Mathématiques de l'Assurance de l'Economie et de la Finance - 2ème année de Master

UE Obligatoires

• Stochastic Calculus

  Stochastic Calculus

  Ects : 6

  Enseignant responsable :
  JUSTIN SALEZ
  

  Volume horaire : 24

  Description du contenu de l'enseignement :

  • Probability basics
  • Stochastic processes, Brownian motion, continuous semi-martingales
  • Stochastic integral, Itô’s formula for semi-martingales and Girsanov’s theorem
  • Stochastic differential equations, diffusion processes
  • Feynman-Kac formula and link with the heat equation
  • Probabilistic representation of the Dirichlet problem.

  Compétences à acquérir :

  The first part of the course presents stochastic calculus for continuous semi-martingales. The second part of the course is devoted to Brownian stochastic differential equations and their links with partial differential equations. This course is naturally followed by the course “Jump processes”.

   

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• Stochastic Control

  Stochastic Control

  Ects : 6

  Enseignant responsable :
  PIERRE CARDALIAGUET
  

  Volume horaire : 18

  Description du contenu de l'enseignement :

  Relationship between conditional expectations and parabolic linear PDEs. Formulation of standard stochastic control problems: dynamic programming principle. Hamilton-Jacobi-Bellman equation Verification approach Viscosity solutions (definitions, existence, comparison) Application to portfolio management, optimal shutdown and switching problems

  Teacher : Bruno BOUCHARD

  Compétences à acquérir :

  PDEs and stochastic control problems naturally arise in risk control, option pricing, calibration, portfolio management, optimal book liquidation, etc. The aim of this course is to study the associated techniques, in particular to present the notion of viscosity solutions for PDEs.

• Monte-Carlo and Finite Differences Methods with Applications to Finance

  Monte-Carlo and Finite Differences Methods with Applications to Finance

  Ects : 6

  Enseignant responsable :
  JULIEN CLAISSE
  

  Volume horaire : 30

  Description du contenu de l'enseignement :

  Generalities on Monte-Carlo methods 1. Generalities on the convergence of moment estimators 2. Generators of uniform law 3. Simulation of other laws (rejection method, transformation, …) 4. Low discrepancy sequences

  Variance reduction 1. Antithetical control 2. Payoff regularization 3. Control Variable 4. Importance sampling

  Process simulation and payoff discretization 1. Black-Scholes model 2. Discretisation of SDEs 3. Diffusion’s bridges and applications to Asian, barrier and lookback options.

  Calculation of sensitivities (greeks) 1. Finite differences 2. Greeks in the Black-Scholes model 3. Tangent process and Greeks 4. Malliavin calculus, Greeks, conditional expectations and pricing of American options

  Calculation of conditional expectations and valuation of American options. 1. Nested Monte Carlo approach 2. Regression Methods (Tsitsiklis Van Roy, Longstaff Schwartz) 3. Rogers’ Duality

  Finite difference methods: the linear case 1. Construction of classical schemes (explicit, implicit, theta-scheme) 2. Conditions for stability and convergence

  Finite difference methods: the non-linear case 1. Monotonous schemes: General conditions of stability and convergence 2. Examples of numerical schemes: variational problems, Hamilton-Jacobi-Bellman equations.

  Compétences à acquérir :

  This course provides an in-depth presentation of the main techniques for the evaluating of options using Monte Carlo techniques.

UE Optionnelles

• Machine Learning in finance

  Machine Learning in finance

  Ects : 6

  Enseignant responsable :
  PIERRE BRUGIERE
  

  Volume horaire : 21

  Description du contenu de l'enseignement :

  - Introduction to statistical learning: The Vapnik Chervonenkis dimension, PAC learning and the calibration versus prediction paradigm. - Supervised learning: SVM, Mercer’s theorem and the kernel trick, C-SVMs, mu-SVMs, a few words on SVMs for regressions. - Unsupervised learning: Single class SVMs, clustering, anomaly detection, equivalence of different approaches via duality. - Introduction to random forests and ensemble methods: bias variance tradeoff, bootstrap method - A few words on neural networks: backpropagation, deep learning. - Remarks on parsimony and penalisation : Ridge and Lasso regressions, dual interpretation of Lasso.

