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

People Research Publications Teaching

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Summary

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:

• Sandra Boschert
• Matthias Ehrhardt (Leader)
• Michael Günther
• Christian Hendricks
• Alexej Nesmejanow
• Long Teng

Slovakian team:

• Darina Graczová
• Daniel Ševčovič
• Beata Stehliková (Leader)
• Zuzana Bučková (formerly Zikova)
• Magdaléna Zitnanská

German institutions:

• Department of Mathematics and Informatics, Faculty of Mathematics and Natural Sciences, University of Wuppertal.

Slovakian institutions:

• Department of Applied Mathematics and Statistics, Faculty of Mathematics, Physics and Informatics, Comenius University Bratislava

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

2013

• Darina Graczova, Dynamic Accumulation Model with Fat-tailed Distributed Returns, ISCAMI 2013 (International Student Conference on Applied Mathematics and Informatics)
• Zuzana Bučková, Workshop on Computational and Algorithmic Finance in the framework of ICCS Conference, June 5-7, 2013, Barcelona, Spain.
• Long Teng, Workshop on Computational and Algorithmic Finance in the framework of ICCS Conference, June 5-7, 2013, Barcelona, Spain.
• Long Teng, 6th General AMaMeF and Banach Center Conference Advances in Mathematics of Finance, June 10-15, 2013, Warzaw, Poland.
• Christian Hendricks, Research Seminar in Energy Economics, Department of Economics at the University of Cologne, Chair of Energy Economic, July 11, 2013. (invited talk)
• Christian Hendricks, Research Seminar, E.ON Global Commodities, August 28, 2013. (invited talk)
• Matthias Ehrhardt, Minisymposium Computational Finance XIII Mathematical Modelling in Engineering & Human Behaviour, Institute of Multidisciplinary Mathematics (IMM), Universitat Politècnica de València, Spain. TALK: A General Approach for Stochastic Correlation using Hyperbolic Functions September 3-6. (invited talk)
• Matthias Ehrhardt, European Science Foundation OPTPDE Workshop Modeling and Control of Large Interacting Dynamical Systems, Université Paris-Dauphine. TALK: A General Approach for Stochastic Correlation using Hyperbolic Functions September 10-12 (invited talk)
• Matthias Ehrhardt, Minisymposium Computational Finance in the framework of SciCADE 2013 - International Conference on Scientific Computation and Differential Equations, September 16-20, 2013, University of Valladolid, Spain. (invited talk)
• Christian Hendricks, Seminar at Chair for Energy Trading and Finance, University of Duisburg-Essen, November 20, 2013. (invited talk)
• Long Teng, Oberseminar Stochastik, Heinrich-Heine Universität Düsseldorf, December 4, 2013. (invited talk)

• Daniel Ševčovič, Riccati transformation method for solving Hamilton-Jacobi-Bellman equation, STRIKE - Winter School 2013/14 - Methods for Nonlinear Problems January 6-10, 2014, University of Greenwich, UK. (invited lecture)
• Matthias Ehrhardt, Stable Explicit Time Stepping for Fully Non-Linear Option Pricing Problems, Seminar, University of Wuppertal, January 14, 2014.
• Zuzana Bučková, Minisymposium Computational Finance, ECMI 2014, Taormina, Sicily, June 9-13, 2014.
• Christian Hendricks, Minisymposium Mathematical Modelling in Energy Markets, ECMI 2014, Taormina, Sicily, June 9-13, 2014.
• Matthias Ehrhardt, Project management and soft skills tutorials (3 invited talks), Summer School on Computational Finance, Smolenice Castle, September 8-12, 2014.
• Long Teng, Option Price with dynamically correlated Stochastic Interest Rate, 18th International Conference on Mathematical Methods in Economy and Industry, Smolenice Castle, September 8-12, 2014.
• Christian Hendricks, High order Combination Technique for the efficient Pricing of Basket Options, 18th International Conference on Mathematical Methods in Economy and Industry, Smolenice Castle, September 8-12, 2014.
• Zuzana Bučková, Numerical Analysis of the Alternating Direction Explicit Method and its Application in Finance, 18th International Conference on Mathematical Methods in Economy and Industry, Smolenice Castle, September 8-12, 2014.

Posters related to the Project

• Zuzana Bučková, Numerical methods for stochastic differential equations, TU Vienna, September 2-4, 2013, Vienna, Austria.
• Long Teng, Summer School Numerical methods for stochastic differential equations, TU Vienna, September 2-4, 2013, Vienna, Austria.
• Magdaléna Zitnanská, Numerical methods for stochastic differential equations, TU Vienna, September 2-4, 2013, Vienna, Austria.

Participation in Online Conferences

• Zuzana Bučková, online MATLAB Computational Finance Virtual Conference, September, 19, 2013.

Activities related to the Project

• Marie Curie ITN STRIKE
• Special Interest Group (SIG) Computational Finance
• Minisymposium Computational Finance, ECMI 2014, The 18th European Conference on Mathematics for Industry, Taormina, Sicily, June 9-13 2014.
• Minisymposium Mathematical Modelling in Energy Markets, ECMI 2014, The 18th European Conference on Mathematics for Industry, Taormina, Sicily, June 9-13, 2014.
• Summer School on Computational Finance, Smolenice Castle, September 8-12, 2014.
• 18th International Conference on Mathematical Methods in Economy and Industry, Smolenice Castle, September 8-12, 2014.

Former Projects

• Fin-Diff-Fin: Finite Differences for Financial derivative models, financed by the DAAD (01/2007-12/2008)