MathMods :: Joint MSc

Sem3 UCA Finance

uca large logoApplications  @  UCA  30 ECTS credits

Mathematical modelling with applications to finance

The semester in Nice (UCA) “Mathematical Modelling with Applications to Finance" aims at providing rigorous mathematics and tools from the specific application field, together with good knowledge in informatics. Students will be supplied with a strong theoretical and numerical mathematical 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. This track aims at producing highly qualified modellers able to apply sophisticated mathematical tools to describe, analyze and simulate trading markets. The three mandatory courses are “Stochastic calculus”, “Probabilistic numerical methods”, and “Advanced Stochastics and applications to Mathematical Finance”. Optional courses deal mainly with statistical methods (three of the fours optional courses) and one course on numerical methods.

 

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 [6 credits]

    Advanced statistics

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

       

      The aim of the course is to go deeper into the theory of mathematical statistics and to address the problems of comparing estimators, multiple testing, model choice and model selection. This program will give us the opportunity to discuss various notions of quality for an estimator among which optimality and robustness.  

    • Topics

       

      Estimation, Multiple Tests, Model Selection, Robustness


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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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  • 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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