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
4762.
Vinod, Vivin; Zaspel, Peter
QeMFi: A Multifidelity Dataset of Quantum Chemical Properties of Diverse Molecules
20244761.
Ackermann, Julia; Kruse, Thomas; Urusov, Mikhail
Reducing Obizhaeva-Wang-type trade execution problems to LQ stochastic control problems
Finance and Stochastics, 28 (3) :813–863
2024
Herausgeber: Springer Verlag4760.
Erbay, Mehmet; Jacob, Birgit; Morris, Kirsten
On the Weierstraß form of infinite dimensional differential algebraic equations
20244759.
Ackermann, Julia; Kruse, Thomas; Urusov, Mikhail
Self-exciting price impact via negative resilience in stochastic order books
Annals of Operations Research, 336 (1) :637–659
2024
Herausgeber: Springer Netherlands4758.
Andersen, Kim Allan; Boomsma, Trine Krogh; Efkes, Britta; Forget, Nicolas
Sensitivity Analysis of the Cost Coefficients in Multiobjective Integer Linear Optimization
Management Science
20244757.
Bartel, Andreas; Schaller, Manuel
Goal-oriented time adaptivity for port-{H}amiltonian systems
20244756.
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 Informatik4755.
Costa, G Morais Rodrigues; Lobosco, Marcelo; Ehrhardt, Matthias; Reis, Ruy Freitas
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
251–2704754.
Hastir, Anthony; Jacob, Birgit; Zwart, Hans
Linear-Quadratic optimal control for boundary controlled networks of waves
20244753.
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
20244752.
Song, Yongcun; Wang, Ziqi; Zuazua, Enrique
FedADMM-InSa: An Inexact and Self-Adaptive ADMM for Federated Learning
20244751.
Vinod, Vivin; Zaspel, Peter
Investigating Data Hierarchies in Multifidelity Machine Learning for Excitation Energies
20244750.
Jamil, Hamza
Intrusive and non-intrusive uncertainty quantification methodologies for pyrolysis modeling
Fire Safety Journal, 143 :104060
2024
ISSN: 0379-71124749.
Bolten, Matthias; Doganay, Onur Tanil; Gottschalk, Hanno; Klamroth, Kathrin
Non-convex shape optimization by dissipative {H}amiltonian flows
Engineering Optimization
20244748.
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
20244747.
Erbay, Mehmet; Jacob, Birgit; Morris, Kirsten; Reis, Timo; Tischendorf, Caren
Index concepts for linear differential-algebraic equations in finite and infinite dimensions
20244746.
Schäfers, Kevin; Finkenrath, Jacob; Günther, Michael; Knechtli, Francesco
Hessian-free force-gradient integrators
20244745.
Bartel, Andreas; Schaller, Manuel
Goal-oriented time adaptivity for port-Hamiltonian systems
Preprint
2024- 2023
4744.
Haussmann, N.; Stroka, S.; Mazaheri, S.; Clemens, M.
Using Point Clouds for Material Properties Smoothing in Low-Frequency Numerical Dosimetry Simulations
21st Biennial IEEE Conference on Electromagnetic Field Computation (CEFC 2024)
Jeju, South Korea
Dezember 20234743.
Kähne, B.; Clemens, M.
A GPU Accelerated Semi-Implicit Method for Large-Scale Nonlinear Eddy-Current Problems Using Adaptive Time Step Control
21st Biennial IEEE Conference on Electromagnetic Field Computation (CEFC 2024)
Jeju, South Korea
Dezember 20234742.
Gernandt, Hannes; Hinsen, Dorothea; Cherifi, Karim
The difference between port-Hamiltonian, passive and positive real descriptor systems
Mathematics of Control, Signals, and Systems
Dezember 20234741.
Abel, Ulrich; Acu, Ana Maria; Heilmann, Margareta; Raşa, Ioan
Voronovskaja formula for Aldaz–Kounchev–Render operators: uniform convergence
Analysis and Mathematical Physics, 14 (1)
Dezember 2023
ISSN: 1664-235X4740.
Stroka, S.; Kasolis, F.; Haussmann, N.; Clemens, M.
Efficient Low-Frequency Human Exposure Assessment with the Maximum Entropy Snapshot Sampling
21st Biennial IEEE Conference on Electromagnetic Field Computation (CEFC 2024)
Jeju, Korea
November 20234739.
Xuan, Mingjun; Fan, Jilin; Khiêm, Vu Ngoc; Zou, Miancheng; Brenske, Kai-Oliver; Mourran, Ahmed; Vinokur, Rostislav; Zheng, Lifei; Itskov, Mikhail; Göstl, Robert; Herrmann, Andreas
Polymer Mechanochemistry in Microbubbles
Advanced Materials, 35 (47) :2305130
November 2023
ISSN: 1521-40954738.
Stroka, S.; Haussmann, N.; Clemens, M.
Efficient Assessment of High-Resolution Low-Frequency Magnetic Field Exposure Scenarios Using Reduced Order Models
15th Scientific Computing in Electrical Engineering (SCEE 2024)
Darmstadt, Germany
November 2023