ECTS credits
8 credits
Semester
Spring
Prerequisites
 Content of the course unit Finance in the DDEFi track (see syllabus)
 Probabillity at Master level (1st year): theory of probability and it is recommended to have knowledge about stochastics processes in discrete or continuous time.
Learning objectives
 Apply stochastic calculus to price financial products such as options.
 Learn the standard models used in mathematical finance.
 Know the basic data science models and their usage
Description of the programme
This course unit consists of three courses (of 24 hours each): Stochastic calculus, Interest rate models, and Volatility modeling, 24h each and is complemented by the third part of the data project (9 hours course and 12 hours project) devoted to models and their validation.
Stochastic calculus
 Gaussian variable and stochastic processes
 Brownian motions
 Stochastic integration and semimartingales
 Stochastic differential equations
 Parabolic partial differential equations and semigroups
 Measure change and Girsanov theoremIntroduction to financial mathematics
Interest rate models
 A Mathematical Toolkit
 Interest rates, swaps and options
 Onefactor ShortRates Models
 Twofactor ShortRates Models
 The HealthJarrowMorton (HJM) Model
 The change of numeraire
 Derivatives Pricing under the Libor Market Model
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
Data science projects. Part 3: Models and validation
 Projects and models
 The BiasVariance tradeoff
 Feature Selection
 Feature Engineering
 Defining a metric
 Models and applications
 Regressions (linear, polynomial, penalized et logistic)
 Decision trees (random forest and gradient boosting)
 Focus on Natural Language Processing (NLP)
Generic central skills and knowledge targeted in the discipline
 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
 Know how to use data science models (Natural Language Processing in particular) in business projects.
How knowledge is tested
 Stochastic calculus (written exam): 25%
 Interest rate models (project): 25%
 Volatility models (project): 25%
 Data project (project): 25%
Bibliography
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
Interest rate models

Brigo, D., & Mercurio, F. (2007). Interest rate modelstheory and practice: with smile, inflation and credit. Springer Science & Business Media

Privault, N. (2012). An elementary introduction to stochastic interest rate modeling. World Scientific.
Volatility models

El Karoui, N. (2004) Couverture des risques dans les marchés financiers. Ecole Polytechnique
Data science projects
 Zeng, A and Casari, A. Feature Engineering for Machine Learning. O'Reilly Media.
 Müller, A. and Guido, S. Introduction to Machine Learning with Python. O'Reilly Media.
Teaching team
 Stochastic calculus: Sébastien Darses (AixMarseille Université)
 Interest rate models: Abderrahim Ben Jazia (RSM Paris)
 Volatility models: Ismail Akil (Morgan Stanley)
 Data science projects: Alexandre Chirié et Maxilimilen Défourné (Mantiks)
Sustainable Development Goal
Partnerships for the goals
 Total hours of teaching100h
 Master class81h
 19h