
There are three different formulations of Quantum mechanics: canonical quantization (or geometrical quantization), star product quantization and path integral quantization. It is well known that the first one manifests the ordering ambiguity: to properly define the model it is not sufficient to produce a Hamiltonian, also one must fix the socalled normal ordering, the way to pass from coordinates at the phase space to the operators. Different choice of ordering can be compensated by quantum corrections to the Hamiltonian, therefore a kind of "gauge symmetry" present. This symmetry also appears when one uses the starproduct approach. However, in the path integral approach it is hidden. Our suggestion is that this ambiguity appears in the definition of the classical action. Usually the Lagrangian is a (polynomial) function of coordinates and velocities. But functions we integrate on are nondifferentiable (w.r.t. time). Therefore we need to define a proper extension of the classical action. It can be done with the help of stochastic calculus and is not unique  this is here where the ordering ambiguity appears. We consider several examples which come from financial word: the stochastic calculus is conventional tool to study the financial market properties. We translate this toolkit to the path integral language and will see in details how ordering ambiguity (known in the stochastic calculus community as Ito and Stratonovich integrals) appears. Also we will see why different ordering schemes are natural for Quantum Mechanics and Probability Theory (Wick rotation is not, therefore, enough to pass from quantum mechanical to thermodynamical problem).
