# Introduction to derivative products - 4MMIPD6

• #### Number of hours

• Lectures 16.5
• Projects -
• Tutorials 16.5
• Internship -
• Laboratory works -
• Written tests -

ECTS 3.0

## Goal(s)

The goal of this course is to present the concepts of pricing and hedging products in continuous time. At the end of the lecture, students will know how to handle vanilla options in the one dimensional Black Scholes model. In particular they shall master the following notions :
Brownian Motion;
Continuous time martingale;
Wienner's and Itô's Integral;
Itô's formulas.

Responsible(s)

Herve GUIOL

## Content(s)

• Stochastic processes and Brownian motion
• Gaussian vectors
• Continuous time stochastic processes
• Brownian motion : definition and first properties
• Continuous time martingales
• Filtrations and stopping times
• Martingales : stopping time theorem, applications to Brownian motion, strong Markov property
• Stochastic integral and Itô formula
• Wiener's integral
• Itô's integral
• Itô's formula
• Applications of Itô's formula : representing the Brownian martingales and the Cameron Martin theorem
• Stochastic differential equations
• The Black Scholes model
• The underlying asset
• Replicating strategies
• Pricing and hedging options

Prerequisites

A first course in probability and the Stochastic processes and application to Finance lecture of the previous semester.

Test

Continuous assessment : presence, participation, homework.

Regular & Second Session :
A Written Exam : 2h

1. Lexique
CC = note de contrôle continu
E1 = note de l'examen de session 1
E2 = note de l'examen de rattrapage
1. Notes transmises à la scolarité
N1=(E1+CC)/2 # Note finale de session 1 si examen en présenciel
ou
N1=CC # Si pas d'examen en présenciel

N2=(E2+CC)/2 # Note de session 2

Calendar

The course exists in the following branches:

• Curriculum - Financial Engineering - Semester 8
see the course schedule for 2023-2024

Additional Information

Course ID : 4MMIPD6
Course language(s):

The course is attached to the following structures:

You can find this course among all other courses.

Bibliography

El Karoui N. & Gobet E. Les outils stochastiques des marchés financiers, Edition de l'Ecole Polytechnique. ISBN 978-2-7302-1579-4

Lamberton D & Lapeyre B. Introduction au calcul Stochastique appliqué à la finance, Ellipse. ISBN 2-7298-4782-0