Master Probabilité et Finance

Core courses

5 mandatory courses [14 weeks, 4 hours a week, 6 ECTS / Course]

• Elements of functional analysis and measure theory: topology, measure and integration theory – Lebesgue measure, Lebesgue integral abstract measure spaces, classical functional spaces – Banach, Hilbert, bounded and unbounded operators, weak topologies
• Probability theory and stochastic process :  Conditional expectations, discrete-time stochastic processes [Kolmogorov extension theorem, canonical processes, stopping times], martingales [convergence theorems, main stopping theorems, applications], continuous state-space Markov chains [transition kernels, main convergence theorems for atomic chains]
  1° Discrete time and space Markov chains : the Markov and strong Markov property, applications; stationary measures, recurrence and transience, ergodic theorem and convergence to equilibrium, applications and simulations, MCMC.
  2° Conditional expectation, main properties, martingales in discrete time : stopping theorem, convergence theorems, applications and simulations, optimal stopping.
• Mathematical Statistics : introduction to decision theory, estimators (M- and Z- estimators, sufficiency, optimality), tests (constructions, Uniformly Most Powerful test for simple and composite alternatives), asymptotic statistics (consistency, asymptotic normality), asymptotic optimality of maximum likelihood estimation, generalized likelihood ratio tests, introduction to linear and non-linear regression, Bayesian statistics.
• Optimization: Optimization in Euclidean spaces (general case and convex case), Optimization under equality and inequality constraints, KKT optimality conditions, convex case, alternative statements (Farkas Lemma), duality in linear and convex optimization, first and second order descent algorithms, subdifferential, subgradient methods, proximal point method, conjugate functions, necessary and sufficient optimality conditions, modeling applications as optimization problems. The course provides the necessary prerequisites in Optimization for advanced courses like Machine Learning or Financial Mathematics.
• Python for Data Sciences: Python is a high-level programming language, which is primarily used in thefields of data science and artificial intelligence, among others. Its design iscentered around readability, which lowers learning barriers for newcomers.
  The objective of this course is to introduce Python in the context of data sciences.
  Foundational topics students should be familiar with before enrolling:
  1.      Familiarity with Basic Programming Concepts
  2.      Basic Mathematical Foundation (Algebra and Statistics)
  3.      Intellectual curiosity

3 mandatory courses :

• Markov Processes and applications - 5 ECTS: In this course, we study Markov processes in continuous time, taking values in discrete spaces. Markov processes are random processes where the evolution of the process depends only on its current state and not on its past. These processes are widely used in various fields, including physics, biology, chemistry, economics, and information theory. They allow us to model the random evolution of variables that, in the context of this course, can take values in a finite or countable state space.
  Specifically, we explore Poisson processes, jump processes, the Markov property, Kolmogorov equations, the ergodic theorem, and long-term behavior. Additionally, the simulation of these processes is addressed through numerical sessions.
• Introduction to Machine Learning - 5 ECTS: Introduction to Supervised Learning, Linear Models, Generalizes Linear Models, Support Vector Machines, Generative Learning (Gaussian discriminant analysis, Mixture models, Naïve Bayes model), Tree-based and Ensemble Models (Decision tree, random forests, boosting, bagging), learning theory (Union bound, Hoeffding inequality, Empirical risk, Probably approximately correct framework), Introduction to Unsupervised Learning (Gaussian Mixture Model, Expectation-Maximization, k-means clustering, Hierarchical clustering), Dimensionality Reduction (Principal component analysis (Eigenvalues/vectors, Spectral theorem, PCA algorithm) ● Independent component analysis (Bell and Sejnowski ICA algorithm,  Linear discriminant analysis, Factor analysis), Neural Networks (Basic architecture: multi-layer perceptron, Activation functions -Sigmoid, tanh, relu, lrelu etc.-, Losses (cross-entropy, l1/l2 loss, binary cross entropy, focal loss, hinge-loss etc.-, Back propagation, learning rate
  Introduction to supervised/unsupervised learning, classification vs regression; Least square regression (ridge/LASSO); Classification with logistic regression; Stochastic gradient methods for training; Unsupervised learning with principal component analysis; Classification with support vector machines; Classification and regression with trees and ensemble methods; Classification and regression with Neural networks; Clustering methods for unsupervised learning
• Databases - 5 ECTS: ● Introduction, Data Models: Relational Model ● Relational Algebra, Relational Calculus ● SQL (Structured Query Language) ● Schema Refinement and Normal Forms ● Storage and Indexes ● Query Processing (Evaluation of Operators)Query Optimization (query plans, algebraic  ● equivalence, plan space, cost estimation)
• FLE/Languages - 3 ECTS