  Pré-requis recommandés :

  Linear algebra, optimisation, differential calculus

  Pré-requis obligatoire :

  Linear algebra, optimisation, differential calculus

  Compétences à acquérir :

  Some Statistical Learning results are presented and applied to credit rating, anomalies detection and yield curves modeling. The principal notions are presented in the context of these case studies in finance.

  Mode de contrôle des connaissances :

  Final exam

  Bibliographie-lectures recommandées

  web.stanford.edu/~hastie/ISLR2/ISLRv2_website.pdf

• Optimisation

  Optimisation

  Ects : 6

  Enseignant responsable :
  ANTONIN CHAMBOLLE
  

  Volume horaire : 24

  Description du contenu de l'enseignement :

  This course will review the mathematical foundations for Machine Learning, as well as the underlying algorithmic methods and showcases some modern applications of a broad range of optimization techniques. Optimization is at the heart of most recent advances in machine learning. This includes of course most basic methods (linear regression, SVM and kernel methods). It is also the key for the recent explosion of deep learning which are state of the art approaches to solve supervised and unsupervised problems in imaging, vision and natural language processing. This course will review the mathematical foundations, the underlying algorithmic methods and showcases some modern applications of a broad range of optimization techniques. The course will be composed of both classical lectures and numerical sessions in Python. The first part covers the basic methods of smooth optimization (gradient descent) and convex optimization (optimality condition, constrained optimization, duality). The second part will features more advanced methods (non-smooth optimization, SDP programming, interior points and proximal methods). The last part will cover large scale methods (stochastic gradient descent), automatic differentiation (using modern python framework) and their application to neural network (shallow and deep nets).

  Compétences à acquérir :

  The objective of this course is to learn how to recognize, manipulate and solve a relatively large class of emerging convex problems in areas such as, for example, learning, finance or signal processing.

• Valuation of financial assets by absence of arbitrage

  Valuation of financial assets by absence of arbitrage

  Ects : 6

  Enseignant responsable :
  PAUL GASSIAT
  

  Volume horaire : 24

  Description du contenu de l'enseignement :

  • Financial markets in discrete time

  – Reminders on the Cox-Ross-Rubinstein model – Evaluation and hedging of European options – Evaluation and hedging of US options, optimal exercise strategy – Passage to the limit and approximation of the Black-Scholes model

  • Financial markets in continuous time

  – Mathematical modeling – Basic theorems of no arbitrage opportunity: Arbitration vs. No Free Lunch with Vanishing Risk – Hedging of European options – Probabilistic and PDE approaches for vanilla, Asian and barrier options – US Options (Probabilistic and PDEs approaches) – Optimal portfolio management and risk minimization (probabilistic approach) – Modeling of volatility and hedging (Dupire, tracking error, Heston, CEV, SABR,…) – Change of numéraire techniques and forward neutral measure – Semi-static hedging (example of the variance swap) – Evaluation by Fourrier transformation

  Compétences à acquérir :

  The objective of this course is to provide students with the mathematics needed to understand the models and techniques for the evaluation of options and risk hedging.

• Jump processes

  Jump processes

  Ects : 6

  Enseignant responsable :
  JULIEN POISAT
  

  Volume horaire : 18

  Description du contenu de l'enseignement :

  Poisson process, compound Poisson process, Infinitely divisible distributions, Random measures of Poisson, Lévy process, Decomposition of Lévy-Khintchine, Itô's formula for Lévy processes, Stochastic differential equations driven by a Lévy process, Equivalence of measures, Doleans-Dade exponential, Girsanov's theorem Merton’s Model Hawkes' Process

  Compétences à acquérir :

  This course aims to master the techniques of analysis and stochastic calculation specific to jump processes. It complements the "Stochastic Calculation" course which is limited to processes with continuous paths.

