syllabus of master iei: qfrm quantitative finance...

Syllabus of Master IEI: QFRM

Quantitative Finance and Risk Management

(version 2014-09-05)

Syllabus of Master Quantitative Finance and Risk Management (QFRM)

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Table of Contents

Table of Contents ........................................................................................................................ 2

Syllabus of M1 ........................................................................................................................... 3

Mathematics Applied to Insurance ......................................................................................... 3

Functional Analysis ................................................................................................................ 5

Partial Differential Equations ............................................................................................... 10

Contingent Claims Valuation................................................................................................ 12

Portfolio Management and Financial Risks ......................................................................... 13

Risk Management in a mono-period Financial Market and Derivatives.............................. 14

Optimization ......................................................................................................................... 18

Jump Processes + Applications ............................................................................................ 20

Stochastic Processes (Discrete and Continuous Time)......................................................... 22

Monte-Carlo Simulation ....................................................................................................... 24

Bloomberg Trading Room .................................................................................................... 26

Object Oriented Design ........................................................................................................ 27

C++ and VBA Programming ................................................................................................ 28

Introduction to Quantitative Finance .................................................................................... 29

Syllabus of M2 ......................................................................................................................... 30

Business Evaluation ............................................................................................................. 33

Portfolio Management .......................................................................................................... 34

Interest Rate, Exchange and Inflation Markets .................................................................... 35

Imperfect Markets ................................................................................................................ 37

Dynamic Hedging and Risk Measures ................................................................................. 38

Mathematical Tools in Finance ............................................................................................ 42

Practical Portfolio Management in Fixed Income ................................................................ 44

Jump Processes and Aplications ........................................................................................... 46

Mathematical Statistics ........................................................................................................ 47

Theory of Contingent Claims ............................................................................................... 49

Simulation ............................................................................................................................ 50

Advanced spreadsheet Programming ................................................................................... 51

Practical Equity Portfolio Managment ................................................................................. 51


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Syllabus of M1

Mathematics Applied to Insurance

Responsible: Danielle Florens

Teachers: Danielle Florens

Teaching Objectives: to introduce the stochastic processes necessary for insurance modeling.

Content of Teaching: Renewal Process, Poisson Process, Modeling of ‘small risks’, Modeling of extreme risks

Total Hours (lecture, tutorials, practical lab): 30hours

ECTS: 3

Program (sessions of 3 hours):

1- Basic mechanisms and vocabulary in insurance

- insurance premiums, tarification

- claim charge for the insurer.

- individual and collective risk models

2- Modeling the frequency of claims

3- Modeling the accumulated amount from claims

- Modeling the cost of claims

- Mixed composed Poisson Model

4- Credibility

- Bulhman and Bulhman-Straub Models

5- Ruin Theory

Poisson Process


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6- Renewal Process

7- Cramer- Lundberg Model

8- Small Risk cases

- extreme risk cases

Evaluation:

Exam

Bibliography:

T. Mikosch: Non life Insurance Mathematics Springer -2000

M. Denuit, A. Charpentier: Mathématiques de l’assurance non-vie Economica 2004

P. Embrechts, C. Klüppelberg, T. Mikosch: Modelling Extremal Events for Insurance and Finance, Springer -1997


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Functional Analysis

Responsible: Irina Ignatiouk

Teachers: Irina Ignatiouk

Teaching Objectives:

To give the students a new, deeper knowledge of Topology and the properties of Topological and Metric spaces, as well as to add breadth to the notions and theorems of functional spaces and their applications.

Content of Teaching:

Compact topological spaces – theory of Bolzano Weierstrass

Continuous Functions on a compact; locally compact spaces; Riesz Theory

Continuous linear applications between normed vector spaces. Topological duality. Theories of Banach - Alaoglu, Ascoli and Stone-Weierstrass. Theory of Baire and Banach - Steinhaus.

Total Hours (lecture, class, tutorials): 30 hours

ECTS: 3


1. Reminders: Topological Spaces, Product Topology, Metric Space Topology, Normed Vector Spaces. Compactness: Theory of Bolzano, Weierstrass. Continued Functions on a compact.

2-3. Continuation of uniformly continuous applications

4-5. Banach Space of continued functions on a compact. Theory of Ascoli and Stone-Weierstrass. Continuous linear applications between normed vector spaces.

6-7. Topological Duality. Theory of {Banach - Alaoglu}.

8-9-10. Theory of Baire, Banach-Steinhaus, of open mappings.

Theory of Riesz-Fréchet. Theory of Hahn-Banach and consequences.

Prerequisites: Algebra Analysis (undergraduate level)

Evaluation: Final exam of 3 hours


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Finite Difference Methods

Responsible: I. Kortchemski

Teachers: I. Kortchemski

Teaching Objectives: This course will teach the students the essential basic knowledge of finite difference techniques necessary for the numerical solutions of partial differential equations.

Content of Teaching: Derivation of difference equations, initial boundary value problemes, explicit and implicit schemes of discretisation, implementation and numerical solutions, stability, consistency.

Total Hours (lecture, tutorials, practical lab): 25 (11, 9, 5)

ECTS: 2.5


1. Mathematical modeling by partial differential equations (PDE). Parabolic, Elliptic and Hyperbolic PDE. Grid on time domain. Discretization of ordinary differential equations. Euler method. Runge-Kutta Method. + Tutorial

2. Grid on time-space domain. Taylor’s Formula. Approximation of differential operators. Explicit and implicit methods for PDE. + Tutorial

3. Discretization of boundary conditions. Dirichlet and Neumann boundary conditions. Periodical boundary conditions. + Tutorial

4. Consistency, Stability, Convergence. Von Neumann criteria. Lax Theorem. + Tutorial

5. Discretization of parabolic PDE. Euler Scheme. Crank-Nicolson Scheme. Thomas’ Algorithm. +Practical Class 1

6. Discretization of hyperbolic PDE. Upwind, Leap-Frog, Lax-Wendroff Schemes. + Tutorial

7. Discretization of elliptic PDE. Discretization in polar coordinates. Problem of singularities. + Tutorial

8. Discretization of non-linear PDE. Linearization method. Method of Mac-Cormack. Burgers’ Equation. Korteweg- de Vries Equation. +Practical Class 2

Practical Class 1: Numerical solution of the heat equation with Neumann boundary conditions.


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Practical Class 2: Numerical solution of the Korteweg- de Vries equation for one and two solitons.

Prerequisits: ‘mathematics for the engineer’, ‘numerical analysis’ (basic level)

References:

1. H. M. Antia, Numerical Methodes for Scietists and Engineers. Birkhauser.

2. M. Rappaz, M. Bellet, M. Deville, Numerical Modeling in Material Science and

Engineering. Springer

3. J.W. Thomas, Numerical Partial Differential Equations

4. W.F. Ames, Numerical Methods for Partial Differential Equations, Nelson and

Sons LTD. London, 1969

5. G.D. Smith, Numerical solution of PDE : Fintite difference methods,Clarendon

Press, Oxford, 1978

6. J.H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics.

Springer, 1996.

