Bergische Universität Wuppertal
Fachbereich Mathematik und Naturwissenschaften
Angewandte Mathematik - Numerische Analysis (AMNA)


Bilateral German-Slovakian Project

NL-BS: Numerical Solution of Nonlinear Black-Scholes Equations

financed by DAAD and the Slovakian Ministry of Education

(01/2013 - 12/2014)


The famous Black-Scholes (BS) equation is an effective and rather simple model for option pricing. While the financial crises of 2008 and 2010 are both clearly outside of the domain of validity of BS theory, it has been rightly pointed out that the market is not Gaussian and it is not frictionless and complete as it had been postulated in the pioneering linear BS model. Contrary, generalized nonlinear option pricing models are capable of capturing several important phenomena like transaction costs, investor's risk from unprotected portfolio, investor's expected utility maximization, illiquid markets, large traders feedback influence, etc.

In this respect, an important part is the development of high-order compact finite difference methods (FDMs) and transformation techniques for numerically solving nonlinear BS equations. We will propose, design and analyze new efficient and robust numerical methods for solving highly nonlinear option pricing problems. Also, we want to study the convergence models of interest rates to investigate the possibility of an analytical approximation of bond prices and the order of accuracy. As an application, we compute Slovak interest rates before the Slovak Republic adopted the Euro currency. Finally, we will use the advantage of having contacts to all 4 major german electricity suppliers; we will design new numerical methods for pricing swing-options which are typical options in the electricity market.

Scientific Objectives

German team:

Slovakian team:

German institutions:

Slovakian institutions:

Publications related to the Project

[Buc13] Z. Bučková,
The mathematical analysis of the multifactor models for pricing of financial derivates, February 2013, Project of the dissertation thesis
[BuEhGu14] Z. Bučková, M. Ehrhardt, M. Günther
Fichera Theory and its Application in Finance, Preprint 14/10, April 2014.
[BuEhGu15] Z. Bučková, M. Ehrhardt, M. Günther
Alternating Direction Explicit Methods for Convection Diffusion Equations, Preprint 15/15, March 2015.
[EhVa13] M. Ehrhardt, R. Valkov,
A stable explicit finite difference scheme for a nonlinear European option pricing problem, Preprint 13/23, October 2013.
[Hen13] C. Hendricks,
Modelling and Numerical Simulation of Clean Spark Spread Options in the German Electricity Market, Master thesis, Bergische Universität Wuppertal, April 2013.
[HeEh13a] C. Hendricks M. Ehrhardt,
Clean Spread Options in the German Electricity Market, Preprint 13/09, July 2013.
[HeEh14] C. Hendricks, M. Ehrhardt,
Evaluating the Effects of Changing Market Parameters and Policy Implications in the German Electricity Market, The Journal of Energy Markets, June 2014.
[HeEhGu14a] C. Hendricks, M. Ehrhardt, M. Günther
High Order Tensor Product Interpolation in the Combination Technique, Preprint 14/25, August 2014.
[HeEhGu14b] C. Hendricks, M. Ehrhardt, M. Günther
High Order Combination Technique for the efficient Pricing of Basket Options, Preprint 14/36, November 2014.
[HeEhGu15] C. Hendricks, M. Ehrhardt, M. Günther
High-Order-Compact ADI schemes for diffusion equations with mixed derivatives in the combination technique, Preprint 15/14, February 2015.
[TEG14b] L. Teng, M. Ehrhardt, M. Günther,
The Dynamic Correlation Model and its Application to the Heston Model, Preprint 14/09, April 2014.
[TEG14a] L. Teng, M. Ehrhardt, M. Günther,
Modelling Stochastic Correlation, Preprint 14/03, February 2014.
[TEEG13] L. Teng, C. van Emmerich, M. Ehrhardt, M. Günther,
A General Approach for Stochastic Correlation using Hyperbolic Functions, Preprint 13/14, August 2013.
[TEG13a] L. Teng, M. Ehrhardt, M. Günther,
Bilateral Counterparty Risk Valuation of CDS contracts with Simultaneous Defaults, Int. J. Theor. Appl. Finan. 16, 1350040 (2013) [20 pages] DOI: 10.1142/S0219024913500404
[TEG13b] L. Teng, M. Ehrhardt, M. Günther,
Numerical Evaluation of Complex Logarithms in the Cox-Ingersoll-Ross Model, Int. J. Comput. Math. Vol. 90, Issue 5 (2013), 1083-1095. DOI: 10.1080/00207160.2012.749348
[TEG15] L. Teng, M. Ehrhardt, M. Günther,
Numerical Simulation of the Heston model with Stochastic Correlation, Preprint 15/01, January 2015.
[KiSe13a] S. Kilianová, D. Ševčovič,
Riccati Transformation Method for Solving Constrained Dynamic Stochastic Optimal Allocation Problem, In: Proceedings of the 13th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2013, June 24-27, 2013. I.P. Hamilton, J. Vigo-Aguiar (Eds.), pp. 852-857.
[KiSe13b] S. Kilianová, D. Ševčovič,
A Transformation Method for Solving the Hamilton-Jacobi-Bellman Equation for a Constrained Dynamic Stochastic Optimal Allocation Problem, ANZIAM Journal, 2013, available on CJO2013. DOI: 10.1017/S144618111300031X
[StCa14] B. Stehliková, L. Capriotti,
An Effective Approximation for zero-coupon bonds and Arrow-Debreu prices in the Black-Karasinski model, Int. J. Theoret. Appl. Finance, Volume 17, Number 06 (September 2014)
[Ste13] B. Stehliková,
A simple analytic approximation formula for the bond price in the Chan-Karolyi-Longstaff-Sanders model, International Journal of Numerical Analysis & Modeling, Series B, Volume 4, Number 3 (2013), 224-234.

Talks related to the Project



Posters related to the Project

Participation in Online Conferences

Activities related to the Project

Former Projects

University of Wuppertal
Faculty of Mathematics and Natural Sciences
Department of Mathematics
Applied Mathematics & Numerical Analysis Group

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