  The price of financial assets are, since Bachelier, 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. In this letter we investigate the
question of scaling transformation of price processes by establishing a new
connexion 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.