• Game theory, applications in economics and finance

  Game theory, applications in economics and finance

  Ects : 6

  Enseignant responsable :
  MIQUEL OLIU BARTON
  

  Volume horaire : 18

  Description du contenu de l'enseignement :

  A- Basics of game theory: 1. Zero-sum games: value, optimal strategies, saddle points, minmax theorem. 2. N-layers normal form games: equilibria in dominant strategies, Nash equilibria, dominated strategies, Nash’s existence theorem. 3. Extensive form: backward induction, subgame perfection, theorem of Kuhn-Zermelo, behavior strategies and Kuhn’s theorem.

  B- Applications: 1. Repeated games and cooperation, folk theorems. 2. Zero-sum repeated games with incomplete information on one side (Aumann-Maschler’s model). Splitting lemma, uniform value. 3. Zero-sum stochastic games: dynamic structure, Shapley operator, theorems of Bewley-Kohlberg and Mertens-Neyman.

  Compétences à acquérir :

  The first part deals with the basics of game theory, the second one with applications in economics and finance. There will only be time to study 2 of the 3 applications (to be decided).

• Fondamentaux Macro-économiques de la gestion de portefeuille

  Fondamentaux Macro-économiques de la gestion de portefeuille

  Ects : 6

  Enseignant responsable :
  OLIVIER DAVANNE
  

  Volume horaire : 18

  Compétences à acquérir :

  Les gestionnaires de portefeuille ont besoin de posséder certaines connaissances macroéconomiques de base pour mieux fonder leurs décisions d’investissement. La valeur dite fondamentale des différents actifs financiers ne peut être analysée sans une prise en compte des évolutions macroéconomiques prévisibles à moyen et long terme. De plus, les performances de court terme des différentes classes d’actifs financiers dépendent crucialement des indicateurs macro-économiques, notamment en matière d’inflation et de croissance. Ce cours présentera notamment les méthodes dominantes utilisées par les praticiens de marché (économistes de marché, gestionnaires…) pour analyser et anticiper les évolutions macro-économiques, ainsi que les inflexions de politique monétaire. Il a une vocation appliquée et vise à donner à de futurs gestionnaires une bonne connaissance des instruments pratiques de prévision macroéconomique ainsi que des indications sur la meilleure façon d’utiliser ces instruments pour améliorer la performance de leur gestion.

• Term Structure and Commodity Derivatives Markets

  Term Structure and Commodity Derivatives Markets

  Ects : 6

  Enseignant responsable :
  DELPHINE LAUTIER
  

  Volume horaire : 18

  Description du contenu de l'enseignement :

  This course in a introduction to the methodology of the research through the prism of prices curves. It presents the theories, their empirical implications, the methodological issues associated with empirical tests, and empirical tests. The course is centered on commodities, with generalization to other assets (more specifically, interest rates).

  1. Overview of derivative markets

  2. The first market models

  3. Term structure models: theory, calibration and empirical tests

  4. Structural models

  Compétences à acquérir :

  At the end of this course, the students must have a broad knowledge about the term structures of derivative prices: the theories, the valuation methods, the econometric techniques, the empirical tests as well as the applications.

  Mode de contrôle des connaissances :

  20% Participation ; 80% Final exam

  Bibliographie-lectures recommandées

  Main references: - Danthine J.P., Donaldson J.B., Intermediate Financial Theory, 2d Ed., Elsevier, 2005. - Hull J., Options, futures and other derivatives, 9th Ed. - Kolb R.W. , Overdahl J.A. , Futures, options, and swaps, 5th Ed., Blackwell, 2007. - Williams J., The economic function of futures markets, Cambridge University Press, 1986 - Wilmott P., Paul Wilmott on Quantitative Finance, 3-volume set, 2nd Ed., Wiley, 2006.

