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
4862.
Jacob, Birgit; Glück, Jochen; Meyer, Annika; Wyss, Christian; Zwart, Hans
Stability via closure relations with applications to dissipative and port-Hamiltonian systems
J. Evol. Equ., 24 :Paper No. 62
20244861.
Ackermann, Julia; Ehrhardt, Matthias; Kruse, Thomas; Tordeux, Antoine
Stabilisation of stochastic single-file dynamics using port-Hamiltonian systems
Preprint
20244860.
Ehrhardt, Matthias; Günther, Michael; Striebel, Michael
Geometric Numerical Integration Structure-Preserving Algorithms for Lattice QCD Simulations- 2024
4859.
Kapllani, Lorenc; Teng, Long
{A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations}
20244858.
Hendricks, C; Ehrhardt, M; Günther, M
High order tensor product interpolation in the Combination Technique
preprint, 14 :254857.
Calvo-Garrido, MC; Ehrhardt, M; Vázquez, C
PDE modeling and numerical methods for swing option pricing in electricity markets
19th European Conference on Mathematics for Industry, Seite 3904856.
Günther, Michael; Wandelt, Dipl Math Mich{\`e}le
Numerical Analysis and Simulation I: ODEs4855.
Ehrhardt, Matthias; Günther, Michael
Numerical Evaluation of Complex Logarithms in the Cox-Ingersoll-Ross Model4854.
Ehrhardt, Matthias; Günther, Michael
Numerical Pricing of Game (Israeli) Options4853.
Ehrhardt, Matthias; Günther, Michael
Numerical Pricing of Game (Israeli) Options4852.
Ehrhardt, Matthias; Farkas, Bálint; Günther, Michael; Jacob, Birgit; Bartel, PD Dr Andreas
Operator Splitting and Multirate Schemes4851.
Ehrhardt, Matthias; Farkas, B{\'a}lint; Günther, Michael; Jacob, Birgit; Bartel, PD Dr Andreas
Operator Splitting and Multirate Schemes4850.
Calvo-Garrido, MC; Ehrhardt, M; V{\'a}zquez, C
PDE modeling and numerical methods for swing option pricing in electricity markets
19th European Conference on Mathematics for Industry, Seite 3904849.
Ehrhardt, Matthias
Positive Schemes for Air Pollution Problems, Optimal Location of Industrial Enterprises and Optimization of their Emissions4848.
Putek, Piotr; PAPLICKI, Piotr; Pulch, Roland; Maten, Jan; Günther, Michael; PA{\L}KA, Ryszard
NONLINEAR MULTIOBJECTIVE TOPOLOGY OPTIMIZATION AND MULTIPHYSICS ANALYSIS OF A PERMANENT-MAGNET EXCITED SYNCHRONOUS MACHINE4847.
Ehrhardt, Matthias
Positive Schemes for Air Pollution Problems, Optimal Location of Industrial Enterprises and Optimization of their Emissions4846.
Carmen Calvo-Garrido, Mar{\i}a; Ehrhardt, Matthias; V{\'a}zquez, Carlos
Pricing swing options in electricity markets with two stochastic factors: PIDE modeling and numerical solution
3rd International Conference on Computational Finance (ICCF2019), Seite 894845.
Carmen Calvo-Garrido, Mar{\i}a; Ehrhardt, Matthias; Vázquez, Carlos
Pricing swing options in electricity markets with two stochastic factors: PIDE modeling and numerical solution
3rd International Conference on Computational Finance (ICCF2019), Seite 894844.
Putek, PA; Ter Maten, EJW; Günther, M
Reliability-based Low Torque Ripple Design of Permanent Magnet Machine4843.
Günther, M; Ehrhardt, M; Knechtli, F; Shcherbakov, D; Striebel, M; Wandelt, M
Symmetric \& Volume Preserving Projection Schemes4842.
Putek, Piotr; Günther, Michael
Topology Optimization and Analysis of a PM synchronous Machine for Electrical Automobiles4841.
Ehrhardt, Matthias; Günther, Michael; Brunner, H; Dalhoff, A
Vorhersage-Modelle am Beispiel des Corona-Virus COVID-194840.
Ehrhardt, Matthias; Günther, Michael; Brunner, H; Dalhoff, A
Vorhersage-Modelle am Beispiel des Corona-Virus COVID-194839.
Acu, A.M.; Heilmann, Margareta; Raşa, I.
Voronovskaja type results for the Aldaz-Kounchev-Render versions of generalized Baskakov Operators
submitted4838.
Maten, E Jan W; Ehrhardt, Matthias
MS40: Computational methods for finance and energy markets
19th European Conference on Mathematics for Industry, Seite 377