family of multiplicative models, studied before in the context of a wide spectrum of physical processes[@schenzle], belongs to the class here considered, through the transformation $x\to x^\beta$.

Summarizing, from Eqs. (\[ss0\])-(\[caseII\]), in general, the asymptotic behaviors below and above the mode are tied, such that, in a log-log scale, if one flattens the other changes rapidly. This explains why models of this class fail to describe empirical volatilities in the whole range of observed data, even under the transformation $v^2\mapsto v$.

Generalized model with additive-multiplicative structure {#Sec:add_mult}
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We analyze in this section, processes that take into account the presence of some additional source of noise. Previous works[@multi1; @multi2; @multi3] show that additive-multiplicative stochastic processes constitute an ubiquitous mechanism leading to fat-tailed distributions and correlated sequences. This extra noise represents a quite realistic feature, since, besides noise modulated by the market, other fluctuations may act directly, additively. From the stream of news, represented by a noisy signal, some are amplified or reduced by cooperative actions, others incorporated unaltered. Related ideas has been discussed in Ref. [@sornette]. Also, a model of financial markets that leads to additive-linear-multiplicative processes has been recently proposed[@spins], where the noises are identified with the fluctuating environment 