M2 Quantitative Finance

Course of Finance of the 1st semester, Stochastic calculus, numerical finance

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

INTEREST RATES:

• basic principles of rate curve modelling; methods used in active bond management and «Bond Picking»; understanding of structured rate products; 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 their use, as part of managing currency risks for companies. 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. 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.

To give an advanced overview in derivative instruments, including in interest rates, FX, volatility modelling and structured products.

44 hours of lectures and 6 hours of tutoring.

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

FX :

S.A. Ross, R.W. Westerfield, J.F. Jaffe, “Finance corporate”, Dunod

VOLATILITY :

• Abi Jaber, Eduardo. Lifting the Heston model. Quantitative Finance, 1-19,2019. - Bayer, Christian; Friz , Peter; Gatheral Jim. Pricing under rough volatility, Quantitative Finance, 16(6), 887-904, 2016. - Gatheral , Jim; Jaisson , Thibault; Rosenbaum , Mathieu. Volatility

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.

• Particle systems
• Propagation of chaos
• Mckean type diffusions
• Mean field games
• Applications to machine learning and finance

The aim of this course is to provide students with a comprehensive understanding of particle systems, McKean type diffusion, mean field games and their applications to machine learning and finance

Course ensured by Prof. Zhenjie REN

Carmona and Delarue. Probabilistic Theory of Mean Field Games with Applications I and II (2018)

STOCHASTIC CONTROL: Probability, Stochastic calculus

CORPORATE FINANCE AND INSURANCE MODELING: Probability, Stochastic calculus

STOCHASTIC CONTROL: I Formulation of control problem

1. Optimal stopping
2. Switching regime and impulse control
3. Regular and singular control II Probabilistic methods 1.Duality methods 2.Optimal stopping with infinite maturity 3.Early exercise premium 4.Optimal dividend distribution III Dynamic programming and Hamilton-Jacobi-Bellman equation 1.Dynamic programming 2.HJB equation and verification theorem 3.Viscosity solutions IV Backward Stochastic Differential Equations 1.Solutions of BSDE 2.Quadratic BSDE and Examples CORPORATE FINANCE AND INSURANCE MODELING: I Corporate Finance
4. Model of firm value
5. Balance sheet and fundings problems
6. Structural models of bankruptcy II Investment strategies
7. Optimal investment strategies
8. Investment, dividends and liquidity III Capital structure
9. Modeling liabilities : equity and debts
10. Issuing capital and bankruptcy IV Models for insurance
11. Ruin theory
12. Dividends strategy under ruin constraints

Formulation of control problem; Dynamic programming and Hamilton-Jacobi-Bellman equation; Corporate finance and insurance modeling

• 21 hours of lecture in Stochastic Control
• 21 hours of lecture in stochastic modelling

Ce cours est pris en charge par l'UEVE à hauteur de 31,5 HETD et le reste par l'ENSIIE).

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

First semester courses related to the topic of the project.

The cutting edge projects in finance allow groups of four/five 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.

Deepen a subject in quantitative finance under the responsibility of a professional

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.

UEVE prend en charge 21,5HETD, le reste est pris en charge par l'ENSIIE.

Provided with the subjects

Courses in finance in First semester M2 and in M1

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.

This course provides advanced techniques in asset management and introduces the concept of sustainable finance.

24 hours of lectures

Roncalli T., Introduction to Risk Parity & Budgeting, Chapman & Hall, 2013. Roncalli, T. (2017), Alternative Risk Premia: What Do We Know?, Chapter 10 of the book Factor Investing: From Traditional to Alternative Risk Premia.

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.

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

Introducing the main relevant applications of very recent machine learning technics in quantitative finance

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).

Statistical machine learning for quantitative finance
Ludkovski, M. Annu. Rev. Stat. Appl. 10 (2023), 271–295.

Deep hedging Buehler, H.; Gonon, L.; Teichmann, J.; Wood, B.
Quant. Finance 19 (2019), no. 8, 1271–1291.

