Applied and Computational Mathematics (ACM)

## 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

### 7063.

Ehrhardt, M.; Günther, M.
Numerik gewöhnlicher Differentialgleichungen : Anwendungen in Technik, Wirtschaft, Biologie und Gesellschaft
Herausgeber: Springer

### 7062.

Ehrhardt, M.; Günther, M.
Numerik gewöhnlicher Differentialgleichungen : Anwendungen in Technik, Wirtschaft, Biologie und Gesellschaft
Herausgeber: Springer
2024

### 7061.

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 :118304
2024
Herausgeber: Elsevier

### 7060.

Jacob, Birgit; Günther, Michael; Ehrhardt, Matthias
Analysis and Numerics of port-Hamiltonian systems Schedule (Start of Seminar: Oct 26, 2022)

### 7059.

Ehrhardt, Matthias; Günther, Michael
Numerical Evaluation of Complex Logarithms in the Cox-Ingersoll-Ross Model

### 7058.

Ehrhardt, M.; Günther, M.
Numerik gewöhnlicher Differentialgleichungen : Anwendungen in Technik, Wirtschaft, Biologie und Gesellschaft
Herausgeber: Springer

### 7057.

Acu, A.M.; Heilmann, Margareta; Raşa, I.
Voronovskaja type results for the Aldaz-Kounchev-Render versions of generalized Baskakov Operators
submitted

### 7056.

Ehrhardt, Matthias; Günther, Michael; Jacob, Birgit; Bartel, PD Dr Andreas; Maten, Jan
Modelling, Analysis and Simulation with Port-Hamiltonian Systems

### 7055.

Ehrhardt, M.; Günther, M.
Numerik gewöhnlicher Differentialgleichungen : Anwendungen in Technik, Wirtschaft, Biologie und Gesellschaft
Herausgeber: Springer

### 7054.

Ehrhardt, Matthias; Günther, Michael
Numerical Pricing of Game (Israeli) Options

### 7053.

Ehrhardt, Matthias; Günther, Michael; Jacob, Birgit; Bartel, PD Dr Andreas; Maten, Jan
Modelling, Analysis and Simulation with Port-Hamiltonian Systems

### 7052.

Maten, E Jan W; Ehrhardt, Matthias
MS40: Computational methods for finance and energy markets
19th European Conference on Mathematics for Industry, Seite 377

### 7051.

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 MACHINE

### 7050.

Günther, Michael; Wandelt, Dipl Math Mich{\`e}le
Numerical Analysis and Simulation I: ODEs

### 7049.

Maten, E Jan W; Ehrhardt, Matthias
MS40: Computational methods for finance and energy markets
19th European Conference on Mathematics for Industry, Seite 377

### 7048.

Kapllani, Lorenc; Teng, Long
{A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations}
2024

### 7047.

Ehrhardt, Matthias; Günther, Michael
Modelling Stochastic Correlations in Finance

### 7046.

Ambartsumyan, I; Khattatov, E; Yotov, I; Zunino, P; Arnold, Anton; Ehrhardt, Matthias; Ashyralyev, Allaberen; Csomós, Petra; Faragó, István; Fekete, Imre; others
Invited Papers

### 7045.

Al{\i}, G; Bartel, A; Günther, M
Electrical RLC networks and diodes

### 7044.

Gjonaj, Erion; Bahls, Christian Rüdiger; Bandlow, Bastian; Bartel, Andreas; Baumanns, Sascha; Belzen, F; Benderskaya, Galina; Benner, Peter; Beurden, MC; Blaszczyk, Andreas; others
Feldmann, Uwe, 143 Feng, Lihong, 515 De Gersem, Herbert, 341 Gim, Sebasti{\'a}n, 45, 333
MATHEMATICS IN INDUSTRY 14 :587

### 7043.

Ehrhardt, Matthias; Zheng, Chunxiong
für Angewandte Analysis und Stochastik

### 7042.

Ehrhardt, Matthias; Zheng, Chunxiong
für Angewandte Analysis und Stochastik

### 7041.

Ehrhardt, Matthias; Günther, Michael; Striebel, Michael
Geometric Numerical Integration Structure-Preserving Algorithms for Lattice QCD Simulations

### 7040.

Hendricks, C; Ehrhardt, M; Günther, M
High order tensor product interpolation in the Combination Technique
preprint, 14 :25

### 7039.

Hendricks, C; Ehrhardt, M; Günther, M
High order tensor product interpolation in the Combination Technique
preprint, 14 :25

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