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
In the programs
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Martingales in financial mathematics
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Martingales in financial mathematics
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Martingales in financial mathematics
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Martingales in financial mathematics
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Martingales in financial mathematics
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional