6.7.4 The Square-Root Law of Market Impact

Basic Prerequisites: 03 · Temporary vs Permanent and Limit Order Book Mechanics (book shape).

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1. Intuition & Practical Objective

Kyle’s linear rule, $P=p_0+\lambda y$, is the correct description of impact at the margin - for small orders over short horizons. Stretch it to large orders and it fails, badly and in a specific direction: real market impact is concave - it grows roughly as the square root of order size. Doubling your order does not double your impact; it multiplies it by about $\sqrt{2}\approx 1.41$.

This single fact - the square-root law - is the most important empirical regularity in execution:

$$I\approx \sigma Y\sqrt{\frac{Q}{V}}(\text{or}\propto \sigma \sqrt{\text{participation rate}}),$$

where $Q$ is the order size, $V$ the average volume, $\sigma$ the daily volatility, and $Y$ an $O(1)$ constant. It is why execution algorithms are not simply “trade the same fraction forever”: they must account for the concavity of cost, which changes the optimal schedule, the marginal cost of size, and the answer to “should we even trade this?”

Why concave, not linear? Two complementary pictures:

1. The book is not a wall - it’s a hill. Depth does not sit uniformly at every price; it thins as you move away from the touch. Squeeze more size into the market and you find progressively less liquidity per price level, so the price you must reach grows sublinearly in size. The canonical econophysics model (Bouchaud, Mézard & Potters 2002) formalises the order book as a diffusive object: the liquidity density at distance $\ell$ from the mid grows linearly in $\ell$, so cumulative liquidity grows as ${\ell }^2$ and the price needed to absorb a volume imbalance grows as $\sqrt{\Delta V}$ (§3 proves this numerically).

2. Volatility sets the yardstick. Empirically, impact scales with volatility and the participation rate (your share of the flow), not with raw size. Tóth et al. (2011) argue the deep reason: the ultimate submitters of large orders are insensitive to price moves of the order of daily volatility during execution, so impact accumulates as a random walk and ends up $\propto \sqrt{Q}$.

The one-sentence essence. “Impact is concave in size - roughly the square root of participation - because liquidity thins with distance and because execution is a race against volatility; a linear-impact assumption understates the cost of small orders and overstates it for large ones.”

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2. Mathematical Ground Truth & Derivations

2.1 The latent-liquidity origin of $\sqrt{Q}$

Model the book by its cumulative liquidity $L(\ell )$: the number of shares resting within a price distance $\ell$ of the mid. Two book shapes:

• Flat book (density $\rho (\ell )={\rho }_0$ constant): $L(\ell )={\rho }_0\ell$, so absorbing a flow $\Delta V$ needs a move $\ell =\Delta V/{\rho }_0$ - linear impact (Kyle).
• Diffusive book ($\rho (\ell )={\rho }_0\ell$, the empirically observed thinning): $L(\ell )=\frac{1}{2}{\rho }_0{\ell }^2$, so

$$\Delta V=\frac{1}{2}{\rho }_0{\ell }^2⟹\boxed{\ell =\sqrt{\frac{2\Delta V}{{\rho }_0}}\propto \sqrt{\Delta V}}\text{- square-root impact.}$$

The square-root law, in this picture, is simply a statement about the shape of the book. (Order signs are long-memory - autocorrelation $C_{\tau }\sim {\tau }^{-\gamma }$ with $\gamma \approx 0.5$ - which is what keeps the book diffusive and reproduces the same concavity under a propagator treatment.)

2.2 The empirical functional form (Almgren et al. 2005)

Fitting power laws $g(v)=c|v{|}^{\alpha }\mathrm{sgn}(v)$, $h(v)=c^{\prime }|v{|}^{\beta }\mathrm{sgn}(v)$ to real metaorders gives

$$I=\gamma \sigma \frac{X}{V}(\frac{\Theta }{V}{)}^{1/4}(\text{permanent}),J=\frac{I}{2}+\mathrm{sgn}(X)\eta \sigma (\frac{X}{VT}{)}^{3/5}(\text{realized}),$$

with $X$ order size, $V$ average daily volume, $\Theta$ shares outstanding, $T$ execution duration in volume time, and fitted coefficients

$$\gamma =0.314\pm 0.041,\eta =0.142\pm \mathrm{0.0062.}$$

Three messages: (i) the permanent exponent is fixed at $1$ (linear, forced by no-arbitrage - Huberman–Stanzl); (ii) the temporary exponent is $3/5$, not the square root $1/2$ - the pure square-root model is rejected at 95%; (iii) the residual $R^2$ is under 1% because volatility dominates the realized impact of any single order, which is exactly why impact estimation requires averaging over many metaorders.

2.3 The Bouchaud–Farmer–Lillo exponent

For a propagator model with order-sign autocorrelation decaying as $C_{\tau }\sim {\tau }^{-\gamma }$, an executed metaorder of $N$ slices has impact scaling as $N^{1-\beta }$ with $\beta =(1-\gamma )/2$. Empirical $\gamma \approx 0.5\Rightarrow \beta \approx 0.25\Rightarrow$ impact $\sim N^{3/4}$. The impact also scales as ${\pi }^{\beta }$ in the participation rate $\pi$: the slower the execution, the smaller the impact (and in the limit of infinitely slow execution, impact $\to 0$). This is the same qualitative message as the temporary/permanent split of page 03, derived from order-flow memory rather than from resilience.