• Practice of structured products in finance and insurance

  Practice of structured products in finance and insurance

  Ects : 6

  Enseignant responsable :
  AYMERIC KALIFE
  

  Volume horaire : 18

  Description du contenu de l'enseignement :

  Participants will lear how financial institutions can build and structured products, how they value and hedge them, and what they are done for.

  Compétences à acquérir :

  The aim of this lecture is to train students in the practical evaluation of derivative products and the control of the associated risks. It also introduces them to the new hybrid structured products that have recently appeared in insurance.

• Computational methods and MCMC

  Computational methods and MCMC

  Ects : 4

  Enseignant responsable :
  CHRISTIAN ROBERT
  

  Volume horaire : 18

  Description du contenu de l'enseignement :

  Motivations Monte-Carlo Methods Markov Chain Reminders The Metropolis-Hastings method The Gibbs Sampler Perfect sampling Sequential Monte-Carlo methods

  Compétences à acquérir :

  This course aims at presenting the basics and recent developments of simulation methods used in statistics and especially in Bayesian statistics. Methods of computation, maximization and high-dimensional integration have indeed become necessary to deal with the complex models envisaged in the user disciplines of statistics, such as econometrics, finance, genetics, ecology or epidemiology (among others!). The main innovation of the last ten years is the introduction of Markovian techniques for the approximation of probability laws (and the corresponding integrals). It thus forms the central part of the course, but we will also deal with particle systems and stochastic optimization methods such as simulated annealing.

UE fondamentale

• Conferences cycle: strategies and actors in portfolio management

  Conferences cycle: strategies and actors in portfolio management

  Ects : 2

  Enseignant responsable :
  CHARLES-ALBERT LEHALLE
  

  Volume horaire : 12

Bloc 1 : Apprentissage pour l'économie et la finance

• High frequency modeling

  High frequency modeling

  Ects : 6

  Enseignant responsable :
  EMMANUEL BACRY
  

  Volume horaire : 9

  Description du contenu de l'enseignement :

  - Financial Markets and Products - Order books - Some basics in estimation theory - High frequency financial series - Stylized facts of daily series - Models - Seasonality of volatility - Intraday stylized facts

  Compétences à acquérir :

  This course introduces the empirical properties of financial data across scales from the perspective of statistical finance.

• Introduction to reinforcement learning

  Introduction to reinforcement learning

  Ects : 3

  Volume horaire : 24

  Description du contenu de l'enseignement :

  1. Multiarmed bandits, Markov Decision Processes and other models 2. Planning: finite and infinite horizon problems, the value function, Bellman equations, dynamic programming, value and policy iteration 3. Probabilistic and statistical tools for RL: Bayesian models, relative entropy and hypothesis testing, concentration inequalities, linear regression, the stochastic approximation algorithm 4. RL algorithms for multiarmed bandits: the explore vs. exploit compromise, bandit algorithms vs. A/B testing, UCB, Thomson sampling, contextual bandits 5. RL algorithms for Markov Decision Processes: off policy and on policy learning, Q-learning, SARSA, Monte Carlo tree search

  Compétences à acquérir :

  This introductory course will provide the main methodological building blocks of reinforcement learning. Some basic notions in probability theory are required to follow the course. The course will imply some work on simple implementations of the algorithms, assuming familiarity with common scientific computing language.

