Kinetic theory for financial Brownian motion from microscopic dynamics

Recent technological development has enabled researchers to study social phenomena in detail, and financial markets have attracted the attention of physicists particularly since key concepts in Brownian motion are applicable to the description of financial systems. In our previous Letter [Kanazawa et al., Phys. Rev. Lett. 120, 138301 (2018)], we presented a microscopic model of high-frequency traders (HFTs) through direct data analyses of individual trajectories of HFTs and revealed its theoretical dynamics by introducing the Boltzmann and Langevin equations for finance. However, the formulation therein was rather heuristic and a more mathematically exact derivation is necessary from the microscopic dynamics of the HFT model. We hereby establish the mathematical foundation of kinetic theory for financial Brownian motion in a manner parallel to traditional statistical physics. We first derive the exact time-evolution equation for the phase-space distribution for the HFT model, corresponding to the Liouville equation in analytical mechanics. By a systematic reduction of the Liouville equation for finance, the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchal equations are generalized for financial Brownian motion. We then derive the Boltzmann and Langevin equations for the order book and the price dynamics by assuming molecular chaos. The asymptotic solutions to these equations are presented for a large number of HFTs, which qualitatively reveal how the strategies of traders at the microscopic level impact the macroscopic dynamics of market price. Our theoretical prediction was numerically examined via Monte Carlo simulations. Our kinetic description highlights the parallel mathematical structure between the financial and physical Brownian motions by a straightforward extension of statistical mechanics.

Figure 1

Hierarchal kinetic description of the conventional Brownian motion (a)–(c) and the financial Brownian motion (d)–(f). In the microscopic hierarchy of physical Brownian motion (a), gas particles and a massive tracer interact with each other, where the dynamics are described by the Liouville equation (2). As the mesoscopic description (b), the full dynamics are reduced to the one-body distribution ${\varphi }^{\left(1\right)}$ for gas particles, which are governed by the Boltzmann equation (6). The macroscopic dynamics of the tracer (c) are described by the master Boltzmann equation (8), or the Langevin equation (9) asymptotically for large system size $M\gg m$. Here we focus on the parallel hierarchal structure of financial markets (d)–(f) to molecular kinetic theory. In the microscopic hierarchy (d), each trader makes decisions to submit or cancel orders. As such, the dynamics of the traders correspond to those of molecules in kinetic theory. In the mesoscopic hierarchy (e), the information on trader identifiers is lost by coarse graining and we obtain the dynamics of the order book (i.e., the quoted price distribution). The order-book profile corresponds to the velocity distribution in the conventional kinetic theory. In the macroscopic hierarchy (f), the dynamics of the market price movement are finally deduced by the coarse graining, which exhibits anomalous random walks. The market price dynamics correspond to those of the Brownian motion in kinetic theory.

Figure 2

Schematic of the price formulation via the double-auction systems. (a) Individual traders make decisions on limit, market, and cancellation orders. In this schematic, the first trader canceled his/her bid quote of price 110.895 JPY at 9:10:10:10.553; the second trader submitted his/her ask limit order of price 110.915 JPY at 9:10:25.001; the third trader submitted a buy market order at 9:11:00.222. (b) Cancellation and limit orders are reflected in the order book as depletion and injection of “blocks.” Here the blocks represent orders of the unit volume (i.e., one million dollars) and are allocated on the lattice with minimum interval 0.005 JPY. (c) A new market price is generated when prices match, as an interaction between limit and market orders.

Figure 3

Notation of the state variables for our market model. (a) Market best bid ${\widehat{b}}_M$ and ask ${\widehat{a}}_M$ prices. The market midprice and market spread are defined by ${\widehat{z}}_M\equiv ({\widehat{b}}_M+{\widehat{a}}_M)/2$ and ${\widehat{L}}_M\equiv {\widehat{a}}_M-{\widehat{b}}_M$, respectively. (b) Best bid ${\widehat{b}}_i$ and ask ${\widehat{a}}_i$ prices for a single trader $i$. Here the light blocks represent the orders submitted by the $i\mathrm{th}$ trader. The midprice and buy-sell spread of the $i\mathrm{th}$ trader are defined by ${\widehat{z}}_i\equiv ({\widehat{b}}_i+{\widehat{a}}_i)/2$ and ${\widehat{L}}_i\equiv {\widehat{a}}_i-{\widehat{b}}_i$, respectively. (c) Schematic of the tick time $T$, incremented every transaction. For the trend-following analysis of individual traders [46], the correlation was studied between future movement of HFT's quoted midprice $\mathrm{\Delta }{\widehat{z}}_i\left[T\right]$ and historical price movement $\mathrm{\Delta }\widehat{p}[T-1]$.