2.4 The no-arbitrage constraint on concavity (Gatheral)

Concavity cannot be arbitrary, or the market admits price manipulation. Gatheral (2010): for a transient-impact model $S_t=S_0+{\int }_0^{t}h({\dot{X}}_s)G(t-s)ds$ with impact function $h(x)=c|x{|}^{\delta }\mathrm{sgn}(x)$ and decay kernel $G(\tau )={\tau }^{-\gamma }$,

$$\boxed{\text{price manipulation exists}⟺\gamma +\delta <1}.$$

Empirically $\delta \approx 0.5$ (square-root impact) and $\gamma \approx 0.5$ (square-root decay) - sitting exactly on the boundary, so observed impact is just barely consistent with no-dynamic-arbitrage. A finite-at-zero kernel such as exponential decay combined with any nonlinear impact function is ruled out (Prop. 22.14) - the price-impact decay must be slow (power-law), a striking theoretical constraint on any “impact decays in seconds” claim.

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3. Computational Implementation - why the book’s shape gives  ​

Sweep a volume imbalance $\Delta V$ through two books and fit the log-log slope. A diffusive book (liquidity per price level $\propto$ distance, $\rho (\ell )=\ell$) gives slope $\frac{1}{2}$; a flat book gives slope $1$. Stdlib only.

The regression recovers the square-root exponent 0.5054 on the diffusive book and 1.0000 on the flat book - confirming the derivation: the concavity of impact is a direct consequence of how liquidity is distributed away from the touch. The analytic check $\sqrt{2\Delta V}=141.4$ matches the simulated 141 exactly.

How a practitioner measures the exponent. Regress $\log I$ on $\log (Q/V)$ across a sample of metaorders (Almgren et al.’s procedure). The slope is your estimate of the exponent: $\approx 1$ means the venue behaves linearly (thin, Kyle-like), $\approx 0.5$ means square-root, and anything below $1$ with slow impact decay is consistent with no-dynamic-arbitrage.

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4. Failure Modes & First-Principles Breakdowns

1. Fitting linear impact to square-root data. The most damaging misspecification: linear models overstate the cost of large orders (they treat impact as proportional, so the fitted line sits above the concave curve at the large sizes) and mis-price the marginal cost of size, which is precisely the decision a large order turns on.
2. Over-fitting the exponent. Almgren et al. reject the pure $\frac{1}{2}$ square root at 95% in favour of $\frac{3}{5}$; quoting “the square-root law” as exact is a simplification. The sign of the concavity is robust; the exact exponent is venue- and size-regime-dependent.
3. Ignoring that impact scales with volatility, not size alone. $I\propto \sigma \sqrt{Q/V}$ means the same order costs twice as much in a high-vol regime. Backtests that fix $\sigma$ under-charge exactly when it matters most.
4. Assuming fast impact decay. A kernel that decays too quickly (e.g. exponential) plus nonlinear impact implies arbitrage (Gatheral) - it lets a strategy buy, wait, sell, and profit risklessly. Empirically impact decays slowly, so any model that “resets” impact within a day is suspect.
5. Confusing the square-root law of temporary impact with permanent impact. The empirical concavity is in the temporary/realized component; the permanent component is (approximately) linear. Applying a square-root rule to permanent impact contradicts no-arbitrage (Huberman–Stanzl).
6. Neglecting the participation-rate dependence. Impact scales with how you cut the order, not just its size: the same order executed at a higher participation rate has strictly higher impact (${\pi }^{\beta }$). A backtest that ignores participation is not measuring impact, it is measuring size.

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5. References

• Almgren, R., Thum, C., Hauptmann, H. & Li, H. (2005), Direct estimation of equity market impact, Risk 18(7), 57–62. The fitted exponents and coefficients ($\gamma =0.314$, $\eta =0.142$, temporary $\beta =3/5$, square root rejected). Primary PDF in corpus (45_Almgren_2005_...).
• Bouchaud, J.-P., Farmer, J. D. & Lillo, F. (2009), How markets slowly digest changes in supply and demand. Order-flow long memory, impact concavity, the $N^{1-\beta }$ and ${\pi }^{\beta }$ results. Primary PDF in corpus (44_Bouchaud_2009_...).
• Gatheral, J. (2010), No-dynamic-arbitrage and market impact, Quantitative Finance 10(7), 749–759. The $\gamma +\delta \ge 1$ constraint and the exclusion of exponential decay + nonlinear impact. Primary PDF in corpus (42Gatheral2010_...).
• Gatheral, J. & Schied, A. (2013), Dynamical models of market impact and algorithms for order execution. Remark 22.15: $\delta \approx 0.5,\gamma \approx 0.5$; Tóth et al. mechanism. Primary PDF in corpus (46_Gatheral_2013_...).
• Tóth, B., Lempérière, Y., Deremble, C., de Lataillade, J., Kockelkoren, J. & Bouchaud, J.-P. (2011), Anomalous price impact and the critical nature of liquidity in financial markets, Phys. Rev. X 1, 021006. The square-root law across a very large range of sizes and the insensitivity mechanism.
• Bouchaud, J.-P., Mézard, M. & Potters, M. (2002), Statistical properties of stock order books, Quantitative Finance 2(4), 251–256. The diffusive-book model behind the square root. Primary PDF in corpus.

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6. Connected Graph Bridges

• Back: 03 · Temporary vs Permanent · Limit Order Book Mechanics
• Forward: 05 · Failure Modes & Practice · 06 · Advanced Extensions · Index Hub
• Cross-pillar: Almgren–Chriss (Pillar 2) (the scheduler consumes this cost function) · Liquidity Risk & Asset Pricing (Amihud is a square-root-flavoured proxy)