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We explore nature of price formation in financial markets and develop a theory of bid and ask price dynamics in which the two prices form due to quantum-chaotic interaction between buy and sell orders.In this model bid and ask prices are represented by eigenvalues of a 2x2 price operator corresponding to "bid" and "ask" eigenstates, while randomness of price operator results in price fluctuations that destroy oscillatory effects. We show that this theory adequately captures behavior of bid-ask spread and allows to model bid and ask price dynamics in a coordinated way. We also discuss ergodicity properties of price formation and show how directional price movement occurs due to ergodicity violation in a quantum process instead of the commonly believed forces acting on price. This theory has wide range of applications such as trade execution modeling, large order pricing and risk valuation for illiquid securities. Stochastic calculus vs. financial marketsModern quantitative finance is built around stochastic calculus treating price movements as random walk [1][2][3]. In doing so it implicitly relies on certain assumptions, that are supposed to ensure similarity between the two processes. Over the decades, countless models have been produced based on this similarity. However, in practice these models are almost always used without verifying the validity of the underlying assumptions. Such careless application of models can easily lead to lack of control, position mismanagement and financial losses [4][5][6]. Let us therefore examine, what criteria ensure similarity between price movements and random walk.