Crossover from Linear to Square-Root Market Impact

Using a large database of 8 million institutional trades executed in the U.S. equity market, we establish a clear crossover between a linear market impact regime and a square-root regime as a function of the volume of the order. Our empirical results are remarkably well explained by a recently proposed dynamical theory of liquidity that makes specific predictions about the scaling function describing this crossover. Allowing at least two characteristic timescales for the liquidity (“fast” and “slow”) enables one to reach quantitative agreement with the data.

Figure 1

Empirically determined scaling function $\mathcal{F}(\eta )$ vs participation rate $\eta$. The data (blue points) interpolate between a $\sqrt{\eta }$ behavior observed at small participation rates and an asymptotically constant regime $\approx 0.4$ for large $\eta$, i.e., for $\eta \gtrsim {\eta }^{\star }$ with ${\eta }^{\star }\approx 3.15\times {10}^{-3}$. Black line: prediction of LLOB, with an adjusted crossover ${\eta }^{\star }≔J_s/J_f$ allowing for the existence of two categories of agents (fast and slow). The data points are obtained by restricting to metaorders with sufficient large order size, i.e., $\varphi \gtrsim {10}^{-5}$.

Figure 2

(Main panel) Empirically determined scaling function $\mathcal{F}(\eta )$ vs participation rate $\eta$ for metaorders with duration $T$ larger (blue dots) or smaller (green dots) than the median sample duration $\overline{T}\approx 0.09$. Note that the crossover value ${\eta }^{\star }$ is $\sim 10$ times larger in the latter case. Note that the asymptotic value of $\mathcal{F}$ (and hence the impact for a given $Q$) is approximately independent of $T$, as predicted by the theory. Both empirical curves are obtained using metaorders with sufficient large order size, i.e., $\varphi \gtrsim {10}^{-5}$. (Inset) Plot of the crossover participation rate ${\eta }^{\star }$ as a function of the execution time $T$, revealing an approximate $T^{-1/2}$ behavior.