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
5225.
Vinod, Vivin; Zaspel, Peter
Investigating Data Hierarchies in Multifidelity Machine Learning for Excitation Energies
20245224.
Botchev, M. A.; Knizhnerman, L. A.; Schweitzer, M.
Krylov subspace residual and restarting for certain second order differential equations
SIAM J. Sci. Comput., 46 (2) :S223-S253
20245223.
Hastir, Anthony; Jacob, Birgit; Zwart, Hans
Linear-Quadratic optimal control for boundary controlled networks of waves
20245222.
Xu, Zhuo; Tucsnak, Marius
LQR control for a system describing the interaction between a floating solid and the surrounding fluid
Mathematical Control and Related Fields, 14(4) :1477-1500
Dezember 20245221.
Costa, G Morais Rodrigues; Ehrhardt, Matthias
Mathematical analysis and a nonstandard scheme for a model of the immune response against COVID-19
Band 793
Seite 251–270
Herausgeber: AMS Contemporary Mathematics
2024
251–2705220.
Costa, G Morais Rodrigues; Ehrhardt, Matthias
Mathematical analysis and a nonstandard scheme for a model of the immune response against COVID-19
Band 793
Seite 251–270
Herausgeber: AMS Contemporary Mathematics
2024
251–2705219.
Bolten, Matthias; Kilmer, Misha E.; MacLachlan, Scott
Multigrid preconditioning for regularized least-squares problems
SIAM J. Sci. Comput., 46 (5) :s271—s295
2024
ISSN: 1064-82755218.
Schultes, Johanna
Multiobjective optimization of shapes using scalarization techniques
Dissertation
Dissertation
Bergische Universität Wuppertal
20245217.
Bolten, Matthias; Doganay, Onur Tanil; Gottschalk, Hanno; Klamroth, Kathrin
Non-convex shape optimization by dissipative {H}amiltonian flows
Engineering Optimization
20245216.
Zaspel, Peter; Günther, Michael
Data-driven identification of port-Hamiltonian DAE systems by Gaussian processes
Preprint
20245215.
Bolten, M.; Doganay, O. T.; Gottschalk, H.; Klamroth, K.
Non-convex shape optimization by dissipative Hamiltonian flows
Eng. Optim. :1—20
20245214.
Bauß, Julius
On improvements of multi-objective branch and bound
Dissertation
Dissertation
Bergische Universität Wuppertal
20245213.
Abel, Ulrich; Acu, Ana Maria; Heilmann, Margareta; Raşa, Ioan
On some Cauchy problems and positive linear operators
Mediterranean Journal of Mathematics, accepted
20245212.
Lorenz, Jan; Zwerschke, Tom; Schaefers, Kevin
Operator splitting for coupled linear port-Hamiltonian systems
20245211.
Kruse, Thomas; Strack, Philipp
Optimal dynamic control of an epidemic
Operations Research, 72 (3) :1031–1048
2024
Herausgeber: INFORMS5210.
Kruse, Thomas; Strack, Philipp
Optimal dynamic control of an epidemic
Operations Research, 72 (3) :1031–1048
2024
Herausgeber: INFORMS5209.
Vinod, Vivin; Kleinekathöfer, Ulrich; Zaspel, Peter
Optimized multifidelity machine learning for quantum chemistry
Mach. Learn.: Sci. Technol., 5 (1) :015054
20245208.
Frommer, Andreas; Ramirez-Hidalgo, Gustavo; Schweitzer, Marcel; Tsolakis, Manuel
Polynomial preconditioning for the action of the matrix square root and inverse square root
Electron. Trans. Numer. Anal., 60 :381-404
20245207.
Jamil, Hamza
Intrusive and non-intrusive uncertainty quantification methodologies for pyrolysis modeling
Fire Safety Journal, 143 :104060
2024
ISSN: 0379-71125206.
Hosfeld, René; Jacob, Birgit; Schwenninger, Felix; Tucsnak, Marius
Input-to-state stability for bilinear feedback systems
SIAM Journal on Control and Optimization, 62 (3) :1369-1389
20245205.
Schäfers, Kevin; Finkenrath, Jacob; Günther, Michael; Knechtli, Francesco
Hessian-free force-gradient integrators
20245204.
Santos, Daniela Scherer; Klamroth, Kathrin; Martins, Pedro; Paquete, Luís
Ensuring connectedness for the Maximum Quasi-clique and Densest $k$-subgraph problems
20245203.
Zaspel, Peter; Günther, Michael
Data-driven identification of port-Hamiltonian DAE systems by Gaussian processes.
20245202.
Kapllani, Lorenc; Teng, Long
Deep learning algorithms for solving high-dimensional nonlinear backward stochastic differential equations
Discrete and continuous dynamical systems - B, 29 (4) :1695–1729
2024
Herausgeber: AIMS Press5201.
Ackermann, Julia; Jentzen, Arnulf; Kuckuck, Benno; Padgett, Joshua Lee
Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for space-time solutions of semilinear partial differential equations
arXiv:2406.10876 :64 pages
2024