"Probability & Finance" section of the Université Pierre & Marie Curie (Paris 6)

DEA "Probabilités & Applications"

(Directors: N. El Karoui - G. Pagès - M. Yor)


The importance of the financial markets has continuously flourished over the last thirty years. This phenomenon is closely related to the deregulation of the economy, which started in the early Seventies, in particular regarding the floating exchange rates.

This expansion would not have taken place without the parallel development of a Financial Risk Industry: a large number of insurance contracts were implemented to help protect the investments of industrials (businessmen), the States and investors, from unfavorable fluctuations in the market. The most common type of contract is the ‘Call Option’, which allows for the possibility to buy a traded secutity at a given date and at a guaranteed price.

"Stochastic calculus" is the keystone of the development of this industry: in 1973, Black, Scholes and Merton stated the following principle: ‘to avoid a potential risk in the future (1 year), it suffices to hedge the day to day ‘Infinitesimal’ risk. It is possible, as a result of the ‘Ito Calculations’, to effectively compute how to do so in the real world. This led to the birth of the pricing and arbitrage hedging theories.

It allowed for substantial developments in the financial risk industry, in particular the following types of risk: stocks, interest rates, exchange rates, hybrid products, commodities, and, more recently, energy, etc. New activities appeared, such as the development of new instruments (CDS, CDO) to transfer credit risks (mortgage holders, consumption, etc).

In addition, the Official Authorities installed increasingly accurate risk indicators, (Bale II recommendations), including, for the first time, probabilistic reference models (SABR).

More generally, in parallel with the innovations within the financial industry, the development of market risk management has also been an important trend over the past decade. The developemnet of hedge funds, and alternative management methods, has given rise to new problems, with a stronger econometric flavour.

Mathematically speaking, the common ground between all of these activities, whether they are emerging, or already established, is their technicality. This can further be extended in terms of modeling, which is generally stochastic, and in terms of increasing needs of efficient numerical methods, both flexible and adapted to the problems (PDE, Monte Carlo simulation, approximation, etc.). This allows for a large amount of scope regarding inspiration and investigation for applied mathematicians.

The objective of the Master 2 « Probabilités et Finance », is to give the students the tools necessary to evolve effectively in this ever-changing industry: in joint base, theoretical bases, in pricing by arbitrage, the essential of numerical tools (Monte Carlo, PDE), and finally, via various options, to propose themes and methods in development on the financial markets.

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