7. W. Press, S. Teokolsky, W. Vetterling, Brian P. Flannery. Numerical Recipes. The art of

Scientific Computing. Cambridge University Press. 2011.

8. N. Giorgano, H. Nakanishi. Computational Physics. Pearson, Pearson Hall, 2009

9. B. Thaller. Visual Quantum Mechanics. Springer


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Calibration of Financial Models

Responsible:I. Kortchemski

Teacher: I. Kortchemski

Teaching Objectives: In this course, we give the students a general introduction to the calibration of implied volatility, yield curves and risk neutral transition density.

Content of Teaching: Reminders on the evaluation of derivatives and optimization methods. Calibration of implied volatility using Newton’s algorithm. We introduce ill- posed inverse problems.

Total Hours (lecture, tutorials, practical lab): 20 hours (11, 3, 6)

ECTS: 2

Programme (sessions of 3 hours):

1. Introduction: option market, Black and Scholes’ model.

2-3. Implied volatitlity. Vega of un Option. Calibration of implied volatility. Newton’s algorithm. + Tutorial

4. Ill- posed inverse problems and notions of regularization. Inverse linear problems. Breeden-Litzenberger Formula. Construction of risk neutral transition density. + Practical Class

5. Inverse linear problems, construction of yield curves. Parametric methods: interpolation by cubic splines, Nelson Siegel method. + Practical Class

6-7. The trinomial tree in CEV model. Implicit tree of pricing. Local volatility profile. + Tutorial

Prerequisits: Evaluation of derivatives, numerical solution of PDE, optimization.

Evaluation: Final exam of 3 hours, plus graded Tutorials

Bibliography:

D. Lamberton, and B. Lapeyre: Introduction au calcul stochastique appliqué à la finance, Ellipses, Paris (1999).

Y. Achdou, and O. Pironneau: Volatility Smile by Multilevel Last Square, International Journal of Theoretical and Applied Finance vol. 5, No. 2 (20002).


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P. Tankov: Calibration des modèles financiers et la couverture des produits dérivés. Class from the University of Paris VI.


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Partial Differential Equations

Responsible: Abdessalam El Janati

Teacher: Abdessalam El Janati


To give the students the basic elements of PDE theory, and their applications.


Sturm-Liouville Theory. Parabolic / Hyperbolic Partial Differential Equations. Elliptic Distributions.

Total Hours (lecture, tutorials, practical lab): 30 hours (10h, 20, 10)

ECTS: 3


1. Introduction. Classification of PDE of second order

2. Elements of the Sturm-Liouville Theory

3. Parabolic Equations. Part I. Boundary problems.

4. Parabolic Equations. Part II. Boundary problems. Green’s Function.

5. Parabolic Equations. Part III. Fundamental Solution.

6-7. Hyperbolic Equations. Wave Equation. Elliptic Equations. Laplace’s Equation. Green’s Function.

8-9-10. Test Functions. Distributions. Operations on Distributions. Convergence. Derivation and integration of distributions. Convolution operators. Fourier transform for distributions. Fundamental Solution and Sobolev Spaces. Lax-Milgram Theorem.

Prerequisites: Algebra Analysis (Bachelor level)

Evaluation: Exam of 3 hours


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Bibliography:

A. PINSKY: Introduction to partial differential equations with applications, McGrow-Hill Book Company, 1984

A.N. TYCHONOV, A.A. SAMARSKI: Partial differential equations of mathematical physics. Volume I, Holden-day, Inc, 1964

I.G. PETROVSKY: Lectures on Partial Differentiel Equations, Dover Publications, Inc, 1991

J. KEVORKIAN: Partial differential equations. Analytical solution technics, Springer, 2000


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Contingent Claims Valuation

Responsible: Erik Taflin

Teacher: Erik Taflin

Teaching Objectives: to give the students a deepened knowledge of the Theory of Contingent Claims (European and American types) in discrete and continuous time.


ECTS: 3

Program (3 hours sessions):

1. Reminder of the theory of contingent claims in a mono-period; reminder of discrete time processes.

2. Valuation of contingent claims in the binomial model.

3. Valuation of contingent claims in a general discrete time model (complete and incomplete cases).

6. Reminder of continuous time processes (Itô’s lemma, Girsanov’s theorem).

7. Valuation of contingent claims in continuous time in a Brownian filtration.

Prerequisites: Risk Management in a mono-period financial market and derivatives (M1), Discrete and continuous time stochastic processes (M1).


Bibliography:

[1] H. Föllmer and A. Schied: Stochastic Finance, Walter de Gruyter 2002

[2] I. Ekeland, N. Fintz and E.Taflin: Les marchés financiers et gestion de portefeuille, Preprint 2007

[3] T. Björk: Arbitrage Theory in Continuous Time, 2nd ed. Oxford 2004

[4] M. Musiela, R. Rutkowski: Martingale Methods in Financial Modelling, 2nd ed. Springer 2007


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Portfolio Management and Financial Risks

Responsible: Mohammed Mikou

Teacher: Mohammed Mikou

Teaching Objectives: to give the students, within the framework of mono-period and discrete time models of financial markets, knowledge in the evaluation of risk, through VaR (Value at Risk), AVaR (Average Value at Risk) as well as other measures of risk and portfolio management through the maximization of the expected-utility.


1. Risk Measures: VaR (Definition, basic properties, examples, problems with VaR), AVaR

2. Acceptable utility functions, optimal portfolio problems

3. Solution to the problem of portfolio management in a mono-period market (complete and non-complete cases)

4. Portfolio management, generale case in discrete time

5. Portfolio management in continuous time


ECTS: 3

Prerequisites: Risk Management in a mono-period financial market and derivatives (M1), Discrete and continuous time stochastic processes (M1), Contingent Asset Claim (M1)


Bibliography:


[2] I. Ekeland, N. Fintz and E.Taflin: Les marchés financiers et gestion de portefeuille, Preprint 2007

[3] T. Björk: Arbitrage Theory in Continuous Time, 2nd ed. Oxford 2004


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Risk Management in a mono-period Financial Market and Derivatives

Responsible: Nesim Fintz

Teachers: Nesim Fintz and Mohammed Mikou

Teaching Objectives: To give a general introduction to the financial markets, products (assets, bonds, derivatives etc.), risk management and portfolio management. To define the most simple model of financial markets (the mono-period model) and the simplest risk measure (variance). In the mono-period model: Evaluation of derivatives through arbitrage and portfolio management (Markowitz).


In this course, the following chapters are tackled:

1. General Introduction to Financial Markets:

Equity Market: Definition and role of different types of shares and equity, the different actors, the functioning of the stock market and quotation (bid price, ask price and the determination of a transaction’s price), quotation frequency, frictions (constant and proportional commissions, stock market indexes, society evaluation, different prices (spot, forward, future), long and short positions. Bond Market: definition and role of different types of bonds, the different actors, bond quotations (bid price, ask price, determination of transaction price), quotation frequency, frictions (constant and proportional commissions, indexes, updating, different rates (spot, actuarial etc.), sensitivity, duration, convexity. Derivatives Market, main options on shares, the problem of price evaluation. Notions of risk in finance: risk measures, ‘utility function’, variance expectancy method.

2. Mono-period Financial Market

Fundamental concepts, such as hedge portfolio, portfolio switching, arbitrage, AOA, dublicable products, neutral risk measure, complete and incomplete markets, price of arbitrage for a derivative, Fundamental Theorem, call-put parity. Portfolio Management in a mono-period financial market, Markowitz’ Model of variance expectancy.


ECTS: 4

Prerequisites: No


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Bibliography:

[1] I. Ekeland, N. Fintz and E.Taflin: Les marchés financiers et gestion de portefeuille,

Preprint 2005

[2] S.R. Pliska: Introduction to Mathematical Finance, Blackwell 2000


[4] J.C. Hull: Options, Futures & Other Derivatives, Prentice-Hall 2000

[5] R. Cobbault: Théorie Financiére, Editions Economica, Paris 1992


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Measure and Integration

Responsible: M.Manolessou

Teachers: M. Manolessou + Abdessalam El Janati

Teaching Objectives: To give the students a deeper understanding of the theory of Measure and Integration, following Lebesgue’s work.

Content of Teaching: Introduction to Lebesgue’s Theory of Measure and Integration, Fundamental Theorems, Applications to spaces of Probability Measures.


ECTS: 2


1. s-fields-Measurable Spaces- Filtrations – Measurable Applications. Borel s-Fields. + Tutorial Introduction to the theory of Measure– Properties -Sets of Measure Zero.

2. Equivalent Measures (Radon-Nikodym Theorem), Measured Spaces, Probability Measures + Tutorial.

3. Integral and Lebesgue Measure – Lebesgue Integral of positive step functions, positive measurable functions and general measurable functions functions.

4. Fundamental Theorems: Lebesgues:Monotone and Dominated convergence

5. Product Spaces. Fubini’s Theorem + Tutorial

6. – 7. Lp -Functional Spaces,Orthogonal Projection Properties (+ Applications)+Tutorial

Prerequisites: Analysis + Algebra (Bachelor level), Probabilities

Evaluation: Final examen of 3 hours +Continuous control

Bibliography:

1. P.Billingsley Probability and Measure

2 J. Dieudonné: Foundations of Modern Analysis, Academic Press New York


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3. B. Friedman: Principles and Techniques of Applied Mathematics, J.Wiley and Sons

4. A. Guichardet: Intégration-Analyse Hilbertienne, Paris X- Polytechnique- Ellipses

5.W. Rudin: Principles of Mathematical Analysis, (MacGraw- Hill Company 1964)

6.A.J.Weir Lebesgue Integration and Measure Cambridge University Press 1988


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Optimization


Teachers: M. Manolessou, Irina Kortchemski

Teaching Objectives: To give the students the necessary skills to apply the methods of linear and non-linear optimization linked to Duality. Methods of Descent, Conjugate Gradient Method.

Total Hours (lecture, tutorials, practical lab): 25 (10, 7, 8 )

ECTS: 2,5


1. Linear Optimization (a) Reminders (+Tutorial), (b) Duality (+Tutorial)

2-3. Convexity and non-linear optimization (with and without constraints) + Tutorial

Lagrangian Duality - Multiplicateurs of Lagrange-Kuhn Tucker + Tutorial

4. Dynamic Programming following Bellmann. Determinist cases, discrete and continuous cases. + Tutorial

5. Optimization without constraints. Direct methods in one dimension: dichotomy. Rules of Or + Wolfe + Tutorial

6. Methods of descent: gradient, conjugate gradient, Newton and quasi-Newton methods. + Practical Lab

7. Numerical minimization algorithms with constraints: projection method, Lagrange-Newton method, penalization method, Uzawa’s Algorithm + Practical Lab

8. Stochastic numerical algorithms: particle methods, simulated annealing, genetic algorithms + Practical Lab

Prerequisites: linear and non-linear algebra (bachelor level), basic functional algebra (bachelor level)

Evaluation: Final exam of 2 hours + graded Practical Labs

Bibliography:


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G.Dantzig: Linear programming and Extensions, Princeton, N.J.Princeton, University Press, 1963

R.Faure: Précis de Recherche Opérationnelle, Dunod (Paris 1979)

M. Minoux: a) Programmation Mathématique (Dunod 1975), b) Programmation Linéaire (classes from l'Ecole Nationale Supérieure des Télécommunications, Paris 1975)

C. Papadimitriou and K.Steigliitz: Combinatorial Optimization: Algorithms and Complexity, Englewood Cliffs, N.J. Prentice-Hall 1982

Operations Research R.E.A. 1983, Problem Solvers

A. W. Tucker: Recent advances in Mathematical Programming (Mc GrawHill, New York)

W.L.WIinston: Operations Research: Applications and Algorithms, PWS-KENT (1991)

J.F.Bonnans, J.C. Gilbert, C. Lamarechal, C. Sagatzabal: Optimisation Numérique (Springer, 1998)

A. Soucharev, A. TimokhovI : Cours sur les méthodes d'optimisation, (Litterature of physics and mathematics, Moscow, 2008)

V. Lesine, U. Lisovetz: Bases des méthodes d'optimisation, (MAI, Moscow, 1998)

V. Bonnalie -NoelL: Méthodes d'optimisation, (course from ENS 2005-2006)

S. Chaznoz, A. Daare : Algorithmes de minimization, (course from Paris VII, CEA Saclay 2005)

H. Zidani, P. CiarletT: Optimisation Quadratique, (course from ENSTA 2005)

A. Attenkov, V. Saroubine: Introduction dans les méthodes d'optimisation, (Science, Moscow, 2008)

A. Bjork: Numerical methods for least square problems, (SIAM, 1996)

P.G. Ciarlet: Introduction à l'analyse numérique matricielle et à l'optimisation, (Masson, 1994)

J.B. Hiriart-Urruty , C. Le Marechal: Convex Analysis and Minimization Algorithms, (Springer, 1993)

D.G. Luenberger: Linear and Nonlinear Programming, (Addison-Wesley, 1984)

J. NocedalL, S.J. Wright: Numerical Optimisation, (Springer, 1999)


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Jump Processes + Applications

Responsible: Eva Löcherbach

Teacher: Eva Löcherbach

Teaching Objectives: Jump processes constitute the main tools allowing to model problems linked to the theories of insurance, finance, image analysis etc. The Poisson Process is a starting point for the construction of numerous processes, such as Levy’s, used in finance and for climatic models, the process of coalescence in physics, point processes etc.

