The goal is to investigate qualitative and quantitative properties of solutions to nonlinear generalizations of the Black-Scholes (BS) equation. Nonlinear models can capture several important phenomena like transaction costs, investor's risk from unprotected portfolio, investor's expected utility maximization, illiquid markets, large traders feedback influence, etc.
Such generalizations can be mathematically stated in the form of a nonlinear BS equation in which the volatility is adjusted to be a function of the Gamma of the option. Our approach is based on the analysis of the nonlinear BS equation for the Gamma of the option by means of combination of fully implicit and explicit finite difference methods (FDMs).
Participating universities and intituitions
- BU Wuppertal (Prof. Matthias Ehrhardt), Applied Mathematics and Numerical Analysis, University of Wuppertal, Germany.
- CU Bratislava (Prof. Daniel Sevcovic), Department of Applied Mathematics and Statistics, Comenius University, Bratislava, Slovakia.
- UP Valencia (Prof. Lucas Jódar Sánchez), Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Valencia, Spain.
- U Rousse (Prof. Lyuben G. Vulkov), Department of Numerical Analysis and Statistics, Ruse University Angel Kanchev, Ruse, Bulgaria.
- ISEG Lisboa (Prof. Maria do Rosario Lourenço Grossinho), Instituto Superior de Economia e Gestão (ISEG), Lisbon, Portugal.
- UA Zittau (Prof. Ljudmila Bordag), Fakultät Mathematik/Naturwissenschaften, Hochschule Zittau/Görlitz, Germany.
- TU Wien (Prof. Ansgar Jüngel), Institute for Analysis and Scientific Computing, TU Wien, Vienna, Austria.
- TU Delft (Prof. Kees Oosterlee), Delft Institute of Applied Mathematics (DIAM), TU Delft, The Netherlands.
- U Greenwich (Prof. Choi-Hong Lai), School of Computing and Mathematical Sciences, University of Greenwich, Greenwich, London, UK.
- U Würzburg (Prof. Alfio Borzi), Lehrstuhl für Mathematik IX (Chair Scientific Computing), University of Würzburg, Germany.
- U Antwerp (Prof. Karel in 't Hout), Department of Mathematics and Computer Science, University of Antwerp, Belgium.
Associated Partners
- Université Paris VI (Prof. Olivier Pironneau), Laboratoire Jacques-Louis Lions, U6 Paris, France.
- University of Sussex (Dr. Bertram Düring), Department of Mathematics at University of Sussex, Brighton, UK.
- University of A Coruña (Prof. Carlos Vázquez Cendón), M2NICA Research Group at Mathematics Department, University of A Coruña, Spain.
- MathFinance AG (Prof. Uwe Wystup), Derivative Consulting, Waldems, Germany.
- d-fine GmbH (Dr. Bodo Huckestein), Risk Modelling, Frankfurt am Main, Germany.
- Postbank AG (Dr. Jörg Kienitz), Quantitative Analysis, Bonn, Germany.
- Ortec Finance (Dr. Hens Steehouwer), Risk & Return Management, Rotterdam, the Netherlands.
- ING Bank (Dr. Marc van Balen and Dr. Drona Kandhai), Corporate Market Risk Management, Amsterdam, the Netherlands.
- Rabobank (Dr. Sacha van Weeren), Modelling & Research, Utrecht, the Netherlands.
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Recent preprints
[5] K. Duris, Shih-Hau Tan, Choi-Hong Lai, and D. Sevcovic: Comparison of analytical approximation formula and New-ton’s method for solving a class of nonlinear Black-Scholes parabolic equations, to appear in: Computational Methods in Applied Mathematics
Published papers in reviewed scholarly journals
2016
[4] S. Kilianova and M. Trnovska: Robust Portfolio Optimization via solution to the Hamilton-Jacobi-Bellman Equation, Int. Journal of Computer Mathematics, 2016. DOI: 10.1080/00207160.2013.871542
2015
[3] D. Sevcovic and M. Trnovska: Application of the Enhanced Semidefinite Relaxation Method to Construction of the Optimal Anisotropy Function, IAENG International Journal of Applied Mathematics, 45(3) (2015), 227-234
[2] D. Sevcovic and M. Trnovska: Solution to the Inverse Wulff Problem by Means of the Enhanced Semidefinite Relaxation Method, Journal of Inverse and III-posed Problems, J. Inverse Ill-Posed Problems 23(3) 2015, 263-285
arXiv: 1402.5668 DOI:10.1515/jiip-2013-0069
2013
[1] S. Kilianova and D. Sevcovic: A Transformation Method for Solving the Hamilton-Jacobi-Bellman Equation for a Constrained Dynamic Stochastic Optimal Allocation Problem, ANZIAM Journal (55) 2013, 14-38.
PDF file  arXiv: 1307.3672 DOI: 10.1017/S144618111300031X
Papers in reviewed proceedings
Disclaimer: These papers are available for free download. Their content is identical with author's final versions submitted for publication. The Copyright of published versions has been transferred to publishers.
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