# Martingales in financial mathematics

## Summary

The aim of the course is to apply the theory of martingales in the context of mathematical finance. The course provides a detailed study of the mathematical ideas that are used in modern financial mathematics. Moreover, the concepts of complete and incomplete markets are discussed.

## Content

- Discrete time models and the Fundamental Theorem of Asset Pricing

- Fundamental results
- Binomial- and trinomial model
- The Snell envelope, optimal stopping, and American options

- Geometric Brownian motion and the Black-Scholes model

- Option pricing and hedging
- Exotic options

- On the theory of (no-)arbitrage in continuous time

- Selected topics on

- Local- and stochastic volatility models
- Stochastic interest rates
- Lévy driven models
- New trends in financial mathematics
- Deep hedging

## Keywords

martingales, financial mathematics, theory of (no-)arbitrage

## Learning Prerequisites

## Recommended courses

Stochastic calculation

## Important concepts to start the course

Stochastic calculation

## Learning Outcomes

By the end of the course, the student must be able to:

- Explore in detail the use of martingales in financial mathematics.
- Prove a criteria for absence of arbitrage in a model based on a finite probability space and state an analogous general result.
- Prove a criteria for completeness of a market model based on a finite probability space and state an analogous general result.
- Explain the difference and the resulting consequences between claims and American options.
- Derive prices for some financial derivatives based on several different models.
- Derive different hedging strategies for some financial derivatives based on several different models.
- Analyze the choice of asset price models according to different criteria.
- Optimize the calibration of chosen asset price models.
- Prove a criteria for completeness of a viable market modeled based on a finite probability space and state an analogous general result.

## Assessment methods

Exam oral

Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés.

## Supervision

Office hours | Yes |

Assistants | No |

Forum | No |

Others |

## Resources

## Bibliography

- Lamberton, D. and Lapeyre, B. (2008), Introduction to Stochastic Calculus Applied to Finance, Second Edition, Chapman and Hall, London.
- Shiryaev, A.N. (1999), Essentials of Stochastic Finance: Facts, Models, Theory, World Scientific Publishing, Singapore.
- Barndorff-Nielsen, O.E. and Shiryaev, A.N. (2015), Change of Time and Change of Measure, Second Edition, World Scientific Publishing, Singapore.
- Eberlein, E. and Kallsen, J. (2019), Mathematical Finance, Springer Finance, Cham.
- Jarrow, R.A. (2021), Continuous-Time Asset Pricing Theory, Second Edition, Springer Finance, Cham.

## Ressources en bibliothèque

- Introduction to Stochastic Calculus Applied to Finance / Lamberton
- Essentials of Stochastic Finance / Shiryaev
- Continuous-Time Asset Pricing Theory / Jarrow
- Mathematical Finance / Eberlein & Kallsen
- Change of Time and Change of Measure / Barndorff-Nielsen

## Moodle Link

## In the programs

**Semester:**Spring**Exam form:**Oral (summer session)**Subject examined:**Martingales in financial mathematics**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Oral (summer session)**Subject examined:**Martingales in financial mathematics**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Oral (summer session)**Subject examined:**Martingales in financial mathematics**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Oral (summer session)**Subject examined:**Martingales in financial mathematics**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

## Reference week

Mo | Tu | We | Th | Fr | |

8-9 | |||||

9-10 | |||||

10-11 | |||||

11-12 | |||||

12-13 | |||||

13-14 | |||||

14-15 | |||||

15-16 | |||||

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20-21 | |||||

21-22 |