Former des ingénieurs spécialisés dans le domaine de la finance de marché, avec une excellente maîtrise des outils mathématiques, statistiques (machine learning), et de programmation associés.
Pour plus d'informations, vous pouvez consulter le site web de cette formation M2 Finance quantitative.
Lieu(x) d'enseignement
EVRY
Pré-requis, profil d’entrée permettant d'intégrer la formation
Compétences à un bon niveau de M1 ou équivalent en probabilités et statistiques, finance de marché, programmation (en C, python)
Compétences
Savoir formaliser mathématiquement un problème quantitatif issu de la finance de marché.
Apprécier les conditions de validités d'un résultat mathématique, les conditions d'applications d'un modèle, le domaine de validité d'un apprenant statistique,.
Mener les calculs mathématiques dans le cadre d'un modèle issu de la finance de marché.
Implémenter informatiquement un modèle mathématique issu de la finance de marché.
Mettre en place et interpréter une stratégie d'apprentissage statistique.
être capable d'évoluer dans un environnement de travail en langue anglaise.
Débouchés de la formation
- ingénieur financier
- analyste quantitatif
- risk manager
- IT-quant
- consultant en finance
- finance de l'assurance,
- data scientists pour la finance
- thèse en finance quantitative
Collaboration(s)
Laboratoire(s) partenaire(s) de la formation
Laboratoire de Mathématiques et Modélisation d'Evry.
Programme
Pour obtenir les 18 ECTS à choix du premier semestre, les étudiants doivent choisir 3UEs à 6. Ils peuvent en choisir une quatrième qui apparaîtra sur un supplément au diplôme.
Intitulé de l’UE en anglais :
IT Project in C++ and VBA
ECTS :
3
Détail du volume horaire :
Cours :2
Travaux dirigés :10
Modalités d'organisation et de suivi :
Coordinateur :
Equipe pédagogique :
Vincent Torri et Pedro Ferreira.
Déroulement et organisation pratique :
Projects in teams of two students.
Objectifs pédagogiques visés :
Contenu :
IT project in C++ interfaced in excel/VBA on quantitative finance topics.
Prérequis :
C++ and VBA programming at the level of the “Programming” course, numerical finance at the level of the “Pricing and calibration methods in finance” course.
Méthodes numériques de pricing et calibration de modèles
Langues d’enseignement :
AN
ECTS :
6
Détail du volume horaire :
Cours :42
Modalités d'organisation et de suivi :
Coordinateur :
Equipe pédagogique :
Stéphane Stéphane.
Déroulement et organisation pratique :
Blackboard course, demos in Matlab, homework in excel, VBA, and C++.
Objectifs pédagogiques visés :
Contenu :
0 Stochastic analysis prerequisites
I Pricing models
II Finite differences pricing schemes
III Monte Carlo Simulation pricing schemes
IV Markov chain pricing schemes
V Pricing path dependent of options
VI Model calibration techniques.
Prérequis :
Probabilities at a good master 1 level. Elements in finance such as provided by the course module “Financial Markets And Actuarial Finance”. C++ and VBA such as provided by the course “Programming”.
Bibliographie :
Mainly:
Crépey, S., Financial Modeling (Springer, 2013), chapters 1 à 9.
Others:
Lamberton, D. and Lapeyre P., Introduction to Stochastic Calculus Applied to Finance. Chapman & Hall, 2nd revised edition, 2007.
Hull, J., Options, Futures, and Other Derivative Securities, Prentice-Hall, last edition.
Glasserman P., Monte Carlo Methods in Financial Engineering, Springer, 2004.
Shreve, S.: Stochastic Calculus for Finance II: Continuous—Time Models, Springer, 2004 or later.
Cont R. et P. Tankov, Modelling with Jump Processes, Chapman & Hall, 2003.
Weekly 3-hour tutorials from September to December
Assessment: oral presentation and written examination.
Objectifs pédagogiques visés :
Contenu :
=-You will study financial articles and comment on Current trends and events affecting trading.
-You will summarize radio / TV financial reports.
-You will consolidate your Economic / financial vocabulary through regular exercises.
-Oral comprehension exercises based on news bulletins will lead You to discuss and write
about contemporary issues.
Prérequis :
Minimum level required : B2 or Toeic score above 750.
