The Square-Root Law of Market Impact
(Published by JP Bouchaud | November 2024)
The square-root law for price impact is arguably one of the most fascinating robust empirical regularities discovered in the last 30 years. It states that when executing a buy (sell) “metaorder” of total size Q, sliced and diced into N child orders of size q=Q/N, the price on average moves up (down) by an amount proportional to sqrt(Q).
Price impact is, remarkably, found to be approximately independent of both N and of the total time T needed to achieve full execution.
In other words, provided the participation rate is not too large, average price impact only depends on the total volume traded Q, and barely on execution schedule.
Such a square-root dependence, and its apparent universality across a wide variety of markets is surprising and non-intuitive. The classical Kyle model would rather predict a linear dependence of impact on Q, with a slope usually called “Kyle's lambda”.
Several theoretical ideas have been put forth in the literature to explain non-linear impact.
Some models predict a concave price impact Q^delta with delta < 1 related to the power-law tail exponent alpha of the executed volume or the power-law tail exponent gamma of the time autocorrelation of the sign of market orders.
However, as shown in a beautiful work by Sato and Kanazawa that just came out using ID-resolved data from the Tokyo Stock Exchange (TSE), the predicted relations between delta and alpha or gamma are not borne out by the data: as Fig. below shows, alpha and gamma significantly differ between stocks, exponent delta remains stubbornly anchored around delta=1/2, i.e. the square-root law.
Yet another line of thought builds upon the classical Glosten-Milgrom model and assumes that metaorders are issued by traders possessing information about future prices, as is traditional in the economics literature. Market makers try to detect these informed traders, but do not know their actual participation rate in the total order flow. Using Bayesian arguments, Saddier & Matteo Marsili derive a universal square-root impact law under certain (weak) conditions. Other aspect of the theory are however inconsistent with data.
The Latent Liquidity Theory (LLT) is based on a dynamical theory of “latent” liquidity, i.e. volumes that are intended to be exchanged but are not necessarily visible in the order book. Plausible assumptions about the dynamics of such liquidity lead to a generic “V-shape” in the vicinity of the current price. Incoming buy (sell) trades then face more and more resistance as the price moves up (down). The linear increase of liquidity then translates into a universal square root price impact, as data suggests – see our book Trades, Quotes & Prices (Cambridge University Press).
Is this the end of the story? Perhaps not, and new work on the TSE data by my student Guillaume Maitrier suggests that one might need to revise somewhat the LLOB story. Stay tuned!