The aim of this course is to give an introduction to Poisson processes in order to get the students familiar with the notion of Poisson processes and point measures. This class is the basis for the M2 course ‘Lévy Process, finance and simulation’ and is strongly recommended to all those who whish to follow the M2 class.


ECTS: 3


1. Introduction: Jump processes as insurance or financial market models (Merton’s model). Link between jump processes and counting processes. Random point measures and random geometry.

2. Exponential times and loss of memory. Definition of standard Poisson process. Law of its jump times.

3. Basic properties of the standard Poisson process: independent and stationary increments.

4. Link between order statistics and uniform variables.

5. Law of large numbers and waiting time paradox.

6. Another characterization of Poisson processes by Markov’s property.

7. Compensated and compound Poisson processes. Fourier Transform.

8. Merton’s (simplified) model in finance, strategies and stochastic integral related to a compound Poisson process.

9. Martingale measures and Girsanov’s theorem for compound Poisson process. Option evaluation and pricing.


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Prerequisites: Probability (bachelor level)


Bibliography:

[1] P. Brémaud: Point processes and queues, martingale dynamics, Springer 1981.

[2] J.F.C. Kingman: Poisson processes, Oxford University Press 1993.

[3] D. Lamberton, B. Lapeyre: Introduction au calcul stochastique appliqué à la finance, Éditions Ellipse, 1997.


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Stochastic Processes (Discrete and Continuous Time)


M. Manolessou, Irina Kortchemski

Teaching Objectives: to give the students a deeper understanding of the theory of discrete and continuous time Stochastic Processes and their applications.


Discrete Time Stochastic Processes: independent and stationary increments, Markovian Processes (Markov Chains, Poisson Process, Martingales, Stop Time, Snell’s Envelope and Applications)

Continuous Time Stochastic Processes with independent and stationary increments, Martingales, Brownian Movement, Doob-Meyer Decomposition, Stochastic Integral (Itô), Stochastic Differential Equations, Stochastic Optimal Control, Bellmann, HJB

Total Hours (lecture, tutorials, practical lab): 55 (29h, 26h, 0)

ECTS: 5,5


1. (Reminders) Convergence of random variables, random Vectors,Tranformation properties, Conditional expectation values, Gaussian vectors, L² Spaces, Orthogonal Projection, Filtrations. Reminders on the measures of the Radon-Nikodyn Theorem. + Tutorial

2-3. Discrete time Stochastic processes with independent and stationary increments. Filtrations; Poisson Process. + Tutorial

4-5. Markov Processes (discrete time), Markov Chains (homogenous chains and associated graphs; transition matrix; dynamic evolution of a system, represented by the process and stability; Equivalence classes (Absorbtive-recurrent and Transitve))+Tutorial

Markov Chains (follow-on): Ergodic Theorems – Foster Criteria, Mean time of first return, Absorbtion probabilities by Recurrent classes + Tutorial

6-7. Discrete time Martingales (Application: Binomial Model); Quadratic Variation; Doob Meyer decomposition of sub-Martingales, Compensated Poisson Process.

8. Stopping Time, Stopped Processes, Snell Envelope and Applications + Tutorial


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9-10. Continuous time Stochastic Processes, Filtrations– Martingales – Stopping Time – Doob Meyer decomposition of sub-Martingales, Quadratic Variation (Compensator) and Crossed Variation (Doob-Meyer) + Tutorial

11-12. Continuous time Stochastic processes with Independent and Stationary Increments. Brownian Motion, Brownian Vectors, Martingale Representation, Exponential Martingale. + Tutorial

13-14. Stochastic Integrals, Diffusion Processes, Stochastic Differential Equations, Itô’s Theorems in one and several dimensions. + Tutorial

15. Girsanov’s Theorem - Applications - Solution- Feynmann Kac. Black Scholes model+ Tutorial

16-17-18. Discrete and Continuous Optimal Control (Deterministic and Stochastic) following Bellmann. Hamilton Jacobi Bellman equation (H.J.B.)

Bibliography: 1. P. Bremaud “Introduction aux probabilités” Edition : Springer et Verlag

2. P. Billingsley “Probability and Measure” (Wiley Series In probability 1995

3. T.Bjork " Arbitrage Theory in continuous time " (Oxford University Press. 4.I Karatzas - St.E.Schreve ``Methods of mathematical Finance'', Springer, 1998

5 I.Karatzas - St.E. Schreeve "Brownian Motion and Stochastic Calculus" (Springer 2me édition) 6. D. Lamberton, B. Lapeyre: ``Introduction au calcul stochastique appliqué à la finance'' Ellipses, Paris (1999) 7. T.L.Saaty “Mathematical methods of Operations Research”

8. Setti- Thompson “ Optimal Control Theory” (Springer) (Dover) 9. P. Wittle

“Time Dynamic Programming and Stochastic Control” (Vol.1 John,-Wiley and Sons N.York (1982)

10. W.L. Winston: Operations Research Applications and Algorithms, (W.S.Kent and

Duxbury Press Belmont 3me ed. 1994)


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Monte-Carlo Simulation


Teachers: M. Manolessou and I. Kortchemski

Teaching Objectives: Learning and application of different types of Stochastic process Simulation.

Content of Teaching: Simulation of discrete and continuous random variables. Monte Carlo Integration. Simulation of the Brownian movement and the associated financial mathematical models.


ECTS: 3


1. Reminders on discrete and continuous random variables + Tutorial

2. The role of the Uniform (Random) law in the simulation of discrete random variables (fundamental theorem) + Tutorial

3. Simulations and estimations of parameters – mean and variance + Tutorial + Practical Lab

4. Simulation of continuous random variables, with an explicit repartition function; variance reduction + Tutorial + Practical Lab


6. Two simulation methods of the Normal random variable + Practical Lab


8. Two simulation methods of the Brownian movement + Practical Lab

9. Tutorial + Practical Lab (Simulation)

10. Tutorial + Practical Lab (Applications)

Prerequisites: Basic Probabilities and Stochastic Processes (Brownian Movement)

Evaluation: Project

Bibliography:


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1. .A.O. Allen: Probability - Statistics and Queuing theory with Computer Scheme

Applications, (Acad. Press 1990)

2. P. Bremaud: Introduction aux probabilities, Edition: Springer et Verlag

3. R. Faure - A. Api: Guide de la Recherche Operationnelle, (Masson)

4. B.D. Ripley ``Stochastic Simulation'' (Wiley 1986)

5. S.M. Ross: Initialisation aux probabilities, (Press Univers. et Polytechniques Romandes

Diffusion)




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Bloomberg Trading Room Teacher: Yalçin Aktar

Teaching Objectives : Using the practical labs with mathematical exercises, give the students some applications of Bloomberg in Option Pricing, Implicit Volatility, Fixed Income, Portfolio Risk Management.