Bibliographie :
Some websites you can visit to help you prepare for the course :
Intitulé de l’UE en anglais :
Programming in C++ and VBA
ECTS :
6
Détail du volume horaire :
Cours :18
Travaux dirigés :24
Modalités d'organisation et de suivi :
Coordinateur :
Equipe pédagogique :
Vincent Torri, Pedro Ferreira.
Déroulement et organisation pratique :
Blackboard course, C++ tutorials.
final examination on paper.
Objectifs pédagogiques visés :
Contenu :
C++:
I) Object model
declaration of a class
definition of a class
definition of methods (inline and outside the class)
this pointer
static members
const method
II) Encapsulation
III) constructor, copy constructor and destructor
IV) overloading operator
internal operator (incremental operators, brackets operators)
external operators (arithmetic operators, stream operators)
V) Friend functions and classes
friend functions
friend classes
VI) Inheritance
single inheritance
multiple inheritance
composition or inheritance ?
VII) Dynamic allocation of memory
new and delete
which methods must be overloaded ?
VIII) Templates
template declaration
template function
template class
template instantiation
Crash course on VBA.
Prérequis :
C language,
additional types (boolean, reference),
additional features of functions : default value of arguments and overloading
function arguments of type reference or const reference
namespaces,
exceptions.
DEEP LEARNING: 6 lessons (50% lesson / 50% practical) and one final oral examination.
FINANCIAL ECONOMETRICS: 4 lecture courses of 3 hours each.
Objectifs pédagogiques visés :
Contenu :
INTEREST RATE MODELING:
In the theoretical part of the course we consider the classical short term interest rate models (Vasicek, Hull-White, Cox-Ingersoll-Ross), the HJM approach, and the Libor market model. The practical part, is focused on the calibration of interest rate models to market data.
DEEP LEARNING:
Deep learning structures have been at the source of the recent Data Science revolution. In this course we will learn the basic architectures that allow performing deep learning analysis of data both for classification and regression problems. The course will be taught in a combination of lectures and simultaneous tutorials on multi-layered perceptrons, recurrent neural networks, convolutional neural networks, autoencoders, generative adverserial networks, regularisation methods and architecture selection.
FINANCIAL ECONOMETRICS:
In this course, we will study time series models related to financial data. We are specifically interested in estimation problems for these models.
Course program :
- Short recap on time series. Basic empirical facts of financial time series.
- GARCH models : existence, properties, estimation of parameters (pseudo-likelihood estimator), volatility forecast. VaR computation, pricing with GARCH, connection with the high frequency sampling of a stochastic volatility model.
- Duration models and trading time modeling (Autoregressive Conditional Duration models, Lo and MacKinlay model).
- Volatility estimation with high frequency data.
DEEP LEARNING: probability, linear regression, penalized regression, python
FINANCIAL ECONOMETRICS: probability, stochastic processes, time series.
Bibliographie :
INTEREST RATE MODELING:
Interest rate models-Theory and Pratice, Brigo and Mercurio
DEEP LEARNING:
http://www.deeplearningbook.org/ , Ian Goodfellow and Yoshua Bengio and Aaron Courville
The Elements of Statistical Learning, T. Hastié, R. Tibshirani, J. Friedman.
Machine learning, a probabilistic perpective, K.P. Murphy
Pattern recognition and machine learning, C. Bishop
FINANCIAL ECONOMETRICS:
[1] Analysis of Financial Time Series (Anglais) Relié – 10 septembre 2010 de Ruey S. Tsay
[2] Statistics of Financial Markets: An Introduction (2015) by Jürgen Franke (Author), Wolfgang Karl Härdle
12 sessions of 3h30 each divided into lessons or practical work as needed.
Objectifs pédagogiques visés :
Contenu :
Data exploitation is now a major challenge in many fields such as industry, finance, society... This course presents theoretical foundations as well as practical application of machine learning models commonly used in regression and supervised classification. Dimension reduction and quantification methods are also studied.
Generalized linear models. Linear models with L1 or L2 penalization (Lasso, ridge).
Bibliographie :
- Hastié, T., Tibshirani, R., Friedman, J. Elements of statistical learning. Springer, 2009.
- Murphy, K. Machine learning: a probabilistic perspective. MIT press, 2012.
Lectures = 6 x 6h = 36h
Training sessions = 5 x 3h = 15h
Final examination = 3h.
