Scaling transformation and probability distributions for financial time series

The price of financial assets are, since [Bachelier L. Annales de lʹEcole Normale Supérieure 1900;3:XVII:21–86], considered to be described by a (discrete or continuous) time sequence of random variables, i.e., a stochastic process. Sharp scaling exponents or unifractal behavior of such processes has been reported in several works [Mandelbrot BB. J Business 1963;36:394–419; Peters EE. Chaos and order in the capital markets. New York: Wiley, 1991; Mantegna RN, Stanley HE. Nature 1995;376:46–49; Evertsz CJG. Fractals. 1995;3:609–616; Bouchaud JP, Potters M. Théorie des risques financiers. Aléa Saclay, 1997]. In this paper we investigate the question of scaling transformation of price processes by establishing a new connection between non-linear group theoretical methods and multifractal methods developed in mathematical physics. Using two sets of financial chronological time series, we show that the scaling transformation is a non-linear group action on the moments of the price increments. Its linear part has a spectral decomposition that puts in evidence a multifractal behavior of the price increments.