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 Math finance [6 credits]

    Stochastic calculus and applications to Math finance

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

       

      This course is devoted to the introduction of the basic concepts of continuous time stochastic processes which are used in many fields : physics, finance, biology, medicine, filtering theory, decision theory. It will consist of a presentation of Brownian motion, Itô integral, stochastic differential equations and Girsanov theorem. Several applications will be given.

    • 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
    • University University of Nice - Sophia Antipolis
    • Semester 3
    • Objectives

       

      Probabilistic numerical methods are widely used in machine learning algorithms as well as 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 methods. Simulation in Scilab of stochastic processes used in mathematical finance, such as Brownian motion and solutions to stochastic differential equations, will be discussed as well

    • 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 statistics and applications [6 credits]

    Advanced statistics and applications

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

       

      This course focuses on three pillars of modern statistical inference: parameter estimation, hypothesis testing, and model selection. Its aim is to provide a good understanding of the current methods via a thorough treatment of the existing theoretical guarantees. A particular emphasis will be placed on the asymptotic setting

    • Topics

       

      Estimation, Multiple Tests, Model Selection, Robustness


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  • Stochastic control and interacting systems in finance [6 credits]

    Stochastic control and interacting systems in finance

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

       

      The course provides the basic knowledge in stochastic control, programming principle, dynamic programming equation, Hamilton-Jacobi-Bellman equation, control for counting processes. A second part addresses the theory of mean-field models. Applications to finance are considered.

    • Topics

       

      Programming principle, dynamic programming equation, Hamilton-Jacobi-Bellman equation, control for counting processes. Mean-field models as many particle limits. Applications to finance.


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  • Numerical Methods for PDEs and applications [6 credits]

    Numerical Methods for PDEs and applications

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

       

      The aim of this course is to introduce some tools in mathematical modelling and numerical simulation. In a first step, we will address finite difference methods for PDEs, with a special focus on notions like consistency, stability, numerical diffusion, numerical dispersion and convergence. Theoretical analysis of the numerical schemes will be addressed in some relevant examples. In a second step, we will present some principles and results of the variational approach for stationary models. Meanwhile, we will address element finite methods, in the 1d case and possible in higher dimension. We will code examples using the software Freefem (http://www.freefem.org)


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