Objectifs pédagogiques visés :
Contenu :
It is an advanced course in financial risk management which includes 6 lectures of 6 hours. Each lecture is organized around a specific topic which is related to the regulation. Four lectures concern (1) market risk, (2) credit risk, (3) counterparty credit risk, credit valuation adjustments and collateral risk, (4) liquidity risk, (5) asset liability management risk and (6) model risk. Two additional lectures are more technical and concern the following tools: (7) copulas and dependence modeling, (8) extreme value theory, (9) stress testing and scenario analysis and (10) credit scoring models.
Prérequis :
Course M1 Finance or Financial Mathematics or equivalent.
Bibliographie :
Roncalli, T. (2020), Handbook of Financial Risk Management, Chapman & Hall/CRC Financial Mathematical Series, 1400 pages, forthcoming.
Roncalli, T. (2020), Handbook of Financial Risk Management - Companion Book (Solutions of Exercises), Chapman & Hall/CRC Financial Mathematical Series, 410 pages, forthcoming, available at http://www.thierry-roncalli.com/download/frm-companion.pdf.
The course is an introduction to the financial aspects of insurance companies. It provides methods regarding how to price traditional insurance products (Life and Death Insurance, Fixed Annuities, etc.) and more advanced insurance products (e.g. CPPI, Variable Annuities). The course also presents some aspects of the Asset and Liability Management of an Insurance company and how to mitigate the risks inherent to insurance business.
FINANCIAL MARKETS AND ACTUARIAL FINANCE :
The purpose of this lesson is to train students in the key issues of financial markets (arbitrage, hedging, and speculation) and familiarize them with basic financial instruments (interest rate hedging, swaps, forwards and futures, and options). Numerous examples come to illustrate the contents of the lesson and aim to put students in conditions similar to those of market professionals.
1)Hedging interest Rate Risk
-With duration
-Beyond duration
2)Swaps, Forwards and Futures
Description, Different Uses and Pricing
3)Plain Vanilla Options
Option description, strategies and pricing (CRR & Black Scholes models)
4)Introduction to Structured Products and Investment Portfolio Strategies.
Prérequis :
FINANCE OF INSURANCE:
Course M1 Finance or Financial Mathematics or equivalent
FINANCIAL MARKETS AND ACTUARIAL FINANCE:
None.
Bibliographie :
FINANCE OF INSURANCE:
M. FROMENTEAU & P. PETAUTON, Théorie et pratique de l’assurance vie (Dunod)
A. TOSETI et al., Assurance, Comptabilité, Réglementation, Actuariat (Economica)
J. HULL, Options, Futures and Other Derivatives (Pearson)
D. BRIGO & F. MERCURIO, Interest Rate models – Theory and Practice (Springer)
FINANCIAL MARKETS AND ACTUARIAL FINANCE:
L. Martellini, P. Priaulet et S. Priaulet, «Fixed-Income Securities: Valuation, Risk Management and Portfolio Strategies», Wiley, 2003.
The purpose of these lectures is to provide the mathematical background to apprehend a wide class of models appearing in finance. We focus on continuous processes through the study of Brownian motion, Itô's formula, Brownian driven Stochastic Differential Equations (SDEs), their correspondence with some appropriate Partial Differential Equations (PDEs). We will also investigate some associated strategies of dynamic/static pricing and hedging of options.
Here are the following key-points of the course.
- Preliminary results on Gaussian vectors
- Characterizations of Brownian motion (as a Gaussian process or as a process with stationary independent Gaussian increments)
- Construction of Brownian motion through the approach of Paul Lévy.
- Itô processes and Itô's formula.
- SDEs and some associated Parabolic PDEs : the Feynman-Kac representation formula and Applications to finance
- The Girsanov theorem.
Prérequis :
The prerequisites are undergraduate probability (Markov chains, discrete time martingales, different convergence notions in probability).
Bibliographie :
Some possible companion books to the lectures are the following :
-Le Gall, J.F., Brownian Motion, Martingales, and Stochastic Calculus. Springer.
-Lamberton, D. and Lapeyre, B. Introduction to Stochastic Calculus Applied to Finance, Second Edition.
Chapman & Hall
-Comets, F. and Meyre, T. Calcul stochastique et modeles de diusions : Cours et exercices corrigés. Dunod.
Some more advanced references are :
-Karatzas, I. and Shreve, S. Brownian Motion and Stochastic calculus. Springer.
-Rogers, L.C.G. and Williams, D. Diffusions, Markov Processes, and Martingales : Volume 1, Foundations.