Total Hours (practical labs, project): 30

ECTS: 3


Practical Lab 1 Introduction to Bloomberg

Practical Lab 2 The Stochastic Volatility Model

Practical Lab 3 Option Pricing

Practical Lab 4 Implied Volatility

Practical Lab 5 Fixed Income

Practical Lab 6 Portfolio Risk Management

+

Project

Prerequisites: Mathematics, bachelor level, Stochastic Processes, Finite Difference Methods, Numeric Analysis

Evaluation: graded practical labs + project

Bibliography: Theoretical Courses of Master Quantitative Finance and Risk Management


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Object Oriented Design

Teacher: Bernard Glonneau

ECTS: 2

Objective: to give the students the main methods and techniques which allow the passing from the abstract analysis of a problem and its solution to its implementation in any object-oriented programming language.

Total Hours (lecture, tutorials, practical lab): 20 (8, 12, 0) Evaluation: theoretical exam and project

Prerequisites: UML

Content of Teaching: Reminders of the notions of object and classes

The links between objects of analysis and design

The two main principles of object-oriented design

Reutilisability through inheritance (white box) and composition (black box) From an individual solution to a generic solution: interfaces

The concept of design patterns

Creational Patterns: singleton, factory, abstract factory and builder

Structural Patterns: adapter, composite, façade, decorator, proxy

Behavioural Patterns: chain of responsibility, command, observer, MVC: model-view-controller, iterator

Bibliography:

E. Gamma, R. Helm, R. Johnson, J. Vlissides: Design Patterns - Elements of Reusable Object-Oriented Software, Addison-Wesley

C. Larman: Applying UML and Patterns: An Introduction to Object-Oriented Analysis and Design and Iterative Development, Craig Larman

C++ and VBA Programming

Specifications

Analysis

Design Development


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Teachers: Bernard Glonneau, Bartholomew George

Objective: In the development of software applications of scientific calculations, C++ is frequently used. In this course we will work on the programs of scientific calculations by bringing together the best of both numerical calculation and object approach.

ECTS: 3 Total Hours (lecture, tutorials, practical lab): 30 (15, 15/15)

Evaluation: theoretical exam, project

Prerequisites: C language


Reminders of C language

Function Pointers

C++ class implementation

header folder: definition of body folder class: method definition

constructors and deconstructors

dynamic allocation and new operator

operator overloading

templates

STL template library

Bibliography: B. Stroustrup: The C++ Programming Language (Third Edition and Special Edition), Addison-Wesley

A. Alexandrescu: Modern C++ Design Generic Programming and Design Patterns Applied, Addison-Wesley Professional

Introduction to Quantitative Finance


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Syllabus of M2


30

Model Calibration

Responsible: I. Kortchemski



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Teaching Objectives: In this class we show the main aspects of the calibration of financial models, used for evaluating and hedging derivatives.

Content of Teaching: Reminders of the evaluation and hedging of derivatives. Monte Carlo simulations of Discrete hedging, Calibration of implied volatility, Cubic splines interpolation, Calibration of local volatility, Dupire’s equation. Introduction to ill-posed inverse problems. Numerical solution of inverse problem. Calibration (via optimization) of the CEV model.


ECTS: 3


1. Hedging and the deduction of Black and Scholes’ equation. Self-financing portfolio strategies. Delta and Gamma-Hedging in the complete market. Hedge error analysis. Final Profit and Loss statement studies. Robustness of Black and Scholes’ formula. Hedging with dividends. Hedging with transaction cost. Leland’s model.

2-3. Practical Lab 1 (TP1). Delta-hedging simulation in complete and incomplete markets.

Content of TP1: Simulation of hedging portfolio. Discrete Delta and Gamma-hedging. Study of the mean and variance of the final Profit and Loss. Simulations of hedging with stochastic volatility.

4. Practical Lab 2 (TP2). Calibration of implied volatility in Black and Scholes’ model with and without dividends. Construction of Volatility smile profile by cubic spline interpolation. Contstruction of implicit volatility surface. Nadaraya-Watson Estimator.

5. Fokker-Planck Equation. Deduction of Dupire’s Equation. Link between local and implicit volatility. Dupire’s formula.

6. Practical Lab 3 (TP3). Calibration of local volatility with Dupire’s formula.

7. Inverse Problems. Regularization of ill- posed problems. Tikhonov’s regularization.

8. Calibration and optimization. Algorithms: Newton, simulated annealing, evolutionary algorithms.

9. Practical Lab 4 (TP4). Numerical solution of Dupire’s equation and calculation of the option’s Vega via the implicit Crank-Nicolson method. Calibration of local volatility in the CEV (Constant Elasticity of Variance) model. Levenberg-Marquardt algorithm.

10. Heston’s stochastic volatility model. Hedging in Heston’s model. Risk premium. Analytical solution of Heston’s PDE. Characteristic Function. Calculation of integrals. Numerical solution of Heston’s equation. Calibration of Heston’s model.


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Prerequisites: Evaluation of derivatives, Numerical solution of PDE, optimization.

Evaluation: graded practical labs

Bibliography:

D. Lamberton, B. Lapeyre: Introduction au calcul stochastique appliqué à la finance, Ellipses, Paris (1999).

Y. Achdou, O. Pironneau: Volatility Smile by Multilevel Last Square, International Journal of Theoretical and Applied Finance vol. 5, No. 2 (20002).

R. Cont, P. Tankov: Nonparametric calibration of jump-diffusion option pricing models, Journal of Computational Finance, 7(3), 2004.

P. Tankov: Calibration des modèles financiers et la couverture des produits derives, class from Paris VI.

L. Andersen, J. Andersen: Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Method, Review of Derivatives Research, 4, 231-262, 2000

S.B. Hamida, R. Cont: Recovering volatility from option prices by evolutionary optimization, Journal of Computational Finance Vol.8, Number 4, 2005

S. Mikhailov, U. Nodel: Heston’s Stochactique Volatility, Model Implementation, Calibration

Coleman, Li, Verma: Reconstructing the unknown volatility function, Journal of Computational Finance Vol.2, Number 3, 1999

Jackson, Suli, Hiwison: Computation of deterministic volatility surfaces, Journal of Computational Finance Vol.2, Number 2, 1998-1999

Lagnado, Osher: A tecnique for calibrating derivative security pricing models: numerical solution of an inverse problem, Journal of Computational Finance Vol.1, Number 1, 1997

Dupire: Pricing with smile. RISK ,1994

Business Evaluation

Responsible: Philippe FOULQUIER

Teacher: Philippe FOULQUIER


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Teaching Objectives: To give the students a deepened knowledge of the valorization methods of industrial and financial companies.