Figure 4

Summary of the empirical characters of real HFTs based on our data set. (a) The number of submissions, typical number of orders, typical volumes designated in one order, depending on the ranking of the trader. For this figure, we analyzed representative numbers for every two traders for anonymization. Top: We sorted the traders by their total number of submissions to define their rankings. The top 135 traders were defined as HFTs, while the remaining 788 traders were defined as LFTs in this paper. We plotted the average of their total submissions for every two traders. Center: We investigated the number of total orders in the bid (ask) side at every bid (ask) order submission and take its median, first, and third quartiles every two traders. Bottom: We studied the volumes designated in one order at every order submission and take its median, first, and third quartiles for every two traders. (b) PDF for the numbers of orders maintained by a single HFT for one side (purple) and volumes designated in a single order of HFTs (green). For this figure, we studied medians as representative numbers for every single HFT. (c) CDF for the numbers of orders maintained for one side (purple) by a single LFT and the volumes designated in a single order of LFTs (green). For this figure, we studied medians as representative numbers for every single LFT. (d) Typical trajectories of the top HFT, continuously maintaining both sides as key liquidity providers. (e) PDF for volumes filled in a single transaction. The percentage of one-to-one transaction is 81.5% of all transactions. Transactions within 5 volumes occupy 98.2%.

Figure 5

Schematics of the dynamics of the trend-following HFT model. (a) Traders maintain two-sided quotes with constant buy-sell spreads $L_i$ and $L_j$ for traders $i$ and $j$. Their midprices move according to a deterministic trend following and random movement. (b) When bid and ask prices coincide, transaction occurs at that price. (c) Both traders requote their bid and ask prices at a distance from the market price after transaction.

Figure 6

Schematics of the microscopic and mesoscopic dynamics of the HFT model. (a) Sample trajectory of an HFT, showing the bid ${\widehat{b}}_i$ and ask ${\widehat{a}}_i$ quoted prices of the $i\mathrm{th}$ trader, the c.m. ${\widehat{z}}_{\mathrm{c}.\mathrm{m}.}$, and the market transaction price $\widehat{p}$. (b) Sample trajectory of the relative price ${\widehat{r}}_i$ from the c.m. ${\widehat{z}}_{\mathrm{c}.\mathrm{m}.}$, showing that ${\widehat{r}}_i$ stationarily fluctuates around zero. (c) Collective motion of the order book, showing herding behavior of traders. This collective motion is minimally implemented as trend following.

Figure 7

Schematic of the two-body collision. When the prices match between the traders $i$ and $j$, they requote their prices far from the market price. Note that the c.m. also moves through a distance of $\mathrm{\Delta }{\widehat{z}}_{\mathrm{c}.\mathrm{m}.}=-(L_i-L_j)/2N$ during this requotation.

Figure 8

Schematic of the three-body collision term ${\mathcal{L}}^{\left(ijk\right)}$. Let us assume that there is a collision between the traders $j$ and $k$. Because of the assumption of the binary interaction, the midprice ${\widehat{z}}_i$ of the $i\mathrm{th}$ trader does not move during this collision. On the other hand, the c.m. of this system ${\widehat{z}}_{\mathrm{c}.\mathrm{m}.}$ moves through a short distance of $\mathrm{\Delta }{\widehat{z}}_{\mathrm{c}.\mathrm{m}.}\equiv {\widehat{z}}_{\mathrm{c}.\mathrm{m}.}^{\mathrm{pst}}-{\widehat{z}}_{\mathrm{c}.\mathrm{m}.}=-\mathrm{sgn}({\widehat{z}}_j-{\widehat{z}}_k)(L_j-L_k)/2N$ because of the requotation. The relative price ${\widehat{r}}_i$ of the $i\mathrm{th}$ trader indirectly moves through a short distance of $\mathrm{\Delta }{\widehat{r}}_i\equiv {\widehat{r}}_i^{\mathrm{pst}}-{\widehat{r}}_i=\mathrm{\Delta }{r}_{ij;s}^{\left(1\right)}$.

Figure 9

Schematic of the NLO boundary layer solution (61). (a) Schematic of the bid and ask average order-book profiles $f_B^i\left(r\right)$ and $f_A^i\left(r\right)$ for the $i\mathrm{th}$ trader. Most of transactions will occur at the c.m. $r=0$ for infinite $N$ since the market is sufficiently liquid. On the other hand, for a finite large $N\gg 1$, a small portion of bid and ask orders can go beyond the c.m. with low probability, which is reflected as the tail near $r\simeq 0$ with thickness of the boundary layer $\varepsilon =L_{\rho }^{*}/2\sqrt{N}$. (b) Schematic of the midprice average order-book profile ${\varphi }^L\left(r\right)$. The tail near $r\simeq \pm L/2$ rapidly decays with the thickness of the boundary layer $\varepsilon$. The peak is also rounded with the thickness of $\varepsilon$ because requoted orders will be submitted near the peak, after transactions near the tail.