Période(s) et lieu(x) d’enseignement :
Période(s) :
Septembre - Octobre - Novembre - Décembre.
Lieu(x) :
EVRY
Pour obtenir les 16 ECTS à choix du second semestre, les étudiants doivent choisir 4UEs à 4. Ils peuvent en choisir une cinquième qui apparaîtra sur un supplément au diplôme.
Intitulé de l’UE en anglais :
XVAs, FRTB, and regulatory quant analysis
ECTS :
4
Détail du volume horaire :
Cours :18
Travaux dirigés :16
Modalités d'organisation et de suivi :
Coordinateur :
Equipe pédagogique :
Stéphane Crépey, Marc Chataigner, et Bouazza Saadeddine.
Déroulement et organisation pratique :
Beamer slides course, tutorials in python / tensorflow.
Objectifs pédagogiques visés :
Contenu :
Since 2008, investment banks compute various X-valuation adjustments (XVAs) to assess counterparty risk and its capital and funding implications. XVAs matter at two different levels. First, trade incremental XVAs are charged to clients as add-ons to deal entry prices. Second, some of the XVA metrics are also accounting entries that affect the result of the bank. More broadly, the advent of these metrics reflects a shift of paradigm in derivative management, from hedging to balance-sheet optimization.
First generation XVAs (CVA, DVA, and FVA, where C sits for credit, D for debt, and F for funding) pose the challenge of a proper understanding of the distinction between firm and shareholder valuation for their purpose. Second generation XVAs involve not only conditional expectations (i.e. prices), but also conditional risk measures: value-at-risk, which underlies MVA (margin valuation adjustment) computations, and expected shortfall, which underlies economic capital based KVA (capital valuation adjustment) approaches.
The course aims at providing a survey of the XVA universe from the triple angle of finance (wealth transfers), stochastic analysis (enlargement of filtration and backward SDE features), and computations (nested Monte Carlo vs. deep learning regression schemes).
The course will also address further FRTB implications in terms of risk measure and model risk (regulatory “Rquant” issues, coming on top of the traditional distinction between “Pquants” and “Qquants”).
Prérequis :
Financial modeling and numerical knowledge and skills, such as provided by the first semester course “Pricing and calibration methods in finance”
Machine learning in finance knowledge and skills, such as provided by the eponymous January module.
Some knowledge of corporate finance is also useful but will be recalled during the course.
Techniques de machine learning pour le pricing d'options, la calibration de modèles et la couverture (2ECTS) ET/OU Données Haute Fréquence et carnets d'ordre (2ECTS) ET/OU Gestion d'actifs avancée
Langues d’enseignement :
FR/AN
Intitulé de l’UE en anglais :
Machine learning in finance (2ECTS) AND/OR High-frequency data and limit order books (2ECTS) AND/OR Advanced asset management
MACHINE LEARNING TECHNIQUES FOR OPTION PRICING, CALIBRATION, AND HEDGING APPLICATIONS:
beamer slides course, tutorials in python / tensorflow (local jupyter notebooks, after local installation of the required packages including anaconda and tensorflow, or notebooks executed online on the google collaborative platform).
HIGH-FREQUENCY DATA AND LIMIT ORDER BOOKS:
A significant part of the course will be dedicated to practical manipulation of empirical data with Python and/or R.
Evaluation includes homeworks, labs and a final exam.
ADVANCED ASSET MANAGEMENT:
Period = January - March
Lectures = 4 x 6 h = 24h
Final examination: home project.
Objectifs pédagogiques visés :
Contenu :
MACHINE LEARNING TECHNIQUES FOR OPTION PRICING, CALIBRATION, AND HEDGING APPLICATIONS:
In recent years, machine learning techniques have emerged as a generic, model-free, financial derivative numerical paradigm. This course module will be devoted to the option pricing, calibration, and hedging applications of machine learning, with a focus on deep neural networks (mainly), and also Gaussian process regression techniques.
HIGH-FREQUENCY DATA AND LIMIT ORDER BOOKS:
Limit order book mechanisms and rules. Empirical stylized facts of high-frequency data and limit order books. Introduction to point processes: Poisson processes, Hawkes processes. Simulation and inference of these processes. Mathematical properties of limit order book models. Limit order book simulation. Introduction to optimal execution and market making.
