80-646 Stochastic Calculus I

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.

Mathematical background (3 weeks)

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
                        Exercises 4


Discrete Time Market Models (3 weeks)

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 

Stochastic Calculus (3 weeks)

                  Chapter 8. Convergence of Random Variable Series

                  Chapter 9. Brownian Motion
                        Browian Motion
                        Exercises 9
                        brownien1.xls      brownien2.m     brownien3.m  

Chapter 10. Ito's Integral
Stochastic integral
                        Exercises 10


Chapter 11. Stochastic Differential Equations and Ito's Lemma
                        Exercices 11  

Applications to Financial Engineering (environ 3 séances)

                    Chapter 12. Girsanov Theorem and Change of Measures
Exercises 12   

Chapter 13. Applications du théorème de représentation des martingales
  Martingale representation theorem and hedging
                          Exercises 13

Chapter 14 Autre application