MATH-470 / 5 crédits

Enseignant: Schmutz Michael

Langue: Anglais

## 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, financial mathematics, theory of (no-)arbitrage

## 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

 Office hours Yes Assistants No Forum No Others

## 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.

## 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

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