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

People Research Publications Teaching

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In diesem Numerik-Seminar werden schnelle 'state-of-the-art', meist analysis-basierte, numerische Methoden behandelt. Diese Algorithmen dienen meist der Lösung von Differential- oder Integralgleichungen. Ein Schwerpunkt liegt in diesem Seminar auf Anwendungen aus der Finanzmathematik speziell finite Differenzen Verfahren und Agenten-basierte Methoden.

Die Einteilung der Gruppen erfolgt auf Grund der Kenntnisse, Interessen und Möglichkeiten der TeilnehmerInnen. Die Zusammensetzung der Gruppen erfolgt unter dem Gesichtspunkt der Komplementarität. Analytisch besonders Interessierte sollen mit numerisch Versierten und Computercracks zusammenarbeiten. Im Idealfall wird jeder Vortrag von einem anderen Gruppenmitglied gehalten. Dabei kommt jedem/jeder TeilnehmerIn in unterschiedlichen Phasen des Seminars eine Führungsrolle zu.

Scheinkriterium:

  Präsentation

• Erläuterung des Anwendungsproblems
• Mathematische Modellierung (evtl. mehrstufig)
• Lösung / Analyse / Numerik
• Diskussion der Resultate und ihrer Relevanz für die Anwendung
• Für die Präsentation wird Prosper bzw. HA-Prosper oder PPower4 empfohlen
  ( Verschiedene Präsentationsprogramme)
• Evtl. numerische Simulationen (möglichst in Matlab, Scilab oder Octave)

Vorkenntnisse:
Basiswissen mathematischer Grundvorlesungen wird vorausgesetzt.
Wünschenswert ist eine erfolgreiche Teilnahme an der Vorlesung Finanzmathematik sowie einige Programmiererfahrung.

Themen: (Literatur wird im Seminar ausgegeben)