4 different elective courses:

CHOICE 1:

• Linear Time Series - ENSAE Paris - École d'ingénieurs pour l'économie, la data science, la finance et l'actuariat - 2,5 ECTS

The first part of this course is devoted to studying univariate time series: first we present the principal statistical concepts, then estimation methods and tests; we examine the non-stationarity problem by studying the main unit root tests from a practical angle. The course is illustrated with practical examples. The second part of the course is devoted to studying stationary VAR models: we briefly present the general framework of multivariate stationary series before developing the specific case of VAR models. Finally, we take a quick look at the principles of cointegration.
The prerequisites of Linear Time Series course are the basic probability and statistics notions, such as conditional expectation, Gaussian vectors, limit theorems, estimation, testing statistical hypotheses and linear regression.

• Financial mathematics - ENSAE Paris - École d'ingénieurs pour l'économie, la data science, la finance et l'actuariat - 2,5 ECTS

The aim of this course is to present the mathematical concepts used to model and value derivatives in finance. The presentation of the course will be mathematically rigorous, but certain results from stochastic calculus will be admitted as read, to be demonstrated later in the third-year stochastic calculus course. After obtaining a mathematical definition of the concept of arbitrage in a financial market, we will study discrete models for each evolutionary tree of assets, giving useful intuitions for the study of continuous-time models. Finally, with the help of the theory of stochastic calculus , we will present asset valuation in the framework of the Black & Scholes model. It is advisable to have taken the course on Markov chains for a better understanding of the concepts from stochastic calculus and to take the simulation course in parallel.

CHOICE 2:

• Financial mathematics - ENSAE Paris - École d'ingénieurs pour l'économie, la data science, la finance et l'actuariat - 2,5 ECTS

The aim of this course is to present the mathematical concepts used to model and value derivatives in finance. The presentation of the course will be mathematically rigorous, but certain results from stochastic calculus will be admitted as read, to be demonstrated later in the third-year stochastic calculus course. After obtaining a mathematical definition of the concept of arbitrage in a financial market, we will study discrete models for each evolutionary tree of assets, giving useful intuitions for the study of continuous-time models. Finally, with the help of the theory of stochastic calculus , we will present asset valuation in the framework of the Black & Scholes model. It is advisable to have taken the course on Markov chains for a better understanding of the concepts from stochastic calculus and to take the simulation course in parallel.

• Emerging Risks and Transformations in Contemporary Insurance - ENSAE Paris - École d'ingénieurs pour l'économie, la data science, la finance et l'actuariat - 2,5 ECTS

This course aims to introduce the main challenges facing the insurance sector in the context of contemporary risks and technological change. It examines how so-called emerging risks (climate risk, pandemic risk, cyber risk, and others) undermine the fundamental mechanisms of insurance, such as the pooling principle, and explores how actuarial methodologies can be adapted to this new landscape, notably through machine learning and Bayesian approaches.

CHOICE 3:

• Numerical Analysis I & 2 - TELECOM Paris  - 5 ECTS

Numerical methods come into play in applied mathematics as soon as the numerical value for some quantity of interest is needed, for which no analytic expression is available.  This quantity may be e.g. anintegral or the solution of a differential equation.
The scope of application of this course thus embraces such diverse fields as Physics, Biology, Economy or Finance.  The role played by numerical
analysis has substantially grown these past few years, together with extensive computing power.