ADVANCED ASSET MANAGEMENT:
1. Risk-Based Investing
a. Modern Portfolio Theory
b. Risk Budgeting Approach
c. Smart Beta & Risk-Based Indexation
d. Application to Bond & Credit Portfolios
e. Risk Parity & Portfolio Allocation
2. Risk Premia Investing
a. The Theory of Risk Premia
b. Skewness Risk Premia versus Market Anomalies
c. Factor Investing in Equities
d. Factor Investing in Corporate Bonds
e. Asset Allocation with Alternative Risk Premia
3. Advanced Topics
a. ESG Investing
b. Large-scale Optimization & Portfolio Optimization
c. Robo-Advisors
d. Machine Learning, Gaussian Processes, Bayesian Optimization & Trading Model Calibration.
Prérequis :
MACHINE LEARNING TECHNIQUES FOR OPTION PRICING, CALIBRATION, AND HEDGING APPLICATIONS:
Financial modeling and numerical knowledge and skills, such as provided by the first semester course “Pricing and calibration methods in finance”
General “machine learning” and “deep learning” knowledge and skills, such as provided by the eponymous first semester courses.
HIGH-FREQUENCY DATA AND LIMIT ORDER BOOKS:
Probability and basic stochastic calculus. Knowledge in Python programming.
MACHINE LEARNING FOR OPTIONS:
Horvath B., Muguruza A., Tomas M. Deep Learning Volatility. Preprint, 2019.
Bühler, H., L. Gonon, J. Teichmann, and B. Wood (2018). Deep hedging. Quantitative Finance. Forthcoming.
LIMIT ORDER BOOKS:
Abergel, F., M. Anane, A. Chakraborti, A. Jedidi, I. Muni Toke, « Limit order books », Cambridge University Press.
Cont, R., Stoikov, S. & Talreja, R. (2010), ‘A stochastic model for order book dynamics’, Operations research 58(3), 549–563.
Laruelle S. and Lehalle, C.-A., « Market microstructure in practice », World Scientific.
STRUCTURED
« Assessment: « Trade idea » take home exam.
Objectifs pédagogiques visés :
Contenu :
INTEREST RATES:
- basic principles of rate curve modelling
- methods used in active bond management and «Bond Picking»
- understanding of structured rate products and timing to buy these products in the appropriate situation
- pricing of options in a rate model in the form of a concrete case
FX:
The goal of this lesson is to present the main characteristics of foreign exchange derivative products and there use, as part of managing currency risks for companies. First of all, we will discuss the description of these products: Financial characteristics, risk factors, current situation on the market and their use by the financial direction of corporate companies to hedge their currency risks. The final objective is to help students, to have a global vision of different currency hedging strategies.
VOLATILITY:
A growing body of empirical research indicates that volatility fluctuates more rapidly than Brownian motion. Fractional volatility models have emerged as compelling alternatives. Even though tractability can be a challenge for these non-Markovian, non-semimartingales models, recent studies have developed numerical methods suitable for their implementation. The main purpose of this course module is to introduce these numerical techniques.
STRUCTURED PRODUCTS:
The objective of the course is to give an all around comprehensive general knowledge and understanding of the theory and the day-to-day use of options for trading, hedging and arbitrage purposes.
Prérequis :
INTEREST RATES: Course of Finance of the 1st semester
FX : Course of Finance of the 1st semester
VOLATILITY : Stochastic calculus, mathematical finance, and numerical finance at MSc level.
STRUCTURED PRODUCTS: Course of Finance of the 1st semester.
Bibliographie :
INTEREST RATES:
L. Martellini, P. Priaulet et S. Priaulet, «Fixed-Income Securities: Valuation, Risk Management and Portfolio Strategies», Wiley, 2003
J. Hull, «Options, Futures and Other Derivatives», Prentice Hall, 9ème Edition, 2017
Intitulé de l’UE en anglais :
Cutting edge finance
ECTS :
4
Détail du volume horaire :
Travaux dirigés :64
Modalités d'organisation et de suivi :
Coordinateur :
Equipe pédagogique :
Stéphane Crépey, Philippe Priaulet.
Déroulement et organisation pratique :
The subjects that relate to market finance, insurance finance but also data mining, machine learning, etc. (in line with the master program curriculum) are defined by the professional. The project starts in December and ends at the end of March. Students meet with their professional manager at least 3 times in the meantime. A co-supervision by an academic member of the pedagogical team of the master program ensures the daily follow-up. The project gives rise to the delivery to the company of a commented code as well as a defense (oral presentation) with beamer slides or jupyter notebook at the end of March.
