Numerische Methoden der Finanzmathematik

Computational Finance

Winter Term 2013/14

Organiser

Prof. Dr. Matthias Ehrhardt

Lecturer: Dr. Jörg Kienitz (Postbank)

Assistant:Dipl.-Math. Long Teng

Type

Lecture and Discuss exercises (4+2 SWS)

The course comprises 4h lectures and 2h discuss 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

6 points

Lab Exercise

Separate course(with 3 credits points):Lab Exercises for Computational Finance

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.

Time and Place

Lecture:
Monday, 8:15-9:45 in room G.15.25.
Monday, 10:00-11:30 in room G.15.25.

Discuss exercises:
Monday, 14:15-16:45 in room G.14.33.

Examination

Oral examinations with duration of about 30 min. for each candidate. As a prerequisite, it is required to work successfully on (at least) 50% of the exercises and (at least) 50% of the programming exercises.

Script

• Matlab Introduction

• Computational Finance , updated November 11.

Exercises

The exercises can be done in groups of 1-2 persons.

• Exercise 1 (updated)Solution hints (updated)

• Exercise 2 (updated)Solution hints

• Exercise 3

• Exercise 4

• Exercise 5 Solution hints

• Exercise 6 Solution hints

• Exercise 7 (updated)Solution hints

• Exercise 8 Solution hints

• Exercise 9 Solution hints

• Exercise 10 (updated)Solution hints

-Please find your examination date as follows:

1. At 10 am on 28 Feb 2014 for the student number ****440

2. At 11 am on 28 Feb 2014 for the student number ****459

3. At 12 am on 28 Feb 2014 for the student number ****003

4. At 13 am on 28 Feb 2014 for the student number ****261

All the examinations will take place in the room WP505 (fifth floor) in Wicküler Park, the location can be found here.

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.