Figure 10

Average order-book profile is given by the superposition (66) of the tent function (63). For the $\delta$-distributed spread (Case 1), the profile is the tent function (69). For the $\gamma$-distributed spread (Case 2), the profile obeys Eq. (70).

Figure 11

Numerical validation of our formula (66) for the average order-book profile for two examples. (a) Numerical average ask order-book profile for the $\delta$-distributed spread ${\rho }_L=\delta (L-L^{*})$, showing the agreement with the theoretical formula (69) for $N\to \infty$. (b) Numerical average ask order-book profile for the $\gamma$-distributed spread ${\rho }_L=L^3{e}^{-L/L^{*}}/6L^{*4}$, showing the agreement with the theoretical formula (70) for $N\to \infty$.

Figure 12

Sample trajectories are plotted for (a) the weak trend-following case $\tilde{c}=0$, (b) the strong trend-following case $(\tilde{c},\mathrm{\Delta }{\tilde{p}}^{*})=(2.0,0.1)$, and (c) the marginal trend-following case $(\tilde{c},\mathrm{\Delta }{\tilde{p}}^{*})=(0.5,2.5)$ for $N=100, {10}^5$ ticks, and the $\gamma$-distributed spread. All parameters are shared except for the trend-following parameters $(\tilde{c},\mathrm{\Delta }{\tilde{p}}^{*})$. As can be seen from the figures, all trajectories seem to be the normal diffusion in the long timescale. The sample trajectories are enlarged 100 times [in the circles in panels (a)–(c)] as panels (${\mathrm{a}}^{\prime }$)–(${\mathrm{c}}^{\prime }$), where the character of each trajectory can be seen. (${\mathrm{a}}^{\prime }$) The price trajectory exhibits a zigzag behavior in the absence of trend following. (${\mathrm{b}}^{\prime }$) The price keeps moving toward the same direction for a certain tick period because of the strong trend following. (${\mathrm{c}}^{\prime }$) The price trajectory exhibits both zigzag behavior and trend following because both effects are in balance.

Figure 13

Numerical study in the absence of trend following $c=0$. (a) Numerical mean transaction intervals ${\tau }^{*}$ and the theoretical line for various $N$. (b) Transaction interval distribution for various $N$ with an exponential guideline by scaling the horizontal and vertical axes. The scaled interval is given by $\tilde{\tau }=c_{\tau }\widehat{\tau }/{\tau }^{*}$, where the fitting parameter $c_{\tau }$ for the decay time was estimated by the least-squares method for the tail as $c_{\tau }=1.34,1.49,1.56,1.58,1.62$, and 1.59 for $N=25,50,100,200,400$, and 800, respectively. (c) MSD plot based on real time $t$ for various $N$ with the theoretical lines, showing the normal diffusion for large $t$ but the slow diffusion $t\sim {\tau }^{*}$. (d) MSD plot based on tick time $K$ for various $N$ with the theoretical line, showing the normal diffusion for large $K$ but the slow diffusion for small $K$. (e) Variance of price difference $\mathrm{\Delta }\widehat{p}$ for various number of traders $N$ with a fitting curve of power-law exponent $N^{-1}$. (f) Plot of the peak of the PDF $P\left(\mathrm{\Delta }\tilde{p}\right)$ for the scaled price movement $\mathrm{\Delta }\tilde{p}\equiv \sqrt{N}\mathrm{\Delta }p/L^{*}$. (g) Log plot of the tail of the PDF $P\left(\mathrm{\Delta }\tilde{p}\right)$ with a Gaussian fitting curve $h\left(\mathrm{\Delta }\tilde{p}\right)$. (h) Autocorrelation function $C_{\mathrm{\Delta }\widehat{p}}\left[K\right]$ with tick time $K$, showing negative correlation at $K=1$. The simulation time was set to be ${10}^5$ ticks except for panels (a), (b), and (e)–(h), whereas the simulation time was set to be ${10}^6$ ticks for panels (c) and (d).

