The course is based on the study of the main tools of probability theory that are used in finance and financial engineering. Although the applications are related to these domains and many examples will be studied in class, it is mainly about mathematics. The main objective of this course is to make the student comfortable with the mathematical concepts commonly used in financial engineering: sigma-field, martingale, stopping time, Brownian motion, stochastic integral, diffusion processes,and risk neutral measures. The course is divided into two main blocks: discrete and continuous models. Each of these blocks is again subdivided in two parts: a more theoretical section where the mathematical concepts are introduced and a second section in which the mathematical tools are used.
Chapter 1. Fundamental set, sigma-field, measurable
function, probability measure
Probability space
Exercises 1
Chapter 2.
Stochastic processes, filtration, stopping time
Stochastic
processes
Exercises 2
Chapter 3. Conditional
expectation
Conditionnal
expectation
Exercises 3
Chapter 4. Discrete time
martingales
Martingales
Exercises 4
Chapter 5. Introduction to risk neutral measures
Binomial model
Exercises 5
Chapter 6. Replication and
risk neutral measures
Discrete
time models
Exercises 6
Chapter. Snell Envelope
American contingent claim
Exercises 7
Chapter 8. Convergence of Random Variable Series
Convergence
Chapter 10. Ito's Integral
Stochastic integral
Exercises 10
Int_stoch.xls
Chapter 11. Stochastic
Differential Equations and Ito's Lemma
SDE
Exercices 11
Chapter 13. Applications du théorème de représentation des martingales
Martingale
representation theorem and hedging
Exercises 13
Chapter 14 Autre application
Bonds