Numerical methods may be divided into two broad classes: deterministic methods on the one hand, based on discrete approximations of integrals or state evolution equations, and stochastic methods on the other hand, which consist in random sampling under an appropriate probability distribution in order to approximate an integral or atrajectory. This may be done by independent sampling (Monte-Carlo methods) or by simulating a well chosen Markov Chain (MCMC methods).

The purpose of this course is to introduce the students to essential methods in Numerical Analysis and to give them the key tools for understanding the mathematical principles behind.

CHOICE 4:

• Transport and Distribution - 2,5 ects

This course, jointly organized by the Applied Mathematics and Mathematics departments, is also referenced MAT567.
The aim of this course is to present models of particle transport and diffusion, which can be found in many energy-relevant fields of application. For example, the chain reaction mechanism in nuclear reactors, the greenhouse effect in climatology, radiative transfer in thermodynamics or astrophysics, and certain models of structured population dynamics in biology all fall within this theme.
After a mathematical presentation of these models, we'll show that diffusion is the limit of transport in a strongly collisional regime, and explain the notion of mass or critical size. Finite-difference and Monte-Carlo methods of numerical resolution will be introduced.

Bibliography :
Dautray R., (1989). Méthodes probabilistes pour les équations de la physique, Eyrolles, Paris
Dautray R., Lions J.-L., (1988). Analyse mathématique et calcul numérique pour les sciences et les techniques, Masson, Paris
Perthame B., (2007). Transport equations in biology, Birkhäuser, Basel
Planchard J. (1995). Mathematical methods in neutronics. Collection de la Direction des Études et Recherches d'EDF, Eyrolles.
Pomraning G., (1973).

• Optimal Structural Design - 2,5 ects

Thanks to immense progress over the last few decades in mechanical modeling, numerical analysis and, above all, the power of computers, it is now relatively easy to design a structure (e.g. a bridge, an aircraft wing, a car body, etc.) using computer simulation and calculation. However, when designing a new structure, it is often necessary to try out several ideas and modify them to improve them. This task can be extremely tedious if it is entirely manual and relies on the designer's intuition alone. This is why it is necessary to propose methods for the automatic optimization of structures.

The aim of this course is to lay the foundations of mathematical models and numerical algorithms for optimal structural design. In particular, we will study the optimization of structural shapes, with a view, for example, to minimizing their weight under minimum mechanical strength constraints. The course will be illustrated by the use of structural optimization software in small classes.

Further information is available at:
http://www.cmap.polytechnique.fr/~allaire/cours_X_majeure.html
http://www.cmap.polytechnique.fr/~MAP562/
https://moodle.polytechnique.fr/course/view.php?id=

Level required: The course will begin with an overview of the mathematical background required.
Evaluation methods: Continuous assessment (one FreeFem++ project/script due each week), no final exam

Internship

Each M1 student must complete an internship spanning a minimum duration of 8 weeks as an integral part of their education, for which they receive 7 ECTS. This internship can be completed at a college / university, in an industrial research laboratory, or in a company. The start of the internship is scheduled for May.

This mandatory internship serves as a crucial bridge between theoretical knowledge and practical application. It not only provides students with invaluable professional insights and networking opportunities, but also helps them refine or solidify their academic and professional goals for their M2 year and beyond. A comprehensive evaluation of this internship is based on a detailed report submitted by the student and a subsequent oral defense. In addition, evaluation from the internship supervisor plays a critical role in the overall assessment.

UE Probabilités et méthodes numériques et optimisation

UE Finance de marché, dérivés et économétrie

Ces cours sont inclus dans l’UE “Finance de marché, dérivés et économétrie”

Options

Quatre cours (au moins) doivent être pris, au choix, dans les modules suivants, dont deux (au moins) dans le même module de spécialisation (majeure).

La validation de deux cours confère 3 ECTS, et celle des deux autres cours confère 3 autres ECTS   

Méthodes numérique avancées

Statistique et trading haute fréquence 

Nouveaux marchés

Ouverture professionnelle

Stage ou rédaction du mémoire

Stage obligatoire d'avril à septembre.