Content of Teaching: Based on the theory of CAPM and APT, valorization methods are studied according to the patrimonial approach (ANC, ANR, EEV), via flux (Free Cash Flow to Equity, Free Cash Flow to the Firm), patrimonial-flux mixes (EVA, MVA) and via comparables. From this basis, we also implement a dashboard in order to better understand the profitability of allocated and surplus capital, the interest of one approach by the sum of the parts. Many case studies will allow us to see how financial analysts implement these theories.


ECTS: 2,5

Prerequisites: Mono-Period Financial Management, Discrete Time Stochastic Processes, Portfolio Management (discrete time)


Bibliography:

Richard A. Brealey, Stewart C. Myers, Franklin Allen: Principles of Corporate Finance, Eighth Edition, McGraw-Hill/Irwin, 2004

S.A. Ross, R.W. Westerfield, J.F. Jaffe: Finance Corporate, Seventh Edition, McGraw-Hill/Irwin, 2005

Aswath Damodaran: Investment Valuation, Second Edition, Wiley, 2002

Portfolio Management



Teaching Objectives: Introducing Dynamic Strategies of portfolio management, and their constructions, in the context of a global financial market with different types of assets:


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shares, bonds (fixed income), change rates and derivatives. A portfolio theory, which brings together all the different instruments, is formulated. The problem of optimal strategies is then solved by methods of duality.

Content of Teaching: Reminders on efficient asset portfolios: mono-period (Markowitz’ portfolio), and discrete time (Samuelson’s portfolio).

Introduction of efficient asset portfolios in continuous time: Merton’s portfolio, theorem of common placement funds.

In the rate models of Heath, Jarrow and Morton, the introduction of efficient portfolios of rate products, in the sense of optimal use expectancy.

Total Hours (lecture, tutorials, practical lab): 25 (12.5, 12.5, 0)

ECTS: 2.5

Prerequisites: M1 and theory of contingent claim


Bibliography:

T. Björk: Arbitrage Theory in Continuous Time, Oxford University Press 1998.

I. Ekeland, N. Fintz, E. Taflin: Les marchés financiers et gestion de portefeuille, Preprint 2004.

I. Ekeland, E. Taflin: Optimal Bond Portfolios, http://arxiv.org/abs/math.OC/0510333

R. Merton: Lifetime Portfolio Selection Under Uncertainty: The Continuous-Time case, Rev. Economics and Stat. 51, 247--257 (1969).

M. Musiela, M. Rutkowski: Martingale Methods in Financial Modelling, Applications of Mathematics 36, Springer-Verlag 1998.

Interest Rate, Exchange and Inflation Markets



Teaching Objectives: Introducing the modeling, in continuous time, of interest rate, exchange and inflation markets, and the evaluation of derivatives.

http://arxiv.org/abs/math.OC/0510333


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Content: Zero-Coupon bonds, different rates (spot, forward), SWAPS, term structure deformation, HJM and No-Arbitrage conditions, spot rate models, affine models (Ho-Lee, CIR), forward measure, stock options, swaptions and other options. Exchange rate and inflation markets.


ECTS: 2,5


1. Zero-Coupon, different rates, deformation of term structure, HJM and No-Arbitrage conditions.

2. Tutorial

3. Affine Models (Ho-Lee, CIR)

4. Tutorial

5. Forward, Futures, Forward Measure and option evaluation

6. Tutorial

7. Exchange Rate Markets

8. Tutorial

9. Inflation Markets

10. Tutorial

Prerequisites: M1, mathematical financial tools, contingent claims valuation


Bibliography:

T. Björk: Arbitrage Theory in Continuous Time, Oxford University Press 1998.

I. Ekeland, N. Fintz, and E.Taflin: Les marchés financiers et gestion de portefeuille, Preprint 2007

A. Lipton: Mathematical Methods for Foreign Exchange, World Scientific 2001

M. Musiela, M. Rutkowski: Martingale Methods in Financial Modelling, Applications of Mathematics 36, Springer-Verlag 1998.


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Imperfect Markets

Responsible: Bruno Bouchard (Cermade, Univ. Paris Dauphine)

Teacher: Bruno Bouchard


The standard models of the Back and Scholes type do not take into consideration many important market imperfections, especially those related to the lack of liquidity in certain


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assets. These imperfections can result in a significant difference between the buying and selling costs (transaction costs), the impossibility of covering certain market factors (incompleteness of the market) and even by certain constraints imposed by the management strategies. The objective of this course is to explain how these imperfections can be taken into consideration in the models, and to present certain techniques of evaluation and associated risk hedging.


ECTS: 2

Dynamic Hedging and Risk Measures




To get the students familiar with the recent concepts and models for the measuring and comparison of risks. In general, risks can’t be eliminated by the possession of a hedge fund.


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This class aims to teach how to construct, in a continuous time market, an optimal hedge for a given risk measurement.


1. The measures of monetary, convex, coherent risks and their representations.

2. Risk of dynamic hedging

3. Minimizing risk


ECTS: 2

Bibliography:

[1] P. Embrecht, C. Klüppelberg, T. Mikosch: Modelling Extremal Events, Springer, 1997 [2] H. Föllmer, A. Schied: Stochastic Finance, Walter de Gruyter 2002

Advanced Numerical Methods for PDE in Finance

Responsible:I. Kortchemski


Objective: This class will teach the students the fundamental elements of numerical analysis which is useful in financial applications.


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Content of Teaching: Finite difference methods applied to Black and Scholes’ equation. Crank-Nicolson Method. Feynman-Kac Theorem and Monte-Carlo methods. Multidimensional PDE. Applications in finance.


ECTS: 2,5


1. Introduction and reminders of Finite Difference Methods. Black and Scholes’ Equation. Vanilla Option. Discretization via Euler’s and Crank-Nicolson’s methods. + Tutorial

2. Implementation of Dirichlet and Neumann’s boundary conditions for Vanilla Option. +Practical Lab 1

3. Feynman-Kac’s Theorem. Link between determinist and stochastic methods. Price calculation of Vanilla Option via the Feynman-Kac Theorem. + Practical Lab 2

4-5. Stochastic differential equations. Discretization. Reminders of Ito’s lemma. Price calculation of Asian and Lookback Options via Monte-Carlo methods. + Tutorial + Practical Labs 3 and 4.

