MathMods :: Joint MSc

Sem3 UNS Finance

Sem3 UNS Finance

Applications  @  UNS  30 ECTS credits

Mathematical modelling with applications to finance

The semester in Nice (UNS) will focus on "Mathematical Modelling Applications to Finance". Given the economic and social landscape, it aims to train engineers provided with a double strong ability, in both rigorous mathematics and tools from the specific application field, together with a deep experience in informatics. Precisely, students will be supplied with a strong theoretical and numerical mathematical and computational background as used in banks and insurance companies and will be given a solid knowledge in financial analysis including a specific course about the related stakes and rules that have emerged since the financial crisis. In particular, major objectives of the master are to train highly qualified engineers able both to apply sophisticated mathematical tools to describe, analyze and simulate complex systems such as trading markets and to keep thinking their activity as connected with real economy. Most of the lectures will be given within the framework of the Master's programme "Mathematics and Interaction" that will start in next September in Nice. Indeed, a branch of this new programme will consist of a continuation of the older Master's programme "Engineering Mathematics in Economy and Actuarial Sciences", referred to as IMEA in the French denomination. In addition to the academic courses, workshops will be organized in collaboration with professionals lecturers from local or national companies or researchers from the Economy Department or from other institutions such as INRIA. Public research institutions and private industrial groups are the principal employers for engineers issued from this track, including banks, universities, companies specialized in simulation and consulting from any industrial field, etc...

 

Below you can find information about the subjects for this semester.

  • Stochastic calculus and applications to maths finance [6 credits]

    Stochastic calculus and applications to maths finance

    • ECTS credits 6
    • Semester 3
    • University University of Nice - Sophia Antipolis
    • Objectives

       

      Stochastic calculus is the right mathematical theory for modelling the dynamics of time-continuous financial markets. Prices of assets are expressed as solutions of stochastic differential equations driven by a Brownian motion and wealth of investors as stochastic integrals. Basic hedging methods rely on a neutral-risk description of the underlying financial markets, corresponding to a new of measuring randomness. Students aiming at working in mathematical finance must develop a strong knowledge in stochastic calculus.

    • Topics

       

      Brownian motion. Filtration and financial information; stopping times. Itô integral, Itô processes and financial strategies. Martingale processes, Girsanov theorem and arbitrage opportunities. Stochastic differential equations and spot prices models. Black-Scholes model.


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  • Probabilistic numerical methods [6 credits]

    Probabilistic numerical methods

    • ECTS credits 6
    • Semester 3
    • University University of Nice - Sophia Antipolis
    • Objectives

       

      Probabilistic numerical methods are widely used in mathematical finance for pricing financial derivatives and computing strategies. The course will present the basic methods used for simulating random variables and implementing the Monte-Carlo method. Simulation of stochastic processes used in mathematical finance, such as Brownian motion and solutions to stochastic differential equations, will be addressed. Several examples of applications to pricing of financial derivatives will be also proposed.

    • Topics

       

      Sampling methods in finite dimension. Discretization of diffusion processes; strong and weak errors. Monte-Carlo methods for option pricing, variance reduction, control variates method, importance sampling. Monte-Carlo methods in risk management.


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  • Advanced stochastics and applications to mathematical finance [6 credits]

    Advanced stochastics and applications to mathematical finance

    • ECTS credits 6
    • Semester 3
    • University University of Nice - Sophia Antipolis
    • Objectives

       

      This course will focus on the theory of stochastic optimal control and its applications in mathematical finance. The lectures will address both time discrete and time continuous models. A special care will be paid to the derivation of the dynamic programming principle and to the analysis of the corresponding Hamilton-Jacobi-Bellman equations. Typical examples of applications will include optimal allocation problems, based on Morgensten and Von Neumann utility functions, and optimal

    • Topics

       

      Stochastic control. Dynamic programming principle; Hamilton-Jacobi-Bellman equations. Optimal allocation problem; Utility functions and mean-variance criterion. Optimal stopping; American options. Cox-Ross-Rubinstein and Black Scholes models.


