**V5F4 Selected Topics in Stochastic AnalysisTopics in financial mathematics**

**Time: **Wednesday 8-10

**Room:** SR 0.011

**Description:**

In this lecture we will discuss some of the fundamental concepts in modelling financial markets. We will first develop the theory of arbitrage and the valuation of options in discrete time market models and then indicate how they adapt to time continuous models. An important example is the Black-Scholes model that admits explicit solutions for many pricing and hedging problems. Other potential topics we might touch upon are optimization of portfolios, more elaborate market models including stochastic volatility, the modelling of interest rates and questions of how to deal with model uncertainty.

**Prerequisites:**

Background in probability theory and stochastic analysis will be assumed. No prior knowledge of finance is requested.

**Exam: ** February 5, 16 (first round); March 19, 20 (second round)

The exam will be an oral examination; the examination takes place in groups of two.

Please have a look at the schedule with available slots and write me an email with **two** preferences to secure a slot.

The list of assigned slots will be available in the office of Birgit Bonn (3.029). Please pass by at your convenience to confirm your slot with your signature.

1st round | 2nd round | ||||
---|---|---|---|---|---|

Mon 05.2. | Fri 16.2 | Mon 19.3. | Tue 20.3. | ||

9:00 | X | X | 9:00 | O | X |

X | X | O | O | ||

9:45 | X | X | 9:45 | X | X |

X | X | O | O | ||

10:30 | X | X | 10:30 | O | O |

X | X | O | O | ||

11:15 | X | X | 11:15 | O | O |

O | X | O | O | ||

12:00 | X | X | 12:00 | O | X |

O | X | O | X | ||

14:00 | O | X | 14:00 | O | O |

O | X | O | O | ||

14:45 | X | X | 14:45 | O | O |

O | O | O | O |

**Programming project: Option pricing**

(after a programming project by Andreas Eberle for the lecture Algorithmic mathematics)

The projects is about numerically computing the fair price of European call and Up-and-out-barrier options in the binomial model and the discrete Black-Scholes model.

You can use the following Mathematica notebook, but of course also any other mathematical software or programming language.

**References:**

- H. Föllmer, A. Schied:
*Stochastic Finance*, de Gruyter - F. Delbaen, W. Schachermayer:
*The Mathematics of Arbitrage*, Springer - S. Shreve:
*Stochastic Calculus for Finance I, II*, Springer - M. Musiela, M. Rutkowski:
*Martingale Methods in Financial Modelling* - Model independent bounds for option prices - a mass transport approach, Mathias Beiglböck, Pierre Henry-Labordere, and Friedrich Penkner
- On a problem of optimal transport under marginal martingale constraints, Mathias Beiglböck and Nicolas Juillet
- An explicit martingale version of Brenier's theorem, Pierre Henry-Labordere and Nizar Touzi
- Complete Duality for Martingale Optimal Transport on the line, Mathias Beiglböck, Marcel Nutz, and Nizar Touzi