Universal scaling and nonlinearity of aggregate price impact in financial markets

How and why stock prices move is a centuries-old question still not answered conclusively. More recently, attention shifted to higher frequencies, where trades are processed piecewise across different time scales. Here we reveal that price impact has a universal nonlinear shape for trades aggregated on any intraday scale. Its shape varies little across instruments, but drastically different master curves are obtained for order-volume and -sign impact. The scaling is largely determined by the relevant Hurst exponents. We further show that extreme order-flow imbalance is not associated with large returns. To the contrary, it is observed when the price is pinned to a particular level. Prices move only when there is sufficient balance in the local order flow. In fact, the probability that a trade changes the midprice falls to zero with increasing (absolute) order-sign bias along an arc-shaped curve for all intraday scales. Our findings challenge the widespread assumption of linear aggregate impact. They imply that market dynamics on all intraday time scales are shaped by correlations and bilateral adaptation in the flows of liquidity provision and taking.

Figure 1

AAPL on NASDAQ in 2016. (a) Rescaled expected return ${\mathcal{R}}_N(\mathcal{Q}/N^{\xi })/N^{\psi }$ conditioned on the volume imbalance $\mathcal{Q}$ for different bin sizes $N$ in arbitrary units [see Eq. (2) ff]. The $x$- and $y$-axis rescaling exponents are $\xi =0.84$ and $\psi =0.53$. (b) Rescaled mean return ${\mathcal{R}}_N(\mathcal{E}/N^{\psi })/N^{\psi }$ conditioned on the sign imbalance $\mathcal{E}$ [see Eq. (9)]. Here ${\xi }_{\epsilon }=0.69$ and ${\psi }_{\epsilon }=0.48$. (c) and (d) Corresponding complementary cumulative distributions. The positive and negative halves were calculated independently and then binned to smooth out noise and discretization steps for small $N$. The largest shown $N$ corresponds to the shortest day in the sample.

Figure 2

AAPL in 2016. (a) Volume scales $Q_N$ for the impact curves [see Eq. (3)] that were fitted to each bin size for the same data shown in Fig. 1. The solid line shows the power-law fit used for the rescaling along the volume-imbalance axis. (b) Same analysis as in (a) but for the return scale $R_N$, i.e., the scaling of the impact curve along the return axis. Note that the previous fitting of the impact curves, yielding $Q_N$ and $R_N$ for each $N$, did not impose any assumptions on their scaling. Relative fitting errors are below $1\%$, but variability across instruments and periods is much larger (discussed below).

Figure 3

(a) Analysis of the distributions of several scaling exponents. Plotted symbols denote the means and vertical lines the standard deviations. Both were calculated over instruments, for each of which the exponents were calculated for all 1-yr periods before averaging. Blue circles correspond to the scaling of a variable that a price impact is typically conditioned on, i.e., the width of an impact curve along the $x$ axis. Green triangles correspond to the scaling of a variable measuring price changes, i.e., the height along the $y$ axis. The variables, from left to right, are volume imbalance and corresponding return rescaling $\xi$ and $\psi$, respectively, sign imbalance and corresponding return rescaling ${\xi }_{\epsilon }$ and ${\psi }_{\epsilon }$, respectively, and Hurst exponents of the volume imbalance $H_q$, returns $H_r$, order signs $H_{\epsilon }$, and return signs $H_{\pm }$ [see Eq. (10)]. (b) Cross correlation calculated over all instruments for the same variables. Only correlations differing from zero by more than three standard deviations are shown in solid colors.

Figure 4

AAPL, 1 May 2016. A typical example of an extreme order-sign imbalance and a “sticky price.” The visible liquidity at each price level just before each trade is drawn in red above and in blue below the midprice (upper solid line). Full saturation corresponds to the 90th percentile for the day. The buy volumes $q^{+}$ for each trade are drawn as bars above the order book reconstruction and the sell volumes $q^{-}$ below. The volume axis is logarithmically scaled. A lower bound for the true number of trades is shown because transactions with the same sign and timestamp were merged (see Sec. 2). Therefore, a series of 100 successive market buys drawn above took at least $100\mathrm{ms}$ in wall time and may have consisted of more than 100 independent market orders submitted to the market. The lower solid curve denotes the 50 trade sign imbalance (causal).

Figure 5

EUROSTOXX on EUREX in 2015. (a) Probability $P(r\ne 0|N^{-1}\mathcal{E})$ that the midprice changes from one trade to the next conditioned on the mean trade sign in the bin containing $N$ trades. (b) Corresponding complementary cumulative distributions.