Numerical Methods for Financial Mathematics

The lecture gives an introduction to some of the most important numerical methods in financial mathematics. In particular, the following is a tentative schedule. We may do some changes to it, but the chore topics will remain.

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.

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 17 programming language and students will be guided to prepare small programming exercises in Java. To this end, and for a better general understanding of the topics faced, a compulsory parallel set of introductory lectures to Java Object Oriented programming is offered at the beginning of the semester.

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 such as

• Software development with Eclipse
• Version control with Git
• Unit testing with jUnit
• Application and checking of coding guidelines with Checkstyle

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 0-470-047224. http://www.christian-fries.de/finmath/book

Eckel, Bruce: (2006) Thinking in Java: The definitive introduction to object-oriented programming in the language of the world wide web. 4th Ed. Prentice Hall International

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), MSc Finanz- und Versicherungsmathematik PO 2011 (WP5), MSc Finanz- und Versicherungsmathematik PO 2019 (P4) and MSc Finanz- und Versicherungsmathematik PO 2021 (WP 22)

Active participation in the exercise courses, thinking through the problems and correcting your solutions is the best preparation for the exam. Exercise sheets will be uploaded during the course. The solutions to exercises need not be submitted, but if you wish to get a feedback, you are very welcome to do that.

The final grade will be the result of three parts:

• Coding assignments to be done at home
• Coding project to be done in groups
• Final 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.