6. Multidimensional PDE. Change of Variables. Dimension reduction of 3D Black and Scholes’ equation. Application: Asian Options with Fixed Strike and Floating Strike. + Tutorial + Practical Lab 5

7. Multidimensional PDE. Application: Lookback Options. Heston’s PDE. Two-asset options. Exchange Option. Margrabe’s Formula. + Tutorial

8. Multidimensional PDE. Direct discretization of three-dimensional Black and Scholes’ Equation for an Asian Option. Implementation of a 3D numerical scheme. Boundary Condition Treatment. +Tutorial

9. Numerical solution methods: PSOR, Newton, Howard, Spliting Algorithms. Application: American Options + Tutorial

10. Portfolio Optimization. Finite difference methods for Hamilton-Jacobi-Bellmann Equation. +Tutorial + Practical Lab 6

Practical Lab 1 Numerical rsolution of Black and Scholes’ equation for the Vanilla option via Euler’s method.

Practical Lab 2 Pricing of the Vanilla option via the Monte-Carlo method.

Practical Lab 3 Pricing of the Asian option via the Monte-Carlo method.


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Practical Lab 4 Pricing of the Lookback option via the Monte-Carlo method.

Practical Lab 5 Numerical solution of Black and Scholes’ equation for an asian option via Crank-Nicolson method.

Practical Lab 6 Numerical solution of the Hamilton-Jacobi-Bellmann equation.

Prerequisites: Mathematiques (bachelor level), Stochastic Processes, Finite Difference Methods, Non-linear Optimization, Numeric Analysis

Evaluation: exam and graded practical labs

Bibliography:

W.F. Ames: Numerical Methods for Partial Differential Equations, Nelson and Sons LTD. London, 1969

H.M. Antia: Numerical methods for Scientists and Engineering (Birkhäuser)

G.D. Smith: Numerical solution of PDE: Finite difference methods, Clarendon Press, Oxford, 1978

M. Rappaz, M. Bellet, M. Deville: Numerical Modeling in Material Science and Engineering, Springer

Y. Achdou, Olivier Pironneau: Computational Methodes for Option Pricing, SIAM

D.J. Higham: An Intoduction to Financial Option Valuation, Springer

J.W. Thomas: Numerical Partial Differential Equations, Springer

Y. Achdou, O. Bokanowski, T. Lelièvre: Partial Differential Equations in Finance. Class from Paris VI

P. Wilmott, J. Dewynne, S Howison: Option pricing: Mathematical models and computation, Oxford Financial Press

F. Dubois, T. Lelièvre: Efficient pricing of Asian options by the PDF approach, Journal of Computational Finance, 8(2):55-64, 2005

L.C.G. Rogers, Z. Shi: The value of an Asian option. Journal of Applied Probability, 32:1077-1088, 1995

G. Barles, Ch. Daher, M. Romano: Convergence of numerical schemes for parabolic equation arising in finance theory, Math. Models Methods Appl. Sci., 5(1)125-143, 1995


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P.A. Raviart and J.M. Thomas: Introduction à l’analyse numérique des équations aux dérivées partielles, Masson, Paris, 1983

L.A. Bordag: Symmetry reductions of a nonlinear option pricing model, arXiv:math/0604207

P. Wilmott: On Quantitative Finance, Wiley

T. Björk: Arbitrage Theory in Continuous Time, Oxford

J. Gatheral: The Volatility Surface, Wiley

F.D. Rouah, G. Vainberg: Option pricing models and volatility: Using EXCEL-VBA, Wiley

P. Wilmott, J. Dewynne, S. Howison: The Mathematics of Financial Derivatives: A Student Introduction, Cambridge University Press

Mathematical Tools in Finance

Responsible: Danielle Florens

Teachers: Danielle Florens, Mohammed Mikou

Teaching Objectives: Although the standard Brownian movement plays a central role in understanding the mechanisms of modern finance, the knowledge of more sophisticated tools, such as Martingale Theory or the Lévy process is indispensable for the studying of financial concepts, such as arbitrage theory, admissible strategies, complete markets etc. This


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program takes elements from the second year study of the Brownian movement, but only deals with non-jump continuous processes.

Content of Teaching: Conditional Expectation, Martingales, Local Martingales, Stochastic Integrals of continuous semi-martingales, quadratic variation. Stochastic exponential. Gisanov’s Theorem. Feyman-Kac Formula.


ECTS: 4

Program:

1. Reminders: Radon-Nikodym Conditional Expectancy as Density. Continuous Martingales. Maximal Inequalities.

2. Reminders on standard Brownian. Notable Brownian Martingales and Brownian Markov properties.

3. Stochastic Integrals of Continuous Semi-Martingales. Local Martingales. Semi-Martingales. Quadratic Variations. Stopped Martingales. Doob Inequalities.

4. Ito’s multidimensional Formula. Integration by parts. Applications in finance: attainable markets. Variable change. Levy’s characterization of Brownian. Burkolder-Davis inequalities.

5. Measure change. Exponential Martingale. Doleans-Dade’s Stochastic Exponential. Novikov’s Condition. Esscher’s Change. Applications in Finance.

6.Girsanov’s multidimensional Theorem. Bayes’ Formula. Martingales Representation Theorem.

7. Stochastic Differential Equations. Existence and Uniqueness of Strong Solution. Markovian Properties.

8. Feyman-Kac’s Formula.

9. Some applications in Finance.

Prerequisites: M1


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Practical Portfolio Management in Fixed Income

Responsible: Adjriou Abdelak

Teacher: Adjriou Abdelak

Teaching Objectives: The main objective of this lecture is to explain the mechanisms, the pricing and the asset management approach to fixed income products.


ECTS: 2


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Program:

1-2. Review of the actuarial calculus and pricing of fixed income products :

Calculation of zero coupons, actuarial and forward rates. Presentation of fixed income products: Money market, Bonds, Swaps, Futures, Futures, Credit default swaps, plain vanilla options. Samples: BUND, EURIBOR, TNOTES and EURIBOR6M swaps

3-4. Presentation of different yield curves:

Bond and swap curves

Calculation of the zero coupon curve by bootstrapping, calculation of the forward rate curve

Samples: SWAP EURIBOR curve and US TNOTE curve

5-6. Presentation of fixed income portfolios:

Liquidity, rates, credit and volatility risks

Rates strategies: bullet, barbell, butterfly. Carry, roll down calculations

Samples: fixed income and credit bond portfolios

Prerequisites: M1


Bibliography:

Martellini, Priaulet, Priaulet: Fixed income securities – Valuation, Risk management and Portfolio Strategies


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Jump Processes and Aplications

Responsible: Eva Löcherbach

Teacher: Eva Löcherbach

Teaching Objectives: After recalling basic facts concerning Brownian motion and the Black-Scholes formula, we turn to processes having jumps and introduce the concept of Lévy processes. A first part of the course is devoted to the investigation of basic properties of Lévy-processes (Lévy-Khintchine formula, the Lévy-Itô decomposition). Then we turn to questions that are at the heart of mathematical finance: the Girsanov theorem and the Itô formula for Lévy processes. We show how to apply these concepts for option pricing. The course is supplied by exercises.