  Bibliographie-lectures recommandées

  Bibliographie, lectures recommandées M. Puterman. Markov Decision Processes: Discrete Stochastic Dynamic Programming. John Wiley & Sons, 1994. R. Sutton and A. Barto. Introduction to Reinforcement Learning. MIT Press, 1998. C. Szepesvari. Algorithms for Reinforcement Learning. Morgan & Claypool Publishers, 2010 J. Myles White. Bandit Algorithms for Website Optimization. O'Reilly. 2012 T. Lattimore and C. Szepesvari. Bandit Algorithms. Cambridge University Press. 2019. downloads.tor-lattimore.com/banditbook/book.pdf

• Python/Pytorch project

  Python/Pytorch project

  Ects : 6

  Enseignant responsable :
  PIERRE ABLIN
  

  Volume horaire : 15

  Description du contenu de l'enseignement :

  The Python and PyTorch languages are commonly used to build ML/IA algorithms. The classroom course is complemented by a practical application thesis in economics or finance (e.g. Deep hedging, rapid calculation of expected shortfall, optimal portfolio management, high-frequency trading, solving semi-linear equations of the second order, variance reduction, etc.).

  Compétences à acquérir :

  • Mastery of Python and PyTorch. Ability to build an ML/IA algorithm.

Bloc 2 : Finance et gestion des risques

• Gestion globale des risques : VAR

  Gestion globale des risques : VAR

  Ects : 3

  Enseignant responsable :
  EMMANUEL LEPINETTE
  

  Volume horaire : 21

  Description du contenu de l'enseignement :

  Introduction. Modèles dynamiques pour les prix d’actifs financiers. Agrégation des risques, normalité, asymétrie, queues de distributions épaisses. La valeur risquée. Définition et méthodologies de calcul de la VaR (historiques, Monte Carlo, analytiques). Présentation de RiskMetrics de J.P. Morgan. Données, méthodologie, interprétations. Application. La cartographie de RiskMetrics, risque sur les instruments financiers comptants et produits dérivés. Estimation des matrices de variances-covariances, volatilités et corrélations.

  Compétences à acquérir :

  Analyse des modèles mathématiques du risque de marché, étude des méthodes de gestion globales du risque de marché lorsque les sources d’incertitude sont multiples.

  Bibliographie-lectures recommandées

  - Jorion Philippe Value at Risk McGraw-Hill, 2006 - Longerstaey, J More, L Introduction to riskmetrics Morgan Guaranty trust Company, 1995 - Hull J, Risk management and Financial - Institutions 5th editions Pearson, 2018 - Hull J, Options, futures and other derivatives Prentice-Hall Pearson Education, 2017 - Portait R, Poncet P Finance de Marché Dalloz , 2014

• Microstructure des marchés financiers

  Microstructure des marchés financiers

  Ects : 6

  Enseignant responsable :
  JEROME DUGAST
  

  Volume horaire : 15

  Description du contenu de l'enseignement :

  The field of market microstructure combines theoretical modeling, institutional knowledge, and empirical analysis to understand how prices result from the interactions of traders in financial markets. The course aims to acquaint students with (i) the canonical models in microstructure, and (ii) econometric models used to test the predictions of microstructure models.

  Course structure:

  1. Trading Mechanisms
  2. Measuring Liquidity
  3. Price Dynamics and Liquidity
  4. Trade Size and Market Depth
  5. Empirical Analysis

  Compétences à acquérir :

  Master the concepts of financial markets microstructure

  Mode de contrôle des connaissances :

  Evaluation: assignement 30%, final exam 70%

  Bibliographie-lectures recommandées

  Foucault, Thierry, Marco Pagano, and Ailsa Röell, Market Liquidity: Theory, Evidence, and Policy, Oxford University Press, 2013.

• Contrôle stochastique et marchés de l'énergie

  Contrôle stochastique et marchés de l'énergie

  Ects : 6

  Enseignant responsable :
  RENE AID
  

  Volume horaire : 15

  Compétences à acquérir :

  Energy markets are a natural field of applications for stochastic control modelling framework. Historical applications go from water management to the pricing of swing and demand-side contracts. With the deregulation of electricity and gas markets, new applications have raised the attention of financial economists. In particular, the question of the optimal investment in generation assets in the context of climate change and the questions linked to retail competition. These domains are conducive to the utilization of stochastic differential games. This course is intended to provide a short introduction to the physics of energy market and extensive applications taken for financial and economical research papers. For their evaluation, students are expected to realize a study of a research paper for which they will provide a critical analysis of their understanding of the model, together with the reproduction of the results of the paper.