1. Einführungen
   • R. Seydel, Tools for Computational Finance, Springer, 4th edition, 2009.
     • Monte Carlo simulations with stochastic differential equations (Chapter 3)
     • Option Pricing with Finite Differences (Chapter 4)
     • Finite Element Methods For American Options (Chapter 5)
     • Pricing of Exotic Options (Chapter 6)
2. Kinetische Modelle des Finanzmarkts
   Contagion and herding seem to play an important role in financial crises. The understanding of these phenomena is of paramount importance to develop strategies which help to avoid or counter balance these effects. Financial markets can be described as a many-agent system in analogy to many-particle systems of gas dynamics.
   • G. Naldi, L. Pareschi, G. Toscani, Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Birkhäuser Verlag, 2010.
     • L. Boudin, F. Salvarani, Modelling opinion formation by means of kinetic equations.
     • E. Cristiani, B. Piccoli, A. Tosin, Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints.
     • B. Düring, D. Matthes, A mathematical theory for wealth distribution.
3. Agenten-basierte Methoden
   • I. Giardina, J.-P. Bouchaud, Bubbles, crashes and intermittency in agent based market models, The European Physical Journal B 31 (2003), 421-437.
   • S.-H. Chen, C.-H. Yeh, Evolving traders and the business school with genetic programming: A new architecture of the agent-based artificial stock market, Journal of Economic Dynamics and Control 25 (2001), 363-393.
   • A.P. Kirman, N.J. Vriend, Evolving market structure: An ACE model of price dispersion and loyalty, Journal of Economic Dynamics and Control 25 (2001), 459-502.
   • T.B. Klos, B. Nooteboom, Agent-based computational transaction cost economics, Journal of Economic Dynamics and Control 25 (2001), 503-526.
   • J. Bower, D. Bunn, Experimental analysis of the efficiency of uniform-price versus discriminatory auctions in the England and Wales electricity market, Journal of Economic Dynamics and Control 25 (2001), 561-592.
   • T. Preis, S. Golke, W. Paul, J.J. Schneider, Multi-agent-based Order Book Model of financial markets, Europhysics Letters 75 (2006), 510-516.
   • P. Albin, D.K. Foley, Decentralized, Dispersed Exchange Without an Auctioneer, Journal of Economic Behavior and Organization 18 (1992), 27-51.
   • D.K. Gode, S. Sunder, Allocative Efficiency of Markets with Zero Intelligence Traders: Market as a partial substitute for individual rationality, Journal of Political Economy 101 (1993), 119-137.
4. Reduzierte Basis Methoden zur Optionsbewertung
   • O. Pironneau, Calibration of options on a reduced basis, Journal of Computational and Applied Mathematics 232 (2009), 139-147.
5. Parallele numerische Methoden zur Optionsbewertung
   • Parallele Binomialbaum Algorithmen
     • A.V. Gerbessiotis, Architecture independent parallel binomial tree option price valuations, Parallel Computing 30 (2004), 301-316.
     • K. Colb, M. Phar, Option pricing on the GPU, GPU Gems 2 (2005), 719-731.
   • Parallele Monte-Carlo Algorithmen
     • J.X. Li, G.L. Mullen, Parallel computing of a quasi-Monte Carlo algorithm for valuing derivatives, Parallel Computing 26 (2000), 641-653.
     • G. Pauletto, Parallel Monte Carlo Methods for Derivative Security Pricing, in: Computing in Economics, Finance 2000, Springer, 650-657.
     • V. Podlozhnyuk, Black-Scholes option pricing, part of CUDA SDK documentation, 2007.
   • Multi-threaded FFT Algorithmen
     • P. Carr, D.B. Madan, Option valuation using the fast Fourier transform, J. Comput. Finance 2 (1999), 61-73.
     • C.C.W. Leentvaar, C.W. Oosterlee, Multi-asset option pricing using a parallel Fourier-based technique, J. Comput. Finance 12 (2009), 1-26.
     • V. Surkov, Parallel option pricing with Fourier space time-stepping method on graphics processing units, Parallel Computing 36 (2010) 372-380.
     • R.K. Thulasiram, P. Thulasiraman, Performance evaluation of a multi-threaded fast Fourier transform algorithm for derivative pricing, Journal of Supercomputing 26 (2003), 43-58.
   • Parallele numerische Methoden zur Bewertung von Asiatischen Optionen
     • K. Huang, R.K. Thulasiram, Parallel Algorithm for Pricing American Asian Options with Multi-Dimensional Assets, 19th International Symposium on High Performance Computing Systems and Applications, 2005, 177-185.
     • H. Sak, S. Özekici, I.Boduroglu, Parallel computing in Asian option pricing, Parallel Computing 33 (2007), 92-108.
   • PDGL-basierte parallele Methoden für (nicht)lineare Black-Scholes Modelle
     • I. Chiorean, Parallel Algorithm for solving the Black-Scholes equation, Kragujevac J. Math. 27 (2005), 91-100.
