=====Course unit: Mathematical finance =====
==== Course metadata ====
  * Title in French: Mathématiques financières
  * Course code: tba
  * ECTS credits: 3
  * Teaching hours: 72h
  * Type: specialized course
  * Language of instruction: English
  * Coordinator: tba
  * Instructor(s): Sébastien Darses (AMU), Ismaïl Akil (tba), Abderrahim Ben Jazia (RSM Paris)
  * //Last update 27/08/2021 by C. Pouet//

==== Brief description ====
The aim of the course is to provide students with mathematical methods that allow valuating financial assets.

This course unit is divided into three parts:
  * ** Stochastic calculus and introduction to the Black-Scholes model ** (24 hours) taught by Sébastien Darses.
  * ** Volatility models ** (24 hours) taught by Ismaïl Akil.
  * ** Interest rate models ** (24 hours) taught by Abderrahim Ben Jazia.

==== Learning outcomes ====

  * Understand stochastic calculus and know how to apply its main results
  * Know how to apply stochastic methods to price financial products
  * Understand the mathematical contexts under which the classical financial mathematics models hold
  * Know and understand the relevance and limits of financial mathematics models
  * Understand the impact of volatility on the profit and losses of a hedged position
  * Know how to build numerical methods for pricing financial products


==== Course content ====


=== Stochastic calculus and introduction to the Black-Scholes model===

  - Gaussian variable and stochastic processes
  - Brownian motions
  - Stochastic integration and semi-martingales
  - Stochastic differential equations
  - Parabolic partial differential equations and semigroups
  - Measure change and Girsanov theorem
  - Introduction to financial mathematics

=== Volatility models====
  - Elementary financial mathematics notions
  - PDE: Black Scholes and risk neutral measure
  - Dupire’s local volatility: advantages and drawbacks
  - Stochastic volatility (Heston and SABR)
  - Tutorial: discretization of the Heston’s model

=== Interest rate models===
  - A Mathematical Toolkit
  - Interest rates, swaps and options
  - One-factor Short-Rates Models
  - Two-factor Short-Rates Models
  - The Health-Jarrow-Morton (HJM) Model
  - The change of numeraire
  - Derivatives Pricing under the Libor Market Model

==== Bibliography ====
Check the availability of the books below at [[https://documentation.centrale-marseille.fr/|Centrale Marseille library]].
 - Stochastic calculus
    * Evans, L. (2010). An Introduction to Stochastic Differential Equation. American Mathematical Society.
    *  Le Gall, J.-F. (2006). Intégration, Probabilités et Processus Aléatoires. Ecole Normale Supérieure de Paris
 - Volatility models
    * El Karoui, N. (2004) Couverture des risques dans les marchés financiers. Ecole Polytechnique
 - Interest rate models
    * Brigo, D., & Mercurio, F. (2007). Interest rate models-theory and practice: with smile, inflation and credit. Springer Science & Business Media 
    * Privault, N. (2012). An elementary introduction to stochastic interest rate modeling. World Scientific.