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, semimartingales, 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.
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
- Change of Time and Change of Measure / Barndorff-Nielsen
- Mathematical Finance / Eberlein & Kallsen
- Continuous-Time Asset Pricing Theory / Jarrow
Moodle Link
Dans les plans d'études
- Semestre: Printemps
- Forme de l'examen: Oral (session d'été)
- Matière examinée: Martingales in financial mathematics
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
- Type: optionnel
- Semestre: Printemps
- Forme de l'examen: Oral (session d'été)
- Matière examinée: Martingales in financial mathematics
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
- Type: optionnel
- Semestre: Printemps
- Forme de l'examen: Oral (session d'été)
- Matière examinée: Martingales in financial mathematics
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
- Type: optionnel
- Semestre: Printemps
- Forme de l'examen: Oral (session d'été)
- Matière examinée: Martingales in financial mathematics
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
- Type: optionnel
- Semestre: Printemps
- Forme de l'examen: Oral (session d'été)
- Matière examinée: Martingales in financial mathematics
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
- Type: optionnel
Semaine de référence
Lu | Ma | Me | Je | Ve | |
8-9 | |||||
9-10 | |||||
10-11 | |||||
11-12 | |||||
12-13 | |||||
13-14 | |||||
14-15 | |||||
15-16 | |||||
16-17 | |||||
17-18 | |||||
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19-20 | |||||
20-21 | |||||
21-22 |