Heston model
In Burnecki, K. and Szwabiński, J. and Teuerle, M., Editor
Springer
In Burnecki, K. and Szwabiński, J. and Teuerle, M., Editor
Herausgeber: Bergische Universität Wuppertal
20245234.
Clevenhaus, Anna; Totzeck, Claudia; Ehrhardt, Matthias
A numerical study of the impact of variance boundary conditions for the Heston model
In Burnecki, K. and Szwabiński, J. and Teuerle, M., Editor
Springer
In Burnecki, K. and Szwabiński, J. and Teuerle, M., Editor
Herausgeber: Bergische Universität Wuppertal
20245233.
Clemens, Markus; Henkel, Marvin-Lucas; Kasolis, Fotios; Günther, Michael
A Port-Hamiltonian System Perspective on Electromagneto-Quasistatic Field Formulations of Darwin-Type
Preprint
20245232.
Clemens, Markus; Henkel, Marvin-Lucas; Kasolis, Fotios; Günther, Michael
A Port-Hamiltonian System Perspective on Electromagneto-Quasistatic Field Formulations of Darwin-Type
Preprint
20245231.
Hoang, Manh Tuan; Ehrhardt, Matthias
A second-order nonstandard finite difference method for a general Rosenzweig-MacArthur predator--prey model
Journal of Computational and Applied Mathematics :115752
2024
Herausgeber: Elsevier5230.
Dächert, Kerstin; Fleuren, Tino; Klamroth, Kathrin
A simple, efficient and versatile objective space algorithm for multiobjective integer programming
Mathematical Methods of Operations Research
20245229.
Kapllani, Lorenc; Teng, Long
{A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations}
20245228.
M., Günther; Jacob, B.; Totzeck, C.
Data-driven adjoint-based calibration of port-Hamiltonian systems in time domain
Math. Control Signals Syst., 36 :957–977
20245227.
Zaspel, Peter; Günther, Michael
Data-driven identification of port-Hamiltonian DAE systems by Gaussian processes
Preprint
20245226.
Allmendinger, Richard; Fonseca, Carlos M.; Sayin, Serpil; Wiecek, Margaret M.; Stiglmayr, Michael
Multiobjective Optimization on a Budget (Dagstuhl Seminar 23361)
2024
Herausgeber: Schloss Dagstuhl – Leibniz-Zentrum für Informatik