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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 time scales for the liquidity ("fast" and "slow") enables one to reach quantitative agreement with the data.Financial markets sputter enormous amounts of data that can now be used to test scientific theories at levels of precision comparable to those achieved in physical sciences (see, e.g. [1] for a recent example). Among the most remarkable empirical findings in the last decades is the "square-root impact law", which quantifies how much prices are affected, on average, by large buy or sell orders, usually executed as a succession of smaller trades. Such a succession of small trades, all executed in the same direction (either buys or sells) and originating from the same market participant, is called a metaorder. A metaorder of total size Q impacts the price as ∼ √ Q and not proportionally to Q as naively expected and actually predicted by classical economics arguments [2]. The square-root law is surprisingly universal: it is found to be to a large degree independent of details such as the asset class, time period, execution style and market venues [3][4][5][6][7][8][9][10][11][12][13][14]. In particular, the advent of electronic markets and High Frequency Trading has not altered the square-root behaviour, in spite of radical changes in the microstructure of markets. The universality of this square-root law, together with its insensitivity to the high frequency dynamics of prices, suggests that its interpretation should lie in some general properties of the low frequency, large scale dynamics of liquidity [15]. In fact, the publicly displayed liquidity at any given time is usually very small -typically on the order of 10 −2 of the total daily transaction volume in stock markets. Financial markets are the arena of a collective hide-and-seek game between buyers and sellers, resulting in a somewhat paradoxical situation where the total quantity that markets participants intend to trade is very large (0.5% of the total market capitalisation changes hands every day in stock markets) while most of this liquidity remains hidden, or "latent". These observa-tions have lead to the development of a physics inspired, "locally linear order book" (LLOB) model for the coarsegrained dynamics of latent liquidity [7,[15][16][17], which naturally explains why the impact of metaorders grows like the square-root of its size in a certain regime of parameters [17]. But this LLOB model also suggests that for a given execution time T , the very small Q regime should revert to a linear behaviour. The model in fact predicts the detailed shape of the crossover between linear and square-root i...

This paper is devoted to the important yet unexplored subject of crowding effects on market impact, that we call "co-impact". Our analysis is based on a large database of metaorders by institutional investors in the U.S. equity market. We find that the market chiefly reacts to the net order flow of ongoing metaorders, without individually distinguishing them. The joint co-impact of multiple contemporaneous metaorders depends on the total number of metaorders and their mutual sign correlation. Using a simple heuristic model calibrated on data, we reproduce very well the different regimes of the empirical market impact curves as a function of volume fraction φ: square-root for large φ, linear for intermediate φ, and a finite intercept I 0 when φ → 0. The value of I 0 grows with the sign correlation coefficient. Our study sheds light on an apparent paradox: How can a non-linear impact law survive in the presence of a large number of simultaneously executed metaorders? 2

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The notion of market impact is subtle and sometimes misinterpreted. Here we argue that impact should not be misconstrued as volatility. In particular, the so-called "square-root impact law", which states that impact grows as the square-root of traded volume, has nothing to do with price diffusion, i.e. that typical price changes grow as the square-root of time. We rationalise empirical findings on impact and volatility by introducing a simple scaling argument and confronting it to data.