5.5 Price impact and the square-root law

5.5.1Metaorders and the square-root law#

The empirical law Across markets, periods and asset classes, the average impact of a metaorder is well described by

Evidence: the BARRA market-impact model (Torre, 1997) used a square-root form; Almgren, Thum, Hauptmann and Li ("Direct estimation of equity market impact", Risk, 2005) estimated a concave temporary-impact exponent of about 0.6 from Citigroup US equity executions; Tóth, Lempérière, Deremble, de Lataillade, Kockelkoren and Bouchaud ("Anomalous price impact and the critical nature of liquidity in financial markets", Physical Review X 1, 021006, 2011) found exponents close to ½ across futures markets; Bouchaud, Bonart, Donier and Gould, Trades, Quotes and Prices (CUP, 2018), survey the evidence in their chapters on metaorder impact. The law holds for metaorders that are a small but non-negligible fraction of daily volume (roughly $10^{-4}$ to $10^{-1}$). It is a statement about averages; individual executions scatter widely.

5.5.2Consequences for size and capacity#

Consequences

1. Concavity favours splitting and patience in size, not time. Doubling the order raises per-share impact by $\sqrt2 \approx 1.41$, not 2. Total impact cost $Q \cdot I(Q) \propto Q^{3/2}$.
2. Impact is set in units of daily volatility. A stock with $\sigma_D = 2\%$ trading 10% of its daily volume moves about $Y \times 0.63\%$.
3. Capacity. A strategy whose per-trade edge is $e$ (as a fraction of price) loses its edge to impact when $Y\sigma_D\sqrt{Q/V_D} \approx e$, i.e. at $Q^* = V_D\,(e/(Y\sigma_D))^2$. Capacity rises with the square of the edge.
4. Retail sizes. For 100 shares of a stock trading 50 lakh shares a day with $\sigma_D = 1.8\%$, (5.13) with $Y = 1$ gives $1.8\% \times \sqrt{2\times10^{-5}} = 0.8$ bp. Here the spread and charges (M01 · 1.7) dominate, and the square-root law is outside the range where it was measured. For retail, impact is not the problem; adverse selection and costs are.

5.5.3Why impact is concave#

Why concave? Several mechanisms have been proposed: the order book's latent liquidity grows linearly away from the price, so a metaorder that eats through it sees impact grow like the square root of size (the "latent order book" models in Bouchaud et al. 2018); and in Kyle-type models with uncertainty about the size of the informed order, impact can be concave. Gatheral (2010, "No-dynamic-arbitrage and market impact", Quantitative Finance 10(7)) shows that the shape of impact and the speed of its decay after the order ends must be consistent, or round-trip trading could make money from impact alone. These are active research topics; treat (5.13) as a well-supported empirical regularity, not a theorem.