Figure 14

Numerical study for the strong trend-following case $\tilde{c}\gg 1$. We adopted $({\tilde{c}}^{*},\mathrm{\Delta }{\tilde{p}}^{*})=(2.0,0.1)$ as the trend-following parameters. (a) Price movement distribution for various $N$ by scaling both horizontal and vertical axes, qualitatively showing the exponential law (104). The least-squares method numerically estimates the decay length as $\kappa /c{\tau }^{*}=0.74,0.68,0.65,0.64,0.64$, and 0.64 for $N=25,50,100,200,400$, and 800, respectively. (b) Autocorrelation function $C_{\mathrm{\Delta }\widehat{p}}\left[K\right]$ with tick time $K$, showing the positive correlation with exponential decay. The fitting parameters were estimated to be $Z_{\mathrm{AC}}=0.62$ and ${\tau }_{\mathrm{AC}}=16.4,14.2,13.1,12.8,12.2$, and 12.6 for $N=25,50,100,200,400$, and 800, respectively. (c) Numerical MSD plot for $N=50$, showing a rapid diffusion of exponent $K^{1.8}$ (almost the ballistic motion of exponent $K^2$) for a short time and a normal diffusion of exponent $K^1$ for a long time. The simulation time was set to be ${10}^5$ ticks for panels (a) and (b), whereas the simulation time was set to be ${10}^6$ ticks for panel (c).

Figure 15

Numerical study for the marginal case $\tilde{c}\sim 1$. (a) The price movement distribution $\mathrm{\Delta }p$ by scaling the horizontal and vertical axes. (b) Autocorrelation function $C_{\mathrm{\Delta }\widehat{p}}\left[K\right]$ based on tick time $K$ (points), showing the negative correlation around $K=1$. This numerical result was consistent with the empirical result obtained from our data set (solid line). (c) MSD plot under the parameters $(\tilde{c},\mathrm{\Delta }{\tilde{p}}^{*})=(0.5,2.5)$, showing a slightly slow diffusion. (d) MSD plot under the parameters $(\tilde{c},\mathrm{\Delta }{\tilde{p}}^{*})=(0.86,1.43)$, showing a slightly rapid diffusion with the Hurst exponent $H=0.65$. The simulation time was set to be ${10}^5$ ticks for panels (a) and (b), whereas the simulation time was set to be ${10}^6$ ticks for panels (c) and (d).

Figure 16

Empirical characteristics of the short-time market volatility $\langle \left|\mathrm{\Delta }\widehat{p}\right|\rangle$. (a) Time series of $\langle \left|\mathrm{\Delta }\widehat{p}\right|\rangle$ and the inverse number of HFTs $1/N$. Both $\langle \left|\mathrm{\Delta }\widehat{p}\right|\rangle$ and $1/N$ had a tendency to take large values during inactive hours of the EBS market, such as the time region just after the market opening (the 5th 18:00–22:00 GMT) and the end of the New York working hours (20:00–22:00 GMT). This figure exhibits the correlation between $\langle \left|\mathrm{\Delta }\widehat{p}\right|\rangle$ and $1/N$ with Spearman's rank correlation coefficient of 0.63. (b) Scattering plot between $\langle \left|\mathrm{\Delta }\widehat{p}\right|\rangle$ and $N$ in the log-log scale. Regression analysis between $log\langle \left|\mathrm{\Delta }\widehat{p}\right|\rangle$ and $logN$ implies a power-law (almost linear) relation $\langle |\mathrm{\Delta }\widehat{p}|\rangle \propto 1/N^{\beta }$ with $\beta =0.86\pm 0.1$. In this analysis, we excluded the two samples after the market opening (the 5th 18:00–24:00 GMT) as outliers. (c) Intraday patterns are studied for $\langle \left|\mathrm{\Delta }\widehat{p}\right|\rangle$ and $N$. We took the averages of $\langle \left|\mathrm{\Delta }\widehat{p}\right|\rangle$ and $N$ conditional on two-hourly intraday time zones from Tuesday to Thursday, with the arrow-type legends showing the working hours for Tokyo (0:00–9:00 GMT), London (8:00–17:00 GMT), and New York (13:00–22:00 GMT). This figure shows that both $\langle \left|\mathrm{\Delta }\widehat{p}\right|\rangle$ and $1/N$ tended to take large values during the end of the New York working hours (20:00–22:00 GMT).

Figure 17

Qualitative picture of the NLO solution (61) depending on the regimes ${\mathit{R}}_0, {\mathit{R}}_1$, and ${\mathit{R}}_2\equiv {\mathit{R}}_2^{-}\cup {\mathit{R}}_2^{+}$. The asymptotic behavior is described by Eq. (A2). The regimes ${\mathit{R}}_0, {\mathit{R}}_1$, and ${\mathit{R}}_2$ represent the peak, transient, and tail regimes, respectively.