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

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

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

• https://go.epfl.ch/MATH-470