     • C.-H. Lai, D. Crane, A. Davies, On a Parallel Time-domain Method for the Nonlinear Black-Scholes Equation, Lecture Notes in Computational Science and Engineering, 2007, 55, Part III, 659-666.
6. Schnelle Finite Differenzen Methoden für Amerikanische Optionen
   • M. Ehrhardt, R.E. Mickens, A fast, stable and accurate numerical method for the Black-Scholes equation of American options, Int. J. Theoret. Appl. Finance 11 (2008), 471-501.
   • D. Ševčovič, An iterative algorithm for evaluating approximations to the optimal exercise boundary for a nonlinear Black-Scholes equation, Canadian Appl. Math. Quarterly 15 (2007), 77-97.
   • D.Y. Tangman, A. Gopaul, M. Bhuruth, A fast high-order finite difference algorithm for pricing American options, Journal of Computational and Applied Mathematics 222 (2008), 17-29.
   • D.Y. Tangman, A. Gopaul, M. Bhuruth, Numerical pricing of options using high-order compact finite difference schemes, Journal of Computational and Applied Mathematics 218 (2008), 270-280.
   • C. Vázquez, An upwind numerical approach for an American and European option pricing model, Appl. Math. Comput. 97 (1998), 273-286.
   • S.-P. Zhu, J. Zhang, A new predictor-corrector scheme for valuing American puts, Appl. Math. Comput. 217 (2011), 4439-4452.
7. Optionsbewertung mit Integraltransformationen
   • J.C.Cortés, L. Jódar, R. Sala, P. Sevilla-Peris, Exact and numerical solution of Black-Scholes matrix equation, Appl. Math. Comput. 160 (2005), 607-613.
   • L. Jódar, P. Sevilla-Peris, J.C.Cortés, R. Sala, A new direct method for solving the Black-Scholes equation, Appl. Math. Lett. 18 (2005), 29-32.
   • A. Mezentsev, A. Pomelnikov, M. Ehrhardt, Efficient Numerical Valuation of Continuous Installment Options, Adv. Appl. Math. Mech. 3 (2011), 141-164.
   • R. Panini, Option pricing with Mellin Transforms, Dissertation, Stony Brook University, 2004.
   • R. Panini, R.P. Srivastav, Option pricing with Mellin Transforms, Math. Comput. Modelling 40 (2004), 43-56.
8. Schnelle und genaue numerische Bewertungsmethoden für nichtlineare Black-Scholes Modelle
   • J. Ankudinova, M. Ehrhardt, The numerical solution of nonlinear Black-Scholes equations, Comput. Math. Appl. 56 (2008), 799-812.
   • J. Ankudinova, M. Ehrhardt, Fixed domain transformations and split-step finite difference schemes for Nonlinear Black-Scholes equations for American Options, Kapitel 8 in M. Ehrhardt (ed.), Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing, Nova Science Publishers, Inc., Hauppauge, NY 11788, 2008, pp. 243-273.
   • E. Dremkova, M. Ehrhardt, A high-order compact method for nonlinear Black-Scholes option pricing equations of American Options, Int. J. Comput. Math. Vol. 88, Issue 13, (2011), 2782-2797.
   • B. Düring, M. Fournier, A. Jüngel, High order compact finite difference schemes for a nonlinear Black-Scholes equation, Int. J. Appl. Theor. Finance 7 (2003), 767-789.
   • B. Düring, Black-Scholes Type Equations: Mathematical Analysis, Parameter Identification & Numerical Solution, Dissertation, University Mainz, 2005.
   • J.R. Pintos, Numerical analysis and computing of nonlinear option pricing models, PhD Thesis, Universidad Politecnica de Valencia, Spanien, 2010.
9. Gitterfreie Methoden der Optionsbewertung
   • G.E. Fasshauer, A.Q.M. Khaliq, D.A. Voss, Using Meshfree Approximation for Multi-Asset American Options, J. Chinese Institute of Engineers 27 (2004), 563-571.
   • Y.C. Hon, A quasi-radial basis functions method for American options pricing, Comput. Math. Appl. 43 (2002), 513-524.
   • Y.C. Hon, X.C. Mao, A radial basis function method for solving options pricing model, J. Financial Engineering 8 (1999), 1-24.
   • U. Pettersson, E. Larsson, G. Marcusson, J. Persson, Option Pricing using Radial Basis Functions, ECCOMAS Thematic Conference on Meshless Methods 2005.
10. Methoden für Asiatische Optionen
    • Approximative analytische Methoden
      • N. Ju, Pricing Asian and basket options via Taylor expansion, J. Comput. Finance 5 (2002), 79-103.
      • G.W.P. Thompson, Fast Narrow Bounds on the Value of Asian Option, Centre for Financial Research, Judge Institute of Management, University of Cambridge, 1998.
      • S. Turnbull, L. Wakeman, A quick algorithm for pricing European average options, J. Financial Quant. Anal. 26 (1991), 377-389.
      • J.E. Zhang, Pricing continuously sampled Asian options with perturbation method, J. Futures Markets 23 (2003), 535-560.
    • Faltungsmethoden
      • A. Carverhill, L. Clewlow, Flexible convolution, Risk 3 (1990), 25-29.
    • Binomialbäume
      • W.W.-Y. Hsu, Y.-D. Lyuu, A convergent quadratic-time lattice algorithm for pricing European-style Asian options, in: Proc. of IASTED International Conference on Financial Engineering and Applications, 2004.
