Double Square-Root Law in Market Impact

• Double Square-Root Law is an empirical scaling rule describing how individual child orders generate a square-root price impact that decays inversely with time.
• The law underscores a mechanical basis for market impact, with both real and synthetic metaorders exhibiting identical scaling, independent of informational content.
• Empirical calibration on high-resolution TSE data validates the universal applicability of the law with strong R² fits and consistent prefactors across various liquid stocks.

The double square-root law characterizes the mechanical origin and temporal propagation of market impact at multiple scales, most notably in the context of metaorder execution in electronic limit order markets. Incorporating results from high-resolution transaction data from the Tokyo Stock Exchange (TSE) for the 2012–2018 period, this empirical law provides an explicit two-stage square-root scaling: (i) an initial square-root dependence of the impact at the microscopic level of single child orders, followed by (ii) an inverse square-root decay of each child order’s impact over time. When these mechanisms are aggregated, they recover the well-established square-root law for the aggregate impact of metaorders. The empirical evidence strongly favors a mechanical, rather than informational, basis for price formation and market impact.

1. Microscopic Impact Law for Single Child Orders

At the microstructural level, the signed price impact of individual child orders obeys a universal square-root law once an appropriate digestion time (post-trade) is allowed for market absorption. Denote $\epsilon = \pm 1$ as the sign of the market order ($+1$ for buy, $-1$ for sell), $q$ as the size of the order, $p_i$ as the mid-price before execution, and $\Delta p_i = p_{i+1} - p_i$ as the price change induced. $\sigma_D$ is the daily volatility and $V_D$ is the daily traded volume. The empirical scaling is:

$$\mathbb{E}[\Delta p \cdot \epsilon \mid q] \simeq C_1\,\sigma_D\,\sqrt{\frac{q}{V_D}}$$

where $C_1 \approx 1$ is a stock-independent prefactor (Fig. 4, inset). This law is valid even for orders with negligible immediate impact, once total traded volume since execution exceeds $q$.

2. Temporal Decay: Inverse Square-Root Law

The subsequent evolution of impact from a single child order $j$ decays as a power law in both physical and volume time. Let $G(\Delta t)$ denote the non-linear propagator quantifying the residual impact at time offset $\Delta t$:

$$G(\Delta t) \propto \frac{1}{\sqrt{\Delta t + s_0}}$$

with $s_0$ representing a short-time cutoff (volume time scale $\sim i_0 \Delta t$, $i_0 \approx 4$). For $\Delta t \gg s_0$, this becomes:

$$\Delta p_j(t_i) \epsilon_j \propto \sqrt{q_j} (\Delta t)^{-1/2}$$

This decay is visually corroborated over timeframes ranging from seconds to tens of minutes, with volume time of order $q$ (Fig. 4, green shaded region).

3. Aggregation: From Micro to Metaorders via the Double Square-Root

A metaorder of total size $Q$ is typically sliced into $N$ equal child orders of size $q=Q/N$ and executed sequentially. The net impact up to slice $N$ aggregates decaying contributions from each preceding order:

$$J(q,N) := \mathbb{E}\left[\sum_{j=1}^N \Delta p_j(t_N) \cdot \epsilon\right] \simeq A\,\sqrt{q}\sum_{j=1}^N \frac{1}{\sqrt{(N-j)\Delta t + s_0}} \simeq A'\,\sqrt{q}\,[\sqrt{N+i_0}-\sqrt{i_0}]$$

with $i_0 \simeq 4$ and coefficients proportional to $\sigma_D/\sqrt{V_D}$. This yields

$$J(q,i) \propto \sigma_D \sqrt{\frac{q}{V_D}}\left[\sqrt{i+i_0} - \sqrt{i_0}\right]$$

Setting $i=N$ and $qN=Q$, the familiar metaorder scaling emerges:

$$I(Q) := \mathbb{E}[\Delta p \cdot \epsilon \mid Q] \propto \sigma_D \sqrt{\frac{Q}{V_D}} \left[\sqrt{1+\frac{i_0}{N}} - \sqrt{\frac{i_0}{N}}\right]$$

For $N \gg i_0$, the bracket tends to $1$, recovering the well-known law $I(Q)\propto\sigma_D\sqrt{Q/V_D}$ (Figs. 1,7).

4. Empirical Methodology and Calibration

• Data: TSE complete order book for top 100 liquid stocks (including 10 ETFs) from 2012–2018, including unique anonymized trader identifiers.
• Metaorder Definition: Consecutive same-sign market orders by the same trader within a session (e.g., 09:00–11:30 or 12:30–15:00), excluding the day’s first/last 10 minutes.
• Size and Slicing: Typical metaorders have $Q/V_D \in [0.1\%, 1\%]$, with $N \sim 10$–$50$, $i_0 \approx 4$.
• Prefactors: For the microscopic law, $C_1 \approx 1$. The partial impact $J(q,i)$ is well-fit by $\propto \sqrt{q}(\sqrt{i+i_0} - \sqrt{i_0})$ with exponent $1/2$ (best fit $1-\beta$, $\beta \approx 0.48$). The metaorder prefactor $Y$ in $I(Q)=Y \sigma_D \sqrt{Q/V_D}$ is found to be $Y \approx 0.8$–$1.0$ across stocks. Regression fits consistently yield $R^2 > 0.99$.
• Universality: All liquid stocks collapse onto the same functional form in normalized coordinates (Fig. 7, left). Synthetic metaorders created by shuffling trader IDs exhibit identical impact curves (Fig. 7, right).

5. Mechanical Versus Informational Interpretation

Informational theories (e.g., Kyle-type Bayesian models, Gabaix et al.) predict that square-root scaling should be present only for metaorders with informational content, not for arbitrary or random order sequences. Empirical analysis demonstrates that any sequence of same-sign market orders—real or synthetic—produces an identical square-root law with statistically indistinguishable prefactors. This is achieved via synthetic metaorders constructed by randomly reassigning trader IDs on the real transaction tape. The observed universality and lack of dependence on order provenance support a purely mechanical explanation, in which impact arises from the interplay of locally linear latent liquidity and a “hot-potato” process among liquidity providers.

6. Limitations, Open Issues, and Further Observations

• For large $Q$ with small $N$, impact saturates due to conditioning and potential front-running.
• Tick-size effects induce a plateau in the impact for small $q$ in select stocks (Fig. 5, right).
• The $1/\sqrt{t}$ propagator decay does not comply with the no-arbitrage (Gatheral) bound for diffusive prices; further discussion appears in the companion paper [34].
• The latent-liquidity and “hot-potato” hypotheses, while consistent with aggregate observations, necessitate direct empirical verification within order book dynamics.
• The refill sequences by liquidity providers exhibit power-law length distributions (exponent $\mu_p \in [1.4,2.4]$) and a refill price skew $K_\ell(i) = C_\ell i^{-\kappa_\ell}$ with power-law decay in $i$ (Figs. 9–11).

In summary, the double square-root law provides a microscopic foundation for the well-known metaorder impact scaling, with explicit and universal applicability across order provenance, emphasizing the mechanical structure inherent to modern financial markets.