Probability Theory and Stochastic Processes for Finance
Docente: Claudio Ortelli
Docente: Fabio Trojani
Tipo di corso: Master +2
Valore in crediti ECTS: 6
Riferimenti bibliografici sul sito della biblioteca (CoRe)
Academic year 2010/2011 - Fall semester
This course is offered within the PhD in Finance programme. Master students interested in quantitative finance are welcome to join the class.
Course description
The objective of this course is to introduce the probability theory and the theory of stochastic processes, so as to provide graduate students of finance with the basic tools necessary to understand a recent research article on, for instance, derivative theory or asset pricing. The course starts by introducing some basic ideas of probability theory, including some key ingredients of measure and integration theory. Subsequently, the definition of a random variable and the formal idea of stochastic independence will be discussed, together with the main properties of conditional expectations and martingales. The lectures will then cover some of the most important stochastic processes in finance, in discrete and continuous time, with a special focus on the Brownian motion process. The course ends with the definition and construction of the stochastic integral and with the proof of Itô´s formula for stochastic differentials.
Contents
1. Introduction to probability theory
a) Sigma algebras, measurable spaces and filtrations
b) Probability and probability spaces
c) Measurability and random variables
d) Some computation rules
e) Lebesgue integral and expected value
f) Stochastic independence
2. Conditioning and conditional expectation
a) Definition
b) Properties
c) Martingales
d) Some theorems on martingales
3. Discrete time and continuous time stochastic processes
a) ARMA processes
b) Random Walks
c) Poisson process
d) Brownian motion
i. Properties of Brownian motion
ii. Functionals of Brownian motions
4. Stochastic integrals
a) Definition
b) Properties
c) Itô´s Lemma
d) Some examples
i. Geometric Brownian motion
ii. Quadratic variation of Brownian motion
Textbooks
S. Shreve, Stochastic Calculus in Finance, Springer-Verlag, New York, 2004.
For probability:
A. F. Karr, Probability, Springer Verlag, New York, 1993.
Other references for Finance:
D. Lamberton and B. Lapeyre, Introduction to Stochastic Calculus Applied to Finance, Chapman and Hall, London,1996.
For Probability and stochastic processes:
G.R. Grimmett and D.R. Stjrzaker, Probability and Random Processes, 2nd ed., Clarendon Press, Oxford, 1995.