Calcul stochastique, Python

1 : High-frequency financial data and limit order books I. Lab: Stylized facts on trade data.
2 : High-frequency financial data and limit order books II. Lab: Stylized facts on quote data.
3 : Introduction to point processes I. Lab : Poisson processes.
4 : Introduction to point processes II. Lab : Hawkes processes.
5 : Hawkes processes in finance.
Lab : Hawkes processes and high-frequency transaction data.
6 : Mathematical modeling of limit order books. Lab : Poisson LOB simulation.
7 : An introduction to market impact.
Lab : Empirical market impact of LOB events.

This course is aimed at students interested in the empirical study, mathematical modeling and numerical simulation of modern financial markets, known as order book markets.

Class (10H30), Python practical (10H30).

Abergel, Frédéric, Anane, Marouane, Chakraborti, Anirban, Jedidi, Aymen, & Muni Toke, Ioane (2016). Limit order books. Cambridge University Press.
Hasbrouck, J. (2007). Empirical market microstructure: The institutions, economics, and econometrics of securities trading. Oxford University Press.

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.

Since 2008, investment banks compute various X-valuation adjustments (XVAs) to assess counterpart 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.

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).

Beamer slides course, tutorials in python / tensorflow.

Related material on https://math.maths.univ-evry.fr/crepey/

Stochastic calculus and Numerical Finance from the first semester

• Discretization of Brownian EDS: strong error and weak error
• Approximation of standard exotic derivatives: barrier options, lookbacks, Asian derivatives
• Integration by parts formula and application to the calculation of Greeks
• Non-asymptotic concentration bounds in the Monte Carlo algorithm

These lectures aim at providing the theoretical basis of the fundamental numerical stochastic analysis techniques that are commonly used in a financial environment.
We will focus on strong and weak error for Brownian SDEs, then address the error associated with the pricing of "usual" exotic derivatives: Barrier/Lookback/Asian options.

We will as well present, through the integration by parts formula, some related applications to the computation of the Greeks.
We will eventually focus on non-asymptotic control bounds for the Monte-Carlo algorithm which is a key tool of all the previously described procedures.

Some additional ouvertures can be related to the convergence analysis of stochastic algorithms of Robbins-Monro type (which can be used in the VaR computation or to the approximation of SDEs with rough coefficients).

4 séances de cours de 3H et 3 séance de TP de

G. Pages "Numerical Probability"
Ikeda Watanabe "Stochastic Differential Equations and Diffusion Processes"

Pour le suivi des stages, l'UEVE prendra en charge 56 HETD (4x14 HETD). Le suivi des élèves bicursus ENSIIE est à la charge de l'ENSIIE.

Minimum level required : B1 + / B2 (Cadre Européen Commun de Référence)

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.

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.

Toeic Practice Exams by Lin Lougheed (Barron’s)
Préparation au Nouveau Toeic by Lin Lougheed (Pearson/Longman)

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.

IT project in C++ interfaced in excel/VBA on quantitative finance topics.

Students will have to work on a programming project in quantitative finance in a team of two.

Projects in teams of two students.

L'UEVE prend en charge 6,5 HETD, et le reste l'ENSIIE.

Probabilities at a good master 1 level, discrete and continuous time stochastic processes, Derivative products and contracts in Finance and programming with Python at a good master 1 level.

I Stochastic analysis prerequisites.
II Monte Carlo methods and variance reduction techniques.
III Hedging financial options.
IV Advanced computation of Greeks.
V Advanced numerical methods for pricing exotic and path dependent of options .
VI Analysis of discretization schemes for stochastic differential equations.
VII Model calibration techniques.

The aim of this course is to introduce advanced numerical methods needed for quantitative work in finance. To this avail, the course will provide a detailed study for calibrating models, pricing and hedging financial options.

42 hours of lectures
Blackboard course, homework coding with Python

L'UEVE prend en charge 31,5 HETD, le reste est à la charge de l'ENSIIE.

Mainly:

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.

probability, linear regression, penalized regression, python

FINANCIAL ECONOMETRICS: probability, stochastic processes, time series

The course starts with a quick reminder of the basics of machine learning, mainly focused on introducing the Perceptron and multi-layer Perceptron algorithms. We will then focus on the Multi-layer perceptron, the backpropagation learning algorithm, the different activation functions and their benefits, and the advantages of regularizations. Finally we will present and apply recurrent neural networks as well as convolutional neural networks.

