# Finanzmathematik II / Stochastic Calculus and Arbitrage Theory in Continuous Time

## Prof. Dr. Thilo Meyer-Brandis, Annika Steibel

## Schedule and Venue

LecturesProf. Dr. Thilo Meyer-Brandis |
Tuesday, 10.15 - 11.45 Thursday, 10.15 - 11.45 |
online online |

Exercise ClassesAnnika Steibel |
Thursday, 08.30 - 10.00 |
online |

Supplementary Exercise ClassesAnnika Steibel |
Friday, 5 February 2021, 10.00 - 13.00 s.t. Monday, 8 February 2021, 10.00 - 12.00 s.t. |
online |

Final Exam |
Monday, 22 March 2021, 10.00 - 12.00 s.t. |
TBA |

Retake Exam |
TBA |
TBA |

**The course will be organised via Moodle (https://moodle.lmu.de) where you can log in using your LMU e-mail address (@campus.lmu.de). If you wish to participate in the course, please sign up as soon as possible by sending an e-mail from your LMU e-mail address to Annika Steibel (steibel@math.lmu.de). **

The final exam will be a written exam which will take place at the Mathematical Institute of the LMU. Please keep in mind that a registration for the exam as well as for the retake exam will be mandatory.

The first lecture will be held on Tuesday, 3 November 2020. (Please note: due to a clash with another module, the Tuesday lectures now start at 10.00 c.t..)

The first exercise class will be held on Thursday, 12 November 2020. (Please note: due to a clash with another module, the exercise class had to be rescheduled and now takes place on Thursdays.)

## Course Description

The lecture provides an introduction to stochastic calculus with an emphasis on the mathematical concepts that are later used in the mathematical modeling of financial markets.

In the first part of the lecture course the theory of stochastic integration with respect to Brownian motion and Ito processes is developed. Important results such as Girsanov's theorem and the martingale representation theorem are also covered. The first part concludes with a chapter on the existence and uniqueness of strong and weak solutions of stochastic differential equations.

The second part of the lecture course gives an introduction to the arbitrage theory of financial markets in continuous time driven by Brownian motion. Key concepts are the absence of arbitrage, market completeness, and the risk neutral pricing and hedging of contingent claims. Particular attention will be given to the the Black-Scholes model and the famous Black-Scholes formula for pricing call and put options.

## References

C. Dellacherie and P. A. Meyer. *Probabilities and Potential B: Theory of Martingales*. North-Holland, Amsterdam, 1982.

I. Karatzas and S. E. Shreve. *Brownian Motion and Stochastic Calculus*. Springer, New York, second edition, 1991.

B. Oksendal. *Stochastic Differential Equations: An Introduction with Applications*. Springer, Berlin, sixth edition, 2003.

Continuous Time Finance:

T. Björk. *Arbitrage Theory in Continuous Time*. Oxford University Press, New York, third edition, 2009.

I. Karatzas and S. E. Shreve. *Methods of Mathematical Finance*. Springer, New York, 1998.

B. Oksendal. *Stochastic Differential Equations: An Introduction with Applications*. Springer, Berlin, sixth edition, 2003.

## For whom is this course?

**Target Participants:** Master students of Financial and Insurance Mathematics or Mathematics.

**Pre-requisites: **Probability Theory.

**Applicable credits: **Students may apply the credits from this course to the Master Finanz- und Versicherungsmathematik (WP12 (PO2011) resp. P1 (PO2019)) or to the Master Mathematik (WP23).

## Exercises

Correcting your answers and thinking through the exercises is the best preparation for the exam.

**Problem Sheets:**During the course, weekly problem sheets will be uploaded on Moodle.

## Final Exams

The exam will take place on Monday, 22 March 2021 (10.00 - 12.00 h) at the LMU Institute of Mathematics.