Section: New Results

Quantitative finance

Participants : Michel Fliess, Cédric Join.

We are settling a longstanding quarrel in quantitative finance by proving the existence of trends in financial time series thanks to a theorem due to P. Cartier and Y. Perrin, which is expressed in the language of nonstandard analysis [78] . Those trends, which might coexist with some altered random walk paradigm and efficient market hypothesis, seem nevertheless difficult to reconcile with the celebrated Black-Scholes model. In [35] , we proved the existence of those trends for financial time series via nonstandard analysis. Moreover, a new model-free definition of the beta coefficient, which plays an important role in systematic risk management is proposed in [45] . This setting, which is based on the work [35] , leads to convincing computer experiments which are easily implementable. These works allow us to define two new technical indicators for trading systems and risk management [46] , where recent fast estimation techniques of algebraic flavor are used [25] . The first indicator tells us if the future price will be above or below the forecasted trend-line and the second one predicts abrupt changes.