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Chair of Applied Mathematics / Numerical Analysis
Bergische Universität Wuppertal
Faculty of Mathematics and Natural Sciences
Gaußstraße 20
D-42119 Wuppertal
Germany

Phone: +49 202 439 5296
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E-Mail: sek-amna{at}math.uni-wuppertal.de

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# Numerische Methoden der Finanzmathematik

Computational Finance

Wintersemester 2012/13

 Organiser Prof. Dr. R. Pulch Type Lecture / Vorlesung (4 SWS). Exercises as well as programming exercises are integrated in this course. The course will be given in English language. Target audience The lecture is dedicated to students within the courses: diploma mathematics, master mathematics, master business mathematics, teaching degree SII, bachelor/master IT (branch of study: computing). The lecture corresponds to the modules SKap.NAaA and SKap.WM within the master course mathematics (branch of study: Numerical Analysis and Algorithms or business mathematics). Die Veranstaltung richtet sich an Personen aus den Studiengängen: Diplom Mathematik, Master Mathematik, Master Wirtschaftsmathematik, Lehramts SII, Bachelor/Master IT (Fachrichtung Computing). Die Veranstaltung deckt die Module SKap.NAaA und SKap.WM im Master-Studiengang Mathematik (Fachrichtung Numerical Analysis and Algorithms oder Wirtschaftsmathematik) ab. Prerequisite Analysis I-II, Lineare Algebra I-II, Introduction to Numerical Mathematics. Abstract Financial derivatives have become an essential tool for the control and hedging of risks. The crucial problem consist in the determination of the fair price of a financial derivative, which is based on mathematical methods. Simple models apply the Black-Scholes equation, where an explicit formula of the corresponding solution exists. More complex models do not allow for an analytic solution. Thus numerical methods are required to solve the models. Both the mathematical modelling and the numerical simulation of financial derivatives is discussed in this lecture. We focus on time-continuous (not time-discrete) models given by stochastic differential equations or partial differential equations. A rough classification of the methods for the determination of the price of a financial derivative yields three types: binomial methods, Monte-Carlo simulations and techniques for partial differential equations possibly including free boundary conditions. We explain each type of method extensively. Corresponding algorithms are implemented in lab exercises using the software package MATLAB. Finanzderivate sind in den letzten Jahren zu einem unentbehrlichen Werkzeug in der Finanzwelt zur Kontrolle und Absicherung von Risiken geworden. Das herausfordernde Problem ist die "faire" Bewertung der Finanzinstrumente, die auf modernen mathematischen Methoden basiert. Die Grundlage für die Bewertung einfacher Modelle ist die Black-Scholes-Gleichung, die eine geschlossene Lösungsformel besitzt. Für komplexere Modelle existieren jedoch keine geschlossenen Formeln mehr, und die Modellgleichungen müssen numerisch gelöst werden. Beide Problemstellungen, die mathematische Modellierung und die numerische Simulation von Finanzderivaten, werden in dieser Vorlesung ausführlich behandelt. Dabei soll ein Bogen von der Modellierung über die Analyse bis zur Simulation realistischer Finanzprodukte geschlagen werden. Wir beschränken uns überwiegend auf zeitkontinuierliche (also nicht zeitdiskrete) Modelle, welche durch stochastische bzw. partielle Differentialgleichungen beschrieben werden können. Für die Bewertung von Finanzderivaten können grob drei Klassen von Methoden unterschieden werden: Binomialmethoden, Monte-Carlo-Simulationen sowie Verfahren zur Lösung partieller Differentialgleichungen und freier Randwertprobleme. Wir erläutern diese Techniken ausführlich und erarbeiten uns deren algorithmische Umsetzung mittels Matlab-Programmen in einem begleitenden Praktikum. The chapters of this lecture are planned as follows:  Modelling Tools for Financial Options, Generating Random Numbers with Specified Distributions, Monte Carlo Simulations with Stochastic Differential Equations, Standard Methods for Options, Finite Element Methods, Pricing of Exotic Options. Time and Place Monday, 16:15-17:45 in room G.14.34. Tuesday, 16:15-17:45 in room G.14.34. First lecture on October 8, 2012. Examination Oral examinations will be done in February 2013. As a prerequisite, it is required to work successfully on (at least) 50% of the exercises and (at least) 50% of the programming exercises. Exercises Remarks Based on the topics of this lecture, diploma theses and master theses can be defined. Thereby, a cooperation with a bank is often possible. Literature R. Seydel, Tools for Computational Finance, 4th edition, Springer, 2009.D. Higham, An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation, Cambridge University Press, 2004. M. Günther and A. Jüngel, Finanzderivate mit MATLAB, Vieweg, 2003.B. Oksendal, Stochastic Differential Equations. An Introduction with Applications , 6th edition, Springer, 2003. D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, Vol. 43, No. 3, pp. 525-546. P. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992.