Objectifs pédagogiques visés :
Contenu :
The cutting edge projects in finance allow groups of four students (including a team leader) to deepen a subject under the responsibility of a professional (team mentor). They offer students and professionals the opportunity for mutually beneficial collaboration. The students deploy their technical expertise in an adventure also mobilizing their creativity, team spirit and professionalism. The professional finds a chance to renew his look on his field and benefits from the students' investment.
Prérequis :
First semester courses related to the topic of the project.
STOCHASTIC CONTROL:
Assessment: Written exam and/or project
CORPORATE FINANCE AND INSURANCE MODELING:
Assessment: Written exam and/or project.
Objectifs pédagogiques visés :
Contenu :
STOCHASTIC CONTROL:
I Formulation of control problem
1. Optimal stopping
2. ChangementRegime switching and impulse control
3. Regular control
4. Singular control
II Probabilistic methods
Duality methods
Optimal stopping with infinite maturity
Early exercise premium
Optimal dividend distribution
III Dynamic programming and Hamilton-Jacobi-Bellman equation
Dynamic programming
HJB equation and verification theorem
Viscosity solutions
IV Backward Stochastic Differential Equations
Solutions of BSDE
Quadratic BSDE
Examples
CORPORATE FINANCE AND INSURANCE MODELING:
I Corporate Finance
1. Model of firm value
2. Balance sheet and fundings problems
3. Structural models of bankruptcy
II Investment strategies
1. Optimal investment strategies
2. Investment and dividends
3. Investments and liquidity
III Capital structure
Modelling liabilities : equity and debts
Issuing capital
Debt and bankruptcy
IV Models for insurance
Ruin theory
Dividends strategy under ruin constraints
Pricing and hedging variable annuities.
CORPORATE FINANCE AND INSURANCE MODELING: Probability, Stochastic calculus.
Bibliographie :
STOCHASTIC CONTROL:
“Stochastic control and optimization with financial applications” H. Pham, Springer 2009
“Optimal stochastic control, stochastic target problems and backward SDE”, N. Touzi, 2013
CORPORATE FINANCE AND INSURANCE MODELING:
“Principle of corporate finance”, R. Brealey, S. Myers,F. Allen, Mcgraw-Hill, 2013
“Investment under uncertainty”, A.K. Dixit and S. Pindyck, Princeton university press 1994
“Stochastic control in insurance”, H. Schmidli, Springer 2007.
Période(s) et lieu(x) d’enseignement :
Période(s) :
Octobre - Novembre - Décembre - Janvier - Février.
Students will be graded through their reports, and through an exam. Student’s reports are used to examine their competence in mathematical article writing. The exam examines the ability to read documents and to understand the semimartingale calculus. It should be noted that the exam is not an end in itself. The exam’s subjects furnish indications for better performing the task of proof- completing.
Objectifs pédagogiques visés :
Contenu :
This course is motivated by the belief that a quantitative financial practitioner at an advanced level should have, not only financial knowledge and expertise, but also some experience in research (in this case, mathematics). In particular, giving the students the ability to know the exact minimal conditions for a mathematical result to be correctly applied is one of the objectives of the course.
The framework chosen in our course is the following : writing a report on the theory of Le?vy processes. Indeed, on the one hand, Le?vy processes and their peers are useful financial modeling tools. On the other hand, it turns out that computations with Le?vy processes are rather easy, compared with other types of processes. Students can, therefore, have a rich experience in stochastic analysis without too much technical difficulty.
Student’s task:
There are a few dozen notions and theorems that form the starting point of the theory of Le?vy processes. Students should learn these notions, study these theorems, and write a report.For each of the theorems, a skeleton of the proof will be presented to the students. Documents on semimartingale calculus will be sent to students.Students read the documents, find appropriate results of the semimartingale calculus, and apply them to fill the skeletons and make the proofs complete.
Prérequis :
Calculus and probability at a good M1 level. Some knowledge of continuous-time processes at the level of the first semester course “stochastic calculus” is a plus but not mandatory.
Weekly 3 hour tutorials from January to March
Assessment : end of semester practise test in class
For further information on the official test, go to https://www.etsglobal.org/Global/Eng.
Objectifs pédagogiques visés :
Contenu :
In class you will train for the test by revising grammar rules, learning vocabulary and improving your reading and listening skills. Full tests will be regularly organised and corrected.
