Numerische Methoden der Finanzmathematik 1

Lecture Details

Summer Term 2018

Organiser

Dr. rer. nat. Long Teng

Igor Kossaczky, M.Sc.

Type

Lecture and exercises(4SWS)

The course comprises 4h lectures integrated with exercises per week. 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.

Credits

9 points

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.

• Modeling of financial markets, options
• Binomial method and its extensions
• risk-neutral valuation, stochastic processes
• Geometric Brownian Motion, Ito Lemma
• exotic options
• stochastic differential equations (SDEs)
• Calibration, jump models
• generating random numbers with specified distributions
• Monte Carlo Methods, variance reduction approaches

Time and Place

Lecture:
Monday, 16:00 - 18:00 room G.14.34
Thursday, 14:00 - 16:00 room G.10.06 (HS06)

Examination

Oral (30 min) or written (120 min) examinations for each candidate. As a prerequisite, it is required to work successfully on at least 50% of the homeworks and at least 2 Labs. The homeworks and the Lab exercises can be done in groups of 1-2 persons.

Exercise Sheets

-

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, 5th edition, Springer, 2012.
• 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, 2nd ed., 2010.
• B. Øksendal, 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.