Emergent quantum mechanics of finances

•The quantum mechanical description for financial markets is provided.•The property of markets’ fractality is taken into account.•The theory becomes non-linear as the efficient market hypothesis is no longer valid.

This paper is an attempt at understanding the quantum-like dynamics of financial markets in terms of non-differentiable price–time continuum having fractal properties. The main steps of this development are the statistical scaling, the non-differentiability hypothesis, and the equations of motion entailed by this hypothesis. From perspective of the proposed theory the dynamics of S&P500 index are analyzed.