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  • Advanced statistics / Statistics with SAS & CART and random forests for high-dimensional data [6 credits]

    Advanced statistics / Statistics with SAS & CART and random forests for high-dimensional data

    • ECTS credits 6
    • Semester 3
    • University University of Nice - Sophia Antipolis
    • Objectives

       

      This course addresses the implementation of advanced numerical methods on softwares that are used in companies, like R or SAS. First, students will learn how to implement various methods (PCA, linear regression and variables selection, clustering...) on real data sets and to interpret the results. The second part will be devoted to decision trees theory, in particular CART and Random Forests algorithms developed by Breiman and his coauthors. These statistical tools define nonlinear models in the regression and classification context and are widely used in companies, including bank and insurance companies.

    • Topics

       

      CART algorithm. Random Forests. Variable selection with CART and Random Forests. Random Forests on high-dimensional data.


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  • Computational methods in statistical analysis [6 credits]

    Computational methods in statistical analysis

    • ECTS credits 6
    • Semester 3
    • University University of Nice - Sophia Antipolis
    • Objectives

       

      This course will focus on several standard computational methods in data sciences and on model selection in statistics. This includes: Concept of statistical model, Supervised model selection, Cross-Validation techniques (Hold-out, V-Fold, Leave-one-out, Leave q-out), Unsupervised model selection, Penalized criteria (AIC, BIC, ICL, slope heuristic), applications of model selection to regression (ridge, lasso), density estimation, model-based classification and clustering, variable selection, dimension reduction (probabilistic PCA).

    • Topics

       

      Supervised and unsupervised model selection, Cross-validation techniques, Penalized criteria, Applications to regression.


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  • Deterministic Numerical Methods and Applications to Modelling [6 credits]

    Deterministic Numerical Methods and Applications to Modelling

    • ECTS credits 6
    • Semester 3
    • University University of Nice - Sophia Antipolis
    • Objectives

       

      The purpose of these lectures is to address deterministic numerical methods for PDEs and applications to modelling. This will include scalar hyperbolic, parabolic, linear and nonlinear parabolic PDEs. Several types of methods will be discussed, together stability, consistency and convergence issues.

    • Topics

       

      Heat equation, Advection equation, Burgers equation, Fischer-KPP, Explicit/implicit methods, Central schemes, Upwind schemes, Numerical diffusion, Numerical dispersion.


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  • Methods in finance [6 credits]

    Methods in finance

    • ECTS credits 6
    • Semester 3
    • University University of Nice - Sophia Antipolis

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  • Statistical learning methods [6 credits]

    Statistical learning methods

    • ECTS credits 6
    • Semester 3
    • University University of Nice - Sophia Antipolis
    • Objectives

       

      The purpose of this course is to provide a self-contained introduction to the field of machine learning, based on a probabilistic approach. Machine learning is indeed a very active field, which has met with great success in academia and in industry. The first part of the lectures will be dedicated to the analysis of the expectation-maximization (EM) algorithm. The second part will address latent linear models and the last one will focus on the notion of kernels.

    • Topics

       

      Machine learning, Bayesian statistics, Information theory, Classification, Regression


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University of L'Aquila, Italy (UAQ)

Department of Information Engineering, Computer Science and Mathematics, via Vetoio (Coppito), 1 – 67100 L’Aquila (Italy)

Autonomous University of Barcelona, Catalonia - Spain (UAB)

Departament de Matemàtiques, Edifici Cc - Campus UAB 08193 Bellaterra – Catalonia

Hamburg University of Technology, Germany (TUHH)

Institute of Mathematics
Schwarzenberg-Campus 3, Building E-10
D-21073 Hamburg - Germany

University of Nice - Sophia Antipolis, France (UNS)

Laboratoire J.A.Dieudonné
Parc Valrose, France-06108 NICE Cedex 2

Vienna Univ. of Technology, Austria (TUW)

Technische Universität Wien
Institute of Analysis & Scientific Computing
Wiedner Hauptstr. 8, 1040 Vienna - Austria

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