Financial Knudsen number: Breakdown of continuous price dynamics and asymmetric buy-and-sell structures confirmed by high-precision order-book information.

We generalize the description of the dynamics of the order book of financial markets in terms of a Brownian particle embedded in a fluid of incoming, exiting, and annihilating particles by presenting a model of the velocity on each side (buy and sell) independently. The improved model builds on the time-averaged number of particles in the inner layer and its change per unit time, where the inner layer is revealed by the correlations between price velocity and change in the number of particles (limit orders). This allows us to introduce the Knudsen number of the financial Brownian particle motion and its asymmetric version (on the buy and sell sides). Not being considered previously, the asymmetric Knudsen numbers are crucial in finance in order to detect asymmetric price changes. The Knudsen numbers allows us to characterize the conditions for the market dynamics to be correctly described by a continuous stochastic process. Not questioned until now for large liquid markets such as the USD-JPY and EUR-USD exchange rates, we show that there are regimes when the Knudsen numbers are so high that discrete particle effects dominate, such as during market stresses and crashes. We document the presence of imbalances of particles depletion rates on the buy and sell sides that are associated with high Knudsen numbers and violent directional price changes. This indicator can detect the direction of the price motion at the early stage while the usual volatility risk measure is blind to the price direction.