• Modélisation stochastique de la courbe de taux

  Modélisation stochastique de la courbe de taux

  Ects : 2

  Enseignant responsable :
  IMEN BEN TAHAR
  

  Volume horaire : 21

  Description du contenu de l'enseignement :

  1. Quelques outils de calcul stochastique : rappels2. Généralités sur les taux d’intérêt3. Produits de taux classiques 4. Modèle LGM à un facteur5. Modèle BGM (Brace, Gatarek et Musiela) / Jamishidian6. Modèles à volatilité stochastique

  Compétences à acquérir :

  Ce cours est consacré aux modèles de taux d’intérêts à temps continu. Au travers de nombreux exemples, on décrira leurs utilisations pour évaluer les produits dérivés sur taux d’intérêt.

Bloc 3 : Economie et jeux

• Mean field games theory

  Mean field games theory

  Ects : 6

  Enseignant responsable :
  PIERRE CARDALIAGUET
  

  Volume horaire : 18

  Description du contenu de l'enseignement :

  Stochastic Control cours (1rst semester) is a necessary prerequisite.

  Mean field games is a new theory developed by Jean-Michel Lasry and Pierre-Louis Lions that is interested in the limit when the number of players tends towards infinity in stochastic differential games. This gives rise to new systems of partial differential equations coupling a Hamilton-Jacobi equation (backward) to a Fokker-Planck equation (forward). We will present in this course some results of existence, uniqueness and the connections with optimal control, mass transport and the notion of partial differential equations on the space of probability measures.

  Pré-requis recommandés :

  Stochastic analysis, Stochastic control.

  Compétences à acquérir :

  Mastering of the mean field games technics.

  Bibliographie-lectures recommandées

  Notes on the course: www.ceremade.dauphine.fr/~cardaliaguet/Enseignement.html

• Variational problems and optimal transport in economy

  Variational problems and optimal transport in economy

  Ects : 6

  Enseignant responsable :
  GUILLAUME CARLIER
  

  Volume horaire : 18

  Description du contenu de l'enseignement :

  1. Duality 2. Optimal Transport 3. Economic Applications of Optimal Transport technics 4. Calculation of variations 5. The principal-agent problem

  Compétences à acquérir :

  Mastering of variational and optimal transport methods used in economy.

• Théorie géométrique du contrôle

  Théorie géométrique du contrôle

  Ects : 6

  Enseignant responsable :
  GUILLAUME CARLIER
  
• Algèbre tropicale en optimisation et en jeux

  Algèbre tropicale en optimisation et en jeux

  Ects : 6

  Volume horaire : 20

  Description du contenu de l'enseignement :

  Algèbre tropicale et structures de « caractéristique un » (semi-corps max-plus). Résolution de problèmes avec paiement moyen via des problèmes spectraux non-linéaires. Représentation des opérateurs de Shapley. Théorie de Perron-Frobenius non-linéaire, dynamiques monotones ou non-expansives. Existence du paiement moyen. Certificats de Collatz-Wielandt. Généralisation non-linéaire de l’algorithme de la puissance. Algorithmes d’itération sur les politiques. Résultats de complexité pour les jeux répétés.

  Compétences à acquérir :

  Ce cours présente un certain nombre d’outils et résultats récents, inspirés de la géométrie tropicale, relatifs aux problèmes de contrôle ou de jeux répétés, déterministes ou stochastiques, avec une attention particulière pour les problèmes de paiement moyen ou en temps long ainsi que pour les aspects combinatoires et algorithmiques. Certains résultats sont illustrés par des exemples issus d’applications (optimisation du référencement, optimisation de la croissance en dynamique de population).