      • J.C. Hull, A. White, Efficient procedures for valuing European and American path-dependent options, J. Derivatives 1 (1993), 21-23.
    • Monte-Carlo Simulation
      • P. Boyle, Options: a Monte Carlo approach, J. Financial Economics 4 (1997), 323-338.
      • J. Corwin, P. Boyle, K. Tan, Quasi-Monte Carlo methods in numerical finance, Management Science 42 (1996), 926-938.
      • A. Kemna, A. Vorst, A pricing method for option based on average asset values, J. Banking and Finance 14 (1990), 13-129.
    • Direkte Integration
      • H. Geman, M. Yor, Bessel processes, Asian options, and perpetuities, Mathematical Finance 3 (1993), 349-375.
    • Partielle Differentialgleichung basierte Methoden
      • B. Alziary, J.-P. Dechamps, P.-F. Koehl, A PDE approach to Asian options: analytical numerical evidence, J. Banking and Finance 21 (1997), 613-640.
      • J. Dewynne, P. Wilmott, Asian options as linear complementary problems, Adv. Futures and Options Research 8 (1995), 145-173.
      • P.A. Forsyth, K.R. Vetzal, R. Zvan, Convergence of numerical methods for valuing path-dependent options using interpolation, Review of Derivatives Research 5 (2002), 273-314.
      • J. Vecer, A new PDE approach for pricing arithmetic average Asian options, J. Comput. Finance 4 (2001), 105-113.
      • J.E. Zhang, A semi-analytical method for pricing and hedging continuously sampled arithmetic average rate options, J. Comput. Finance 5 (2001), 59-79.
      • R. Zvan, P.A. Forsyth, K.R. Vetzal, Robust numerical methods for PDE models of Asian options, J. Computational Finance 1 (1998), 39-78.
    • Fourier- und Laplace Transformation
      • E. Benhamou, Fast Fourier transform for discrete Asian options, J. Comput. Finance 6 (2002), 49-61.
      • G. Fusai, Pricing Asian options via Fourier and Laplace transforms, J. Comput. Finance 7 (2004), 87-105.
    • Finite Volumen Methoden
      • C.-Y. Hsu, Adaptive finite volume methods for pricing European-style Asian options, Master's thesis, Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan, 2005.
11. Finite Volumen Methoden für Optionsbewertung
    • L. Angermann, S. Wang, Convergence of a fitted finite volume method for European and American option valuation, Numer. Math. 106 (2007), 1-40.
    • C.-S. Huang, C.-H. Hung, S. Wang, A fitted finite volume method for the valuation of options on assets with stochastic volatilities, Computing 77 (2006), 297-320.
    • S. Wang, A novel fitted finite volume method for the Black-Scholes equation governing option pricing, IMA J. Numer. Anal. 24 (2004), 699-720.
    • K. Zhang, S. Wang, Pricing options under jump diffusion processes with fitted finite volume method, Appl. Math. Comput. 201 (2008), 398-413.
    • R. Zvan, K.R. Vetzal, P.A. Forsyth, PDE methods for pricing barrier options, J. Economic Dynamics & Control 24 (2000), 1563-1590.
    • R. Zvan, P.A. Forsyth, K.R. Vetzal, A finite volume approach for contingent claims valuation, IMA J. Numer. Anal. 21 (2001), 703-731.
12. Numerische Bewertungsmethoden von Energie-Derivaten
    • Z. Chen, P.A. Forsyth, Implications of a regime-switching model on natural gas storage valuation and optimal operation, Quantitative Finance 10 (2010), 159-176.
    • Z. Chen, P.A. Forsyth, A semi-Lagrangian approach for natural gas storage valuation and optimal operation, SIAM J. Scientific Computing 30 (2007), 339-368.
    • T. Kluge, On the numerical pricing of swing options and the seasonality of derivative securities from the electricity market, Ph.D. Thesis, Oxford, 2006.
    • M.H. Nguyen, M. Ehrhardt, Modelling and Numerical Valuation of Power Derivatives in Energy Markets, im Druck: Adv. Appl. Math. Mech., 2011.
13. Optionsbewertung mittels diskreter singulärer Faltung
    • S. Zhao, G.W. Wei, Option valuation by using discrete singular convolution, Appl. Math. Comput. 167 (2005), 383-418.

Links:

• Computational Finance domain www.compfin.de (R. Seydel)
• Modellierung und Numerische Methoden von Finanzderivaten (Wikiversity-Kurs)
• Financial Numerical Recipes in C++ (Bernt Arne Ødegaard)
• DAAD Project Fin-Diff-Fin: Finite differences for Financial derivative models
• Master's program in Financial Mathematics (Numerical Methods), Halmstad University, Schweden.
• PREMIA - pricing financial derivatives Mathematics (MATHFI-Team at INRIA)
• Black-Scholes in Multiple Languages (Espen Haug)

Didaktische Vortragstipps:
Wie halte ich einen Seminarvortrag (M.Lehn, Mainz)
Artikel 1, Artikel 2, Artikel 3

Ähnliche Vorlesung im Sommersemester 2011

• Einführung in die Numerik (4+2 SWS) (Matthias Ehrhardt)
• Numerische Methoden der Analysis in der Finanzmathematik (2+1 SWS) (Jörg Kampen)