Martingales in financial mathematics

MATH-470 / 5 credits

Teacher: Schmutz Michael

Language: 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
  • 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

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

Reference week

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