Scaling Laws in Stock Markets. An analysis of prices and volumes

In the last years fractal models have become the focus of many contributions dealing with market dynamics modelization. In this context, one of the key points concerns the estimation of the so called self-similarity parameter, that is the scaling exponent for which the finite-dimensional probability distribution functions (pdf’s) relative to different time horizons (time scales) become equal. More formally, the continuous time, real-valued process {X(t), t ∈ T}, with X(0) = 0, is self-similar with index H0> 0 (concisely, H0-ss) if, for any a ∈ R+ and any integer k such that t1,..., tk ∈ T, the following equality holds for its finite-dimensional distributions {X(at1),X(at2),...,X(atk)} d = {aH0X(t1), aH0X(t2),..., aH0X(tk)}. (1) A less restrictive definition considers the second-order stationary, real-valued stochastic process X(t), which is said H0-second order self-similar if − denoted by Y (t, a) its a-lagged increments, namely Y (t, a) = X(t+a)−X(t), and by Y (t,m) = m−1 Ptm τ=(t−1)m+1 Y (τ, 1), m, t ∈ {1, 2,...} the averaged (over blocks of lenght m) sequence − it holds V a