Lecturer

Prof. Dr. Martin Schweizer

Coordinator

Benjamin Kotlov

This is an advanced course on mathematical finance for students with a good background in probability. We want to give an overview of main concepts, questions and approaches, and we do this mostly in continuous-time models.

Topics include:

• semimartingales and general stochastic integration
• absence of arbitrage and martingale measures
• fundamental theorem of asset pricing
• option pricing and hedging
• hedging duality
• optimal investment problems
• and probably others

Prerequisites

Prerequisites are the standard courses

• Probability Theory (for which lecture notes are available)
• Brownian Motion and Stochastic Calculus (for which lecture notes are available)

Those students who already attended "Introduction to Mathematical Finance" will have an advantage in terms of ideas and concepts. This course is the second of a sequence of two courses on mathematical finance. The first course "Introduction to Mathematical Finance" (MF I), 401-3888-00L, focuses on models in finite discrete time. It is advisable that the course MF I is taken prior to the present course, MF II. For an overview of courses offered in the area of mathematical finance, see Link.

Lectures take place live on Tuesdays 8:00-10:00 in HG E1.1 and Thursdays 8:00-10:00 in HG G19.1.

There will be no recordings.

Lecture Notes

The course is based on different parts from different books as well as on original research literature.

Lecture notes will not be available. However, the lecture notes for BMSC are available and can be found here, Appendix A can be found here, Appendix B can be found here, Appendix C can be found here, and Appendix D can be found here. The password will be emailed to the enrolled students.

Note that the lecture notes are protected by copyright, and their dissemination in any form is strictly prohibited.

Exercise classes take place live on Fridays 10:00-12:00 in ML F38.

The first exercise class is on September 22, in the first week of the semester.

Exercise Sheets

New exercise sheets are uploaded below on Wednesdays before the corresponding Friday exercise class, along with solutions to the exercise sheet from the previous week.

While handing in your solutions is not compulsory, experience shows that being able to solve the exercises independently goes a long way towards good exam performance. We strongly encourage you to hand in written solutions. This can be done electronically. Please follow the instructions below.

1. Scan your solution into a single PDF file.
2. Name the created file as {email address}_{exercise sheet number}.pdf (e.g. andrey.kolmogorov@probability.com_1.pdf).
3. Upload your solution using the button below before 12:00 on the indicated Wednesday. The password will be emailed to the enrolled students.

Late submissions will not be considered. In case you submit your solution early and then decide to change something and submit again, we will only consider the most recent submission.

Exercise Sheet	Due by	Solutions
Exercise Sheet 1	September 27	Solution 1
Exercise Sheet 2	October 4	Solution 2
Exercise Sheet 3	October 11	Solution 3
Exercise Sheet 4	October 18	Solution 4
Exercise Sheet 5	November 1	Solution 5
Exercise Sheet 6	November 8	Solution 6
Exercise Sheet 7	November 15	Solution 7
Exercise Sheet 8	November 22	Solution 8
Exercise Sheet 9	November 29	Solution 9
Exercise Sheet 10	December 6	Solution 10
Exercise Sheet 11	December 13	Solution 11
Exercise Sheet 12	December 20	Solution 12
Exercise Sheet 13	December 27	Solution 13

Office hours during the semester with assistants from group 3 are available on Mondays and Thursdays starting from the fourth week of the semester. More information can be found here. Note that you can also ask questions during the exercise classes.

The forum for this course can be found here.

The grade for this course is based solely on the final exam. The exam is oral, 30 minutes (no preparation time) and will cover all the material discussed during the lectures and all the material from the exercise sheets (except if explicitly stated otherwise).

Office Hours

Office hours during the winter break with assistants from group 3 will be available in the last two weeks before the exam session. More information can be found here.

While there are many textbooks on mathematical finance, none of them is ideal to cover the contents of this course. References include the following books:

• T. Björk, Arbitrage Theory in Continuous Time, 4th edition, Oxford Academic (2019)
• J. Cvitanic and F. Zapatero, Introduction to the Economics and Mathematics of Financial Markets, MIT Press (2004)
• F. Delbaen and W. Schachermayer, The Mathematics of Arbitrage, Springer (2006)
• E. Eberlein and J. Kallsen, Mathematical Finance, Springer (2019)
• R. J. Elliott and E. P. Kopp, Mathematics of Financial Markets, 2nd edition, Springer (2005)
• P. Hunt and J. Kennedy, Financial Derivatives in Theory and Practice, revised edition, Wiley (2004)
• M. Jeanblanc, M. Yor and M. Chesney, Mathematical Methods for Financial Markets, Springer (2009)
• G. Kallianpur and R. L. Karandikar, Introduction to Option Pricing Theory, Springer (2000)
• I. Karatzas and C. Kardaras, Portfolio Theory and Arbitrage: A Course in Mathematical Finance, American Mathematical Society (2021)
• I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, Springer (1998)
• D. Lamberton and B. Lapeyre, Introduction to Stochastic Calculus Applied to Finance, 2nd edition, CRC Press (2007)
• A. N. Shiryaev, Essentials of Stochastic Finance: Facts, Models, Theory, World Scientific (1999)
• The asymptotic elasticity of utility functions and optimal investment in incomplete markets by Dmitry Kramkov and Walter Schachermayer, Annals of Applied Probability 9