Trade Execution Flow as the Underlying Source of Market Dynamics

TL;DR

The paper argues that market dynamics are fundamentally driven by the execution flow $I=dV/dt$, and develops a Radon–Nikodym based framework to compute $I$ from transaction data via moment calculations. It introduces a generalized eigenproblem to extract a characteristic time scale and constructs a density-based moving-average with internal degrees of freedom, enabling immediate regime switching and a P&L–oriented interpretation of market activity. A future-impact construct uses the maximal eigenvalue $oldsymbol{λ^{[ ext{maxI}]}}$ to forecast future liquidity swings, which the authors translate into directional price information through a liquidity-trading strategy and P&L evaluation in a density-matrix state. The work is validated on real NYSE TAQ and Nasdaq ITCH data, demonstrates forward-looking indicators with higher fidelity than price-based signals and offers a scale-invariant alternative via the Christoffel function spectrum for distribution-coverage analysis, with potential for real-time multi-asset deployment.

Abstract

In this work, we demonstrate experimentally that the execution flow, $I = dV/dt$, is the fundamental driving force of market dynamics. We develop a numerical framework to calculate execution flow from sampled moments using the Radon-Nikodym derivative. A notable feature of this approach is its ability to automatically determine thresholds that can serve as actionable triggers. The technique also determines the characteristic time scale directly from the corresponding eigenproblem. The methodology has been validated on actual market data to support these findings. Additionally, we introduce a framework based on the Christoffel function spectrum, which is invariant under arbitrary non-degenerate linear transformations of input attributes and offers an alternative to traditional principal component analysis (PCA), which is limited to unitary invariance.

Figures (8)

• Figure 1: An example of regular exponential moving average corresponding to $\tau=128$s and $\tau=512$s. Standard deviation is also calculated with the same $\tau$ and moving average $\pm$ standard deviation is plotted as a thin line in the same color. As $\tau$ increases -- the moving average "shifts to the right" ($\tau$-proportional time delay, lagging indicator). The data is for AAPL stock on September, 20, 2012.
• Figure 2: An example of a higher-order orthogonal polynomial root calculated from the moments $\Braket{P^k I}$, $k = 0 \dots 2n$, is shown for $n = 7$. Seven roots are obtained, with a substantial volume expected to be traded at each corresponding price level. In this example, the actual measure is approximated by a discrete measure with $n = 7$ support points. The figure is reproduced from 2015arXiv151005510G.
• Figure 3: A demonstration of execution flow. We present the original price $P$ and $P^{[\mathrm{maxI}]}$ (\ref{['PIH']}) (light blue). The other plots are shifted to the 693 level and then scaled to avoid cluttering the chart. We also present $T^{[\mathrm{maxI}]}$ (\ref{['TIH']}), the minimal and maximal eigenvalues of (\ref{['GEV']}), and $I_0 = \Braket{\psi_0|I|\psi_0}$ (yellow); the result is obtained for $n=12$ and $\tau=128$s. All execution flows are scaled by a factor of $5\cdot10^{-6}$ to fit the chart. Among the calculated values, only $I_0 = \Braket{\psi_0|I|\psi_0}$ can be regarded as a traditional moving average, since $\psi_0(x)$ (\ref{['psi0def']}) does not change with the data. The others --- $P^{[\mathrm{maxI}]}$, $T^{[\mathrm{maxI}]}$, $\lambda^{[\mathrm{minI}]}$, and $\lambda^{[\mathrm{maxI}]}$ --- can be viewed as moving averages with internal degrees of freedom. One can clearly observe an immediate switch due to these internal degrees of freedom, without the $\tau$-proportional lag typical of regular moving averages shown in Fig. \ref{['MovingAveragePlot']}.
• Figure 4: A demonstration of P&L calculation according to (\ref{['PnLdef']}). The discrete measure $dS$ represents the trader’s actions, and its integral $S$ gives the position held. Integrating $dS$ with the asset price yields the P&L. It is important to emphasize that the P&L depends on both the asset price $P(t)$ and the trader’s actions $dS$.
• Figure 5: The directional information (\ref{['dirOld']}) and (\ref{['DirOur']}) (shifted to 693 to fit the chart), the price, and $P^{[\mathrm{maxI}]}$ (\ref{['PIH']}) are shown above. Below (shifted to 691), we present an indicator of low $I$ -- a possible "entry point", $\Braket{\psi^{[\mathrm{minI}]}|\psi_0}^2$ (if $>0.8$), and an indicator of low $I$ -- a possible "exit point", $\Braket{\psi^{[\mathrm{maxI}]}|\psi_0}^2$ (if $>0.8$), shown below the 691 level in the plot.