Derivation of the Boltzmann Equation for Financial Brownian Motion: Direct Observation of the Collective Motion of High-Frequency Traders

A microscopic model is established for financial Brownian motion from the direct observation of the dynamics of high-frequency traders (HFTs) in a foreign exchange market. Furthermore, a theoretical framework parallel to molecular kinetic theory is developed for the systematic description of the financial market from microscopic dynamics of HFTs. We report first on a microscopic empirical law of traders’ trend-following behavior by tracking the trajectories of all individuals, which quantifies the collective motion of HFTs but has not been captured in conventional order-book models. We next introduce the corresponding microscopic model of HFTs and present its theoretical solution paralleling molecular kinetic theory: Boltzmann-like and Langevin-like equations are derived from the microscopic dynamics via the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy. Our model is the first microscopic model that has been directly validated through data analysis of the microscopic dynamics, exhibiting quantitative agreements with mesoscopic and macroscopic empirical results.

Figure 1

Hierarchies for financial Brownian motions for the microscopic, mesoscopic, and macroscopic dynamics.

Figure 2

(a)–(c) Lifetimes of orders are plotted as trajectories for the top 3 HFTs. Typical traders tend towards continuous two-sided quotes, with the buy-sell spread fluctuating around a time constant unique to the trader. The percentage of two-sided quotes among HFTs was 48.4% (see Supplemental Material [35]). (d) Quantification of trend following for individual traders, where $\mathrm{\Delta }p$ and $\mathrm{\Delta }{z}_i$ are the movements of the market price and the midprice of the $i$th trader, respectively.

Figure 3

(a) Average $\mathrm{\Delta }{z}_i$ assuming previous price movements of $\mathrm{\Delta }p$ and an active trader with $\mathrm{\Delta }{z}_i\ne 0$. The behavior can be fitted by the master curve (1) for the top 20 HFTs by introducing scaling parameters $\mathrm{\Delta }{p}_i^{*}$ and $c_i$. (b) Conditional standard deviation on price movement $\mathrm{\Delta }p$, showing that the randomness associated with trend following is independent of $\mathrm{\Delta }p$.

Figure 4

Schematic of the microscopic model (2). (a) Midprice of each trader obeys trend-following random walks. (b) Transaction takes place after price matching $b_i=a_j$ with market transacted price $p$ and its movement $\mathrm{\Delta }p$ updated. (c) A pair of traders requote their bid and ask prices simultaneously after transactions. (d) Order-book collective motion induced by trend following. (e) Volume change in the bid (ask) order book is positive (negative) near the best price on average for $\mathrm{\Delta }p>0$.

Figure 5

(a) Daily distribution of buy-sell spreads for HFTs with the empirical master curve (8). (b) Daily average order-book profile for the best prices of HFTs, agreeing with our theoretical line (9) without fitting parameters. Here the relative depth is measured from the market midprice instead of the c.m. for simplicity, which did not cause big numerical difference in comparing with our theory (see Supplemental Material [35]). (c) Two-hourly segmented CDFs for the price movement in one-tick precision for the three typical time regions (see Supplemental Material [35]). The CDFs are exponential, consistently with our theoretical prediction (10). (d) Two-hourly segmented CDFs are scaled into the single exponential master curve every 2 hours (62 time regions). (e) Price movement CDF over the whole week obeys a power-law of exponent $\alpha$. (f) Decay length $\kappa$ obeys a power-law $Q(\ge \kappa )\sim {\kappa }^{-m}$. (g) Order-book layered structure by our HFT model. Pearson’s coefficient $C_r^{-}(C_r^{+})$ are numerically plotted between $N_r^{-}(N_r^{+})$ and $\mathrm{\Delta }p$ with crossover point ${\gamma }_{\mathrm{c}}\approx 16.5$ tpip. (h) Linear correlation between the total number change $N_{\text{inner}}$ in the inner layer and the price movement $\mathrm{\Delta }p$ with correlation coefficient of 0.63.