To follow this course, you will need to bring your own computer and have installed the following material: https://github.com/brajard/nn/blob/master/INSTALL.md

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

Continuous monitoring and presentation of articles

• http://www.deeplearningbook.org/ , by Ian Goodfellow, 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

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

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.

L'UEVE prend en charge 27 HETD, l'ENSIIE: 24 HETD

INTEREST RATE MODELING: probability, discrete mathematical finance, stochastic process, stochastic calculus.

DEEP LEARNING: probability, linear regression, penalized regression, python

FINANCIAL ECONOMETRICS: probability, stochastic processes, time series

• Intro to Interest Rates
• Interest Rates Derivatives
• Spot Rate Models
• HW and CIR Models
• HJM and MLM Models
• MLM Models and Review
• Project presentation and assessment

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.

21 hours of lectures

An Elementary Introduction to Stochastic Interest Rate Modeling by Nicolas Privault
Interest Rate Modeling - Theory and Prac- tice by Lixin Wu
Term-Structure Models by Damir Filipovi ́c Interest Rate Models - Theory and Practice
by Damiano Brigo and Fabio Mercurio
Lecture notes of Tomasz Bielecki, Monique Jeanblanc and Marek Rutkowski (available on https://cel.archives-ouvertes.fr/cel-00398075)

Course M1 Finance or Financial Mathematics or equivalent

1. Hedging interest Rate Risk -With duration -Beyond duration 2)Swaps, Forwards and Futures Description, Different Uses and Pricing
2. Plain Vanilla Options Option description, strategies and pricing (CRR & Black Scholes models)
3. Introduction to Structured Products and Investment Portfolio Strategies.

he 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).

Ce cours est dupliqué en anglais (à l'UEVE, pris en charge par l'UEVE) et en français à l'ENSIIE (pris en charge par l'ENSIIE).

L. Martellini, P. Priaulet et S. Priaulet, «Fixed-Income Securities: Valuation, Risk Management and Portfolio Strategies», Wiley, 2003.

Generalized linear models. Linear models with L1 or L2 penalization (Lasso, ridge).

parametric models (bayes, ADL, QDL,..), non-parametric models (KNN, decision trees,..), ensemble methods (bagging, random forest, boosting). Dimension reduction methods (functional PCA, Gaussian mixtures, Spectral clustering, Kmeans,…). Performance metrics. ROC curves.

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.

12 sessions of 3h30 each divided into lessons or practical work as needed.

• Hastié, T., Tibshirani, R., Friedman, J. Elements of statistical learning. Springer, 2009.
• Murphy, K. Machine learning: a probabilistic perspective. MIT press, 2012

he prerequisites are undergraduate probability (Markov chains, discrete time martingales, different convergence notions in probability).

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.

The purpose of these lectures is to provide the mathematical background to apprehend a wide class of models appearing in finance.

42h of lectures

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. Cambridge University press.
-Rogers, L.C.G. and Williams, D. Diffusions, Markov Processes and Martingales : Volume 2, Itô Calculus.Cambridge University press
-Ikeda, N. and Watanabe, N. Stochastic differential equations and diffusion processes. North Holland.

probability, stochastic processes, time series

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.

[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 (Author), Christian Matthias Hafner (Author)
[3] Time Series: Time Series: Theory and Methods (Springer Series in Statistics) (Anglais) Broché – 28 avril 2009 Peter J. Brockwell
[4] Lecture on "The econometrics of high frequency data" by Per Mykland and Lan Zhang, http://tigger.uic.edu/~lanzhang/LaManga022209.pdf

FINANCE OF INSURANCE

Course M1 Finance or Financial Mathematics or equivalent

FINANCIAL MARKETS AND ACTUARIAL FINANCE:

None

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.

The course is an introduction to the financial aspects of insurance companies

This course contains 12 lectures of 3 hours and is mainly taught by a team of professionals from AXA.

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)

Course M1 Finance or Financial Mathematics or equivalent

Market Risk Credit Risk Counterparty Credit Risk and Collateral Risk Operational Risk Liquidity Risk Asset Liability Management Risk Model Risk Copulas and Dependence Modeling Extreme Value Theory Stress Testing and Scenario Analysis Credit Scoring Models.

The objectives of this course is to give an advanced knowledge in risk management and financial regulations.

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.