On the functional equation

The paper concerns with the functional equation $$varphi (x,y,z) = varphi (varphi (x,y,t),t,z)$$ where ?:?₊×?₊×?₊→?₊. This equation has been formulated by Cantelli [1] to investigate general forms of financial laws other than the classic linear and exponential ones. Subsequently some Authors [2,3,4,5,6] have studied the above equation under weaker and weaker assumptions. In the present paper we solve the equation without any assumption of regularity.