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Arbeitsgruppe Angewandte Mathematik / Numerische Analysis
Bergische Universität Wuppertal
Fakultät 04
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D-42119 Wuppertal
Deutschland

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E-Mail: sek-amna{at}math.uni-wuppertal.de

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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.



Publications

Referenzen
7.
Long Teng; Matthias Ehrhardt; Michael Günther
Bilateral Counterparty Risk Valuation of CDS contracts with Simultaneous Defaults
Int. J. Theoret. Appl. Finance, 16(7)
2013
6.
R. Company; Matthias Ehrhardt; Jodar S
Novel Methods in Computational Finance Special Issue of International Journal of Computer Mathematics of selected papers from these fields in Computational Mathematics and it's applications, presented at the International Conference on Mathematical Modell
Instituto Universitario de Matem
2013
5.
Matthias Ehrhardt; Ronald E. Mickens
A fast, stable and accurate numerical method for the Black-Scholes equation of American options
International Journal of Theoretical and Applied Finance, 11(5):471 -- 501
2008
4.
Matthias Ehrhardt
Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing
Herausgeber: Nova Science Publishers, Hauppauge, NY 11788
2008
3.
Matthias Ehrhardt; Julia Ankudinova
Fixed domain transformations and split-step finite difference schemes for Nonlinear Black-Scholes equations for American Options
In Ehrhardt, Matthias, Editor,
Seite 243 -- 273.
Herausgeber: Nova Science Publishers, Hauppauge, NY 11788
2008
2.
Matthias Ehrhardt
Nonlinear Models in Option Pricing - an Introduction, Preface
In Ehrhardt, Matthias, Editor,
Seite 1 -- 19.
Herausgeber: Nova Science Publishers, Hauppauge, NY 11788
2008
1.
Christian Kahl
Modelling and Simulation of Stochastic Volatility in Finance
Herausgeber: Disssertation.com, Florida, USA
2007

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