Workgroup Financial Mathematics

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Numerical Methods for Financial Mathematics

Prof. Dr. Christian Fries, Dr. Alessandro Gnoatto, Dr. Juan Miguel Montes

Schedule and Venue

Programming Lectures
Dr. Alessandro Gnoatto
Dates and Times:
26.03, 27.03, 31.03, and 01.04  (14:00 - 18: 00)
28.03                                        (09:00 - 13:00)
Room B 121
Prof. Dr. Christian Fries
First lecture: 10.04
Thu 14:15 - 15:45 and Fri 08:30 - 10:00
Dr. Juan Miguel Montes
First exercise: 09.04
Wed 14:00 - 16:00
quantLab Tools and Technology Tutorium
Dr. Juan Miguel Montes
Tues 14:00 - 16:00 and Fri 10:00 - 12:00
Mid-term Project Review to be announced
Final Written Exam to be announced

Note: It is obligatory for students to attend the programming lectures, during which an introduction to Object-Oriented Programming in Java will be given.

Course Description

The lecture gives an introduction to some of the most important numerical methods in financial mathematics. A central topic of this lecture is the Monte Carlo method and its applications to stochastic differential equations, as used for example in the valuation of financial derivatives. In this context pseudo-random number generation, Monte Carlo simulation of stochastic processes and variance reduction methods are discussed. For low dimensional models, existing alternatives to derivatives valuation by numerical solutions of partial differential equations (PDEs) will be discussed, albeit with less emphasis.

In addition, numerical methods for financial mathematics are addressed as they are used in the processing of market data, model calibration and calculation of risk parameters.

The lecture also covers the object-oriented implementation of the numerical methods in the context of their application. We will use the Java 8 programming language and students will be guided to prepare small programming exercises in Java. Note: to follow this course it is obligatory to attend the programming lectures on ''Introduction to Object-Oriented Programming in Java''.

During the discussion of the numerical methods and their object-oriented implementation, students will also learn to work with some state-of-the-art / industry standard software developments tools (development with Eclipse, version control with subversion or git, unit testing with jUnit, integration testing with Jenkins).

The lecture has a clear focus on the presentation of mathematical methods with relevance to practical applications.


Glasserman, Paul: Monte-Carlo Methods in Financial Engineering. Springer, New York, 2003. ISBN 0-387-00451-3.
Asmussen, Søren; Glynn, Peter W.: Stochastic Simulation: Algorithms and Analysis. Springer, 2007. ISBN 978-0387306797.
Fries, Christian P.: Mathematical Finance. Theory, Modeling, Implementation. John Wiley & Sons, 2007. ISBN 0-470-04722-4.

For whom is this course?

Target Participants: Master students of Mathematics or Business Mathematics.

Pre-requisites: Probability Theory, Finanzmathematik II (Stochastic Calculus).

Applicable credits: Students may apply the credits from this course to Masterprüfungen Mathematik (WP3) and Wirtschaftsmathematik (WP5).


Active participation in the Exercises is necessary to pass the Exam. Moreover, correcting your answers and thinking through the exercises is the best preparation for the exam. The written solutions to theory-related exercises need not be submitted, but if you wish them to be corrected, please submit your exercise solutions.

The individual solutions to the programming exercises will not be corrected, but students may request a code review together with any of the instructors, during the quantLab tutorium. An example solution, that can be used to do a self-assessment of your programming solutions, will be provided on the SVN repository.

Exercise Handouts: The problem sheets will be uploaded here during the course.

Exercise Sheet 0 (setting-up your SVN account and folder)
Exercise Sheet 1 (deadline: 16.Apr.2014) Solution Sheet 1
Exercise Sheet 2 (deadline: 30.Apr.2014) SolutionSheet 2
Pi Errors.ods
Exercise Sheet 3 (deadline: 07.May.2014) SolutionSheet 3
Exercise Sheet 4 (deadline: 14.May.2014) SolutionSheet 4
Exercise Sheet 5 (deadline: 21.May.2014) SolutionSheet 5
Exercise Sheet 6 (deadline: 28.May.2014) SolutionSheet 6
Exercise Sheet 7 (deadline: 4.Jun.2014) Solution Sheet 7
Exercise Sheet 8 (supplementary exercises ) SolutionSheet 8
Exercise Sheet 9 (deadline: 11.Jun.2014 ) SolutionSheet 9
Exercise Sheet 10 (deadline: 2.Jul.2014 ) Solution Sheet 10 to be uploaded


The exam of this lecture will consist of two parts both of which have to be passed: a successful review of a mid term project and a written exam at the end of the lecture. The final grade shall be computed from 70% of the written exam grade and 30% from the mid term project grade.

Mid term project:

You may now have a look at the preliminary project handout. Note that this is still a preliminary version. We shall update it with corrections, before announcing the final version.

Mid-Term Project (deadline: t.b.a. )

Written exam: The written exam is open-book, that is, all notes, books, solutions of exercises etc. may be used. Personal electronic devices of any kind are not allowed. To participate, please bring to the exam your ID card or passport and your student card. Please be on time.