The fluctuations of prices and other observables in financial markets have non-trivial behaviors, such as long temporal correlations, non-Gaussianity or scaling laws, whose origin is not well understood so far. In this work we have investigated possible deterministic and stochastic mechanisms responsible for the anomalous probability distributions observed for market indexes and volumes of traded shares. In the first case, we consider the Kramers-Moyal expansion as a starting point to describe the evolution of probability densities. For the modelling of trading volumes, we consider the mixed statistics that emerges from the long-scale fluctuations of inner parameters that describe the dynamics on shorter scales. This study provides a consistent demonstration, from empirical analysis of real time series, on how probability density functions with power laws tails may emerge through various mechanisms, such as stochastic processes with additive-multiplicative fluctuations or as a result of mixed statistics.