- Lecturer
- Prof. Dr. Erich Walter Farkas
- Coordinator
- Bálint Gersey

This course gives a first introduction to the main modelling ideas and mathematical tools from mathematical finance. It mainly aims at non-mathematicians who need an introduction to the main tools from stochastics used in mathematical finance. However, mathematicians who want to learn some basic modelling ideas and concepts for quantitative finance (before continuing with a more advanced course) may also find this of interest. The main emphasis will be on ideas, but important results will be given with (sometimes partial) proofs.

Topics to be covered include

- financial market models in finite discrete time
- absence of arbitrage and martingale measures
- valuation and hedging in complete markets
- basics about Brownian motion
- stochastic integration
- stochastic calculus: Itô's formula, Girsanov transformation, Itô's representation theorem
- Black-Scholes formula

**Lecture notes** that will be closely followed during the lectures will be sold after the second lecture on Wednesday 18/09/2019 in HG D1.1 and on Friday 20/09/2019 in HG G47.2 for CHF 15. The lecture notes will *not* be available in electronic form.

Results and facts from measure-theoretic probability theory as given in the book Probability Essentials by Jean Jacod and Philip Protter will be used freely. The book can be downloaded from Springer (within the ETH network or using VPN) for free. Especially participants without a direct mathematical background are strongly advised to familiarise themselves with those tools before (or *very quickly* during) the course.

A possible alternative to the above textbook is the English or German version of the ETH lecture notes for the standard course on *Probability Theory*. These lecture notes can also be purchased during the Präsenz hours.

For those who are not sure about their background, we suggest to have a look at the exercises in Chapters 8, 9, 22-25, 28 in the Probability Essentials book. If these pose problems, you will have a hard time during the course. So be prepared.

Time | Room | Lecturer |
---|---|---|

Tuesday 12:00-13:00 | HG D 1.1 | Prof. Dr. Walter Farkas |

Wednesday 10:00-12:00 | HG D 1.1 | Prof. Dr. Walter Farkas |

Time | Room | Assistant |
---|---|---|

Friday 08:00-10:00 | HG D 7.1 | Hanna Wutte |

Friday 10:00-12:00 | LFW E 13 | Bálint Gersey |

New exercise sheets will be uploaded here on Tuesdays before the exercise class along with a model solution to the exercise sheet from the previous week.

Note that handing in your solutions is not obligatory, but being able to solve the exercises independently goes a long way towards good exam performance. In case you decide to do so, please hand in your solutions by the following Tuesday 18:00 to your assistant's box next to HG G 53.2. Your solutions will be corrected and returned in the following exercise class or, if not collected, returned to the box next to HG G 53.2.

Exercise sheet | Due by | Solutions | Extra material |
---|---|---|---|

Exercise sheet 1
S&P 500 |
September 24, 2019 | Solution sheet 1
R code Python code |
Measure theory |

Exercise sheet 2 | October 1, 2019 | Solution sheet 2 | Basics of financial modelling |

Exercise sheet 3 | October 8, 2019 | Solution sheet 3 | Arbitrage and Equivalent Martingale Measures: definitions |

Exercise sheet 4 | October 15, 2019 | Solution sheet 4 | Arbitrage and Equivalent Martingale Measures: 1st Fundamental Theorem of Asset Pricing |

Exercise sheet 5 | October 22, 2019 | Solution sheet 5 | RN derivatives and density processes, complete and incomplete markets, second FTAP |

Exercise sheet 6 | October 29, 2019 | Solution sheet 6 | Read the lecture notes on attainable payoffs and complete markets. In particular, make sure you understand the binomial model example on pages 55-60. See the links with continous time models, in particular the simple Black Scholes model. |

Exercise sheet 7 | November 5, 2019 | Solution sheet 7 | Stochastic Processes in continuous time |

Exercise sheet 8 | November 12, 2019 | Solution sheet 8 | Normal distributions, square and sharp bracket processes |

Exercise sheet 9 | November 19, 2019 | Solution sheet 9 | Stochastic Integration (notes used during the lecture on 13/11/2019 by Balint Gersey) |

Exercise sheet 10 | November 26, 2019 | Solution sheet 10 | Read chapter 6 on Ito's formula. Make sure you understand the examples given in the lecture concerning Ito's formula. |

Exercise sheet 11 | December 3, 2019 | Solution sheet 11 | Girsanov theorem (gives the ELMM in BS model and therefore proves the arbitrage free property of BS model), Ito's representation theorem (used to prove completeness of BS model and prove existence of hedging strategy) |

Exercise sheet 12 | December 10, 2019 | Solution sheet 12 | Black Scholes model: construction of unique EMM, hedging of general payoffs |

Exercise sheet 13 | December 17, 2019 | Solution sheet 13 | Black Scholes model continued |

Exercise sheet 14 | December 18, 2019 | Solution sheet 14 | Extensions of Black Scholes: stochastic interest rate and local volatility models |

Your grade for the course will be based solely on the written final exam. The exam will cover **all material discussed during the lectures and the exercise classes (except if stated otherwise)**.

Some old exams can be found here on the homepage of Group 3, but students are highly discouraged from preparing from the old exams only. It will not be enough to ensure a passing grade.

For questions before the exam, please use the semester break Präsenz.

During the second and the third week of the semester after the exam, you have the possibility to look at your exam during the regular Präsenz hours.

**Probability Essentials**by Jean Jacod and Philip Protter (Springer, 2003)**Stochastic Finance: An Introduction in Discrete Time**by Hans Föllmer and Alexander Schied (Walter de Gruyter, 2011).**Introduction to Stochastic Calculus Applied to Finance**by Damien Lamberton and Bernard Lapeyre (Chapman-Hall, 2008)

For computational aspects, you can consult for example the following books:

**Finanzderivate mit MATLAB**by Michael Günther and Ansgar Jüngel (Vieweg+Teubner, 2010)**The Concepts and Practice of Mathematical Finance**by Mark Joshi (Cambridge University Press, 2008)**Tools for Computational Finance**by Rüdiger Seydel (Springer, 2009)**Binomial Models in Finance**by John van der Hoek and Robert J. Elliott (Springer, 2006)