Content of Teaching: see program


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Total Hours (lecture, tutorials, practical lab): 20

ECTS: 2


1. Processes in continuous time and Brownian motion

2. General definition of Lévy processes, examples: Poisson process, compound Poisson process

3. Poisson random measures (PRM) and integrals with respect to PRM

4. Lévy-Itô decomposition

5. Properties of sample paths of Lévy processes: finite/infinite activity, bounded/unbounded variation. Subordinators.

6. Lévy measures and moments

7. Martingales and Lévy measures

8. Girsanov's theorem and Esscher transform. Itô formula for Lévy processes

9. Option pricing: Lévy driven Market models, incompleteness, Fourier transform method for option pricing

10. Simulation of Lévy processes

Mathematical Statistics

Responsible: Abdessalam El Janati

Teachers: Abdessalam El Janati- Abderrahim Bourhattas

Teaching Objectives: Allow the students to acquire some knowledge and skills in estimators and the methods of estimation. To give the students all the necessary information of statistical Inference to apply the appropriate methods for a data analysis of a population using qualitative and quantitative measures.

Content of Teaching: Estimation: usual estimators. Confidence Interval. Hypothesis tests Neyman Pearson – Tests of mean- tests of Variance- Regression and Correlation Analysis


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Total Hours (lecture, tutorials, practical lab): 20 (10h, 10, 0)

ECTS: 2

Program (sessions of 3 hours)

1. Estimation: qualities of an estimator – usual estimators (Consistent efficient).

2. Estimation: confidence interval + Tutorial

3. Estimation: Maximum Likelihood + Tutorial

4. Hypothesis Test: Introduction-Generalities (risk hypotheses) + Tutorial

5. Hypothesis Test: reference value tests (mean) + Tutorial

6-7. Hypothesis Test: reference value tests (variance – frequence) + Tutorial

Hypothesis Test: Neyman Pearson + Tutorial

Hypothesis Test: sample comparison (mean) + Tutorial

Regression and Correlation Analysis

Prerequisites: Probabilities (bachelor level)

Evaluation: exam of 2 hours

Bibliography:

[2] T. W. Anderson: An Introduction to Multivariate Statistical Analysis, Wiley, 1958

[3] C. Baskiotis, J. Raymond, A. Rault: Parameter identification and discriminant analysis for jet engine mechanical state diagnosis, IEEE Conf. in Dec \ Control, 1979, p.648

[4] D. A. Belsley, E. Kuh, R. E. Welsch: Regression Diagnostics, Wiley, 1980

[5] J.-P. and F. Benzécri: Pratique de l’analyse des données, Dunod, 1984

[6] F. Cailiez, J.P. Pages: Introduction à l’analyse des données, 1976

[7] CEA: Statistique appliquée à l’exploitation des mesures, Tomes I et II, Masson, 1978

[8] L. L. Chao: Statistics: Methods and Analyses, McGraw Hill, 1969


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Theory of Contingent Claims



Teaching Objectives: Give the students a deepened knowledge of the theory of contingent in continuous time and its application.

Content: Recall the theory of contingent claims in discrete time and deepen its stochastic modeling with interest rates applied to forwards and futures. Introduction of different types of options: European, American and ‘exotic’.

Introduction of continuous time financial markets, type standard Black-Scholes: B-S Equation, Feynman-Kac representation of solutions, B-S price formula for a European option, the Greeks.


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Introduction of generalized continuous time financial markets and arbitrage pricing. No-Arbitrage conditions, Equivalent Martingale Measures, Completeness and Valuation of contingent claims. Price forks in incomplete markets. Examples of option valuation in the case of stochastic interest rates, stochastic volatility (Heston’s model) and local volatility.

Total Hours (lecture, tutorials, practical lab): 45 (22.5, 22.5 0)

ECTS: 4.5

Prerequisites: M1


Bibliography:

T. Björk: Arbitrage Theory in Continuous Time, Oxford University Press 1998

I. Ekeland, N. Fintz, E. Taflin: Les marchés financiers et gestion de portefeuille, Preprint 2004

H. Föllmer, A. Schied: Stochastic Finance, Walter de Gruyter 2002

J.C. Hull: Options, Futures & Other Derivatives, Prentice-Hall 2000

M. Musiela, M. Rutkowski: Martingale Methods in Financial Modelling, Applications of Mathematics 36, Springer-Verlag 1998

R. Pliska: Introduction to Mathematical Finance, Blackwell 2000

Simulation


Teacher: M. Manolessou

Teaching Objectives: Learning more deeply the Stochastic Processes Simulation for application and modelling of different types of problems in Finance.

Content of Teaching: Extension of the course of Simulation Monte Carlo Simulation of discrete and continuous Processes in M1 (Poisson, Binomial Brownian) to more general stochastic mathematical models in Finance.

Volume en heures Total (Cours, TD, TP) : 25 (10h, 15,10) ECTS : 2


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Reminders:

1.1 Three simulation methods of the Normal random variable + Practical Lab

1.2 Simulations and estimations of parameters – mean and variance; Simulation methods of the Brownian movement + Practical Lab

2 - 3 Brownian Bridge. Comparisons and tests of the different limits of the parameters and Graphical representations + Practical Lab

4 -5 The Geometrical Brownian motion Comparisons and tests with The Binomial model aproach and the analytic Solution of Black-Scholes model. Graphical representations + Practical Lab.

6 Ornrstein – Uhlenbeck model. Different aproaches of the solution and graphical representations with comparisons and parameters tests. + Practical Lab.

7- 8 Heston model. Simulation and Graphical representations of the different steps of the solution and

comparison of the two different methods for the construction of the Volatility Smile. + Practical Lab

.(tests of comparison for different values of the associated parameters)

Prerequisites: Basic Probabilities and Stochastic Models; Simulation Monte-Carlo of M1.

Evaluation: Project

Bibliography:

1..A.O. Allen: Probability - Statistics and Queuing theory with Computer Scheme Applications, (Acad. Press 1990)

2. B.Bouchard Méthodes de Monte-Carlo en Finance ( Cours Univ.Dauphine)(2003)

3. P. Bremaud: Introduction aux probabilities, Edition: Springer et Verlag

4. E.Taflin Notes de cours en 3me année (IFI)

5. B.D. Ripley ``Stochastic Simulation'' (Wiley 1986)


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6. S.M. Ross: Initialisation aux probabilities, (Press Univers. et Polytechniques Romandes Diffusion)



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