A nonextensive approach to the dynamics of financial observables

We present results about financial market observables, specifically returns and traded volumes. They are obtained within the current nonextensive statistical mechanical framework based on the entropy <InlineEquation ID="Equ1"> <EquationSource Format="TEX">$S_{q}=k\frac{1-\sum_{i=1}^{W} p_{i} ^{q}}{1-q}\, (q\in \Re) \left(S_{1} \equiv S_{BG}=-k\sum_{i=1}^{W}p_{i}\,\ln p_{i}\right)$</EquationSource> </InlineEquation>. More precisely, we present stochastic dynamical mechanisms which mimic probability density functions empirically observed. These mechanisms provide possible interpretations for the emergence of the entropic indices q in the time evolution of the corresponding observables. In addition to this, through multi-fractal analysis of return time series, we verify that the dual relation q<Subscript>stat</Subscript>+q<Subscript>sens</Subscript>=2 is numerically satisfied, q<Subscript>stat</Subscript> and q<Subscript>sens</Subscript> being associated to the probability density function and to the sensitivity to initial conditions respectively. This type of simple relation, whose understanding remains ellusive, has been empirically verified in various other systems. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007