Sumários

Lévy processes

20 Outubro 2021, 10:00 • João Guerra

Lévy processes: Definition and association to infinitely divisible distributions. 

Lévy-Khintchine formula for a Lévy process and the characteristic exponent of a Lévy process. 
Examples. 
Exercise on the Poisson Process. 
Stable processes. 

Stable Distributions

15 Outubro 2021, 10:00 • João Guerra

Stable distributions. Definition and relationship of sable distributions with the central limit theorem. 

Characteristic functions of stable distributions. 
Moments properties for stable distributions. 
The heavy  tails property of stable distributions. 

Characteristic functions and infinitely divisible distributions

13 Outubro 2021, 10:00 • João Guerra

Characteristic functions. Definitions and basic properties. Examples and exercise. 

Infinitely divisible distributions. Examples. 
Lévy measure. Definition and basic properties. Exercise. 
Lévy-Khintchine formula. Sketch of the proof. 

Introduction to Lévy processes

8 Outubro 2021, 10:00 • João Guerra

Example of how to compute the implied volatility. 

Characteristic function. 
Infinitely divisible distributions and Lévy-Khintchine formula. 
The Lévy characteristic exponent. 
Lévy measures: definition and simple examples. 
Activity, path variation properties and moments of Lévy processes related to the Lévy measure. 
Basic simple Lévy models in Finance: Black-Scholes model, Merton model

Presentation of the course and introduction to Lévy Processes

6 Outubro 2021, 10:00 • João Guerra

Presentation of the course: programme, bibliography, assessment. 

Motivation for Lévy processes and applications. 
Limitations of the Black-Scholes model. 
The implied volatility smile. 
Definition of Lévy processes and simple examples.