# Fiches de cours

## Martingales in financial mathematics

Schmutz Michael

English

#### 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

#### 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 viable market modeled 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 chosen asset price models.
• Optimize the calibration of chosen asset price models.

#### 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 Office hours: Friday, 13:00-14:00

#### Resources

No

##### 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. (2010), Change of Time and Change of Measure, World Scientific Publishing, Singapore.

### Dans les plans d'études

• Ingénierie mathématique, 2018-2019, Master semestre 2
• Semestre
Printemps
• Forme de l'examen
Oral
• Crédits
5
• Matière examinée
Martingales in financial mathematics
• Cours
2 Heure(s) hebdo x 14 semaines
• Exercices
2 Heure(s) hebdo x 14 semaines
• Ingénierie mathématique, 2018-2019, Master semestre 4
• Semestre
Printemps
• Forme de l'examen
Oral
• Crédits
5
• Matière examinée
Martingales in financial mathematics
• Cours
2 Heure(s) hebdo x 14 semaines
• Exercices
2 Heure(s) hebdo x 14 semaines
• Mathématiques - master, 2018-2019, Master semestre 2
• Semestre
Printemps
• Forme de l'examen
Oral
• Crédits
5
• Matière examinée
Martingales in financial mathematics
• Cours
2 Heure(s) hebdo x 14 semaines
• Exercices
2 Heure(s) hebdo x 14 semaines
• Mathématiques pour l'enseignement, 2018-2019, Master semestre 2
• Semestre
Printemps
• Forme de l'examen
Oral
• Crédits
5
• Matière examinée
Martingales in financial mathematics
• Cours
2 Heure(s) hebdo x 14 semaines
• Exercices
2 Heure(s) hebdo x 14 semaines
• Mathématiques pour l'enseignement, 2018-2019, Master semestre 4
• Semestre
Printemps
• Forme de l'examen
Oral
• Crédits
5
• Matière examinée
Martingales in financial mathematics
• Cours
2 Heure(s) hebdo x 14 semaines
• Exercices
2 Heure(s) hebdo x 14 semaines

LuMaMeJeVe
8-9
9-10
10-11
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
En construction
Cours
Exercice, TP
Projet, autre

### légende

• Semestre d'automne
• Session d'hiver
• Semestre de printemps
• Session d'été
• Cours en français
• Cours en anglais
• Cours en allemand