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

Abstract

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.

Price and Volume Dynamics in the Japanese Stock Market

“…Applying the methods in [3,4], we investigated data of stocks listed on Tokyo Stock Exchange. We extracted high correlation coefficients between time series of midprice and that of limit order volume.…”

“…When the midprice increases, volume in the respective layers are declined to increase or decrease as presented in the figure. The negatively (positively) correlated region around the best ask (bid) price limit order is called the inner layer, and the other region is called the outer layer [3,4]. In the appendix we show some other examples observed in the market showing the same behavior.…”

“…To exploit two layered structure and get higher correlation coefficients, we can use accumulated volume series up to the critical depth γ c [4]. An accumulated volume series is calculated by integration of aggregated differences of volume along the depth axis.…”

“…In the study on foreign exchange markets, Yura et al [3,4] revealed that this analogy is generalized to that of the order book dynamics and the origin of a Brownian particle motion in physics. In physics, Brownian particles collide with smaller particles of the solvent randomly.…”

“…In financial markets, price dynamics is driven by creation and annihilation processes of limit orders on the order book. The model is named as Financial Brownian Particle model (FBP model) emphasizing this situation [3,4]. On the path to confirm their model, they have clarified the way to extract high correlation between price changes and volume changes of the limit order book.…”