The Subtle Interplay between Square-root Impact, Order Imbalance & Volatility: A Unifying Framework

Guillaume Maitrier and Jean-Philippe Bouchaud

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Abstract: In this work, we aim to reconcile several apparently contradictory observations in market microstructure: is the famous ''square-root law'' of metaorder impact that decays with time compatible with the random-walk nature of prices and the linear impact of order imbalances? Can one entirely explain the volatility of prices as resulting from the flow of uninformed metaorders that mechanically impact prices? We introduce a new theoretical framework to describe metaorders with different signs, sizes and durations, which all impact prices as a square-root of volume but with a subsequent time decay. We show that, as in the original propagator model, price diffusion is ensured by the long memory of cross-correlations between metaorders. In order to account for the effect of strongly fluctuating volumes $q$ of individual trades, we further introduce two $q$-dependent exponents, which allows us to account for the way the moments of generalized volume imbalance and the correlation between price changes and generalized order flow imbalance scales with $T$. We predict in particular that the corresponding power-laws depend in a non-monotonic fashion on a parameter $a$ that allows one to put the same weight on all child orders or overweight large orders, a behaviour clearly borne out by empirical data. We also predict that the correlation between price changes and volume imbalances should display a maximum as a function of $a$, which again matches observations. Such noteworthy agreement between theory and data suggests that our framework correctly captures the basic mechanism at the heart of price formation, namely the average impact of metaorders. We argue that our results support the ''Order-Driven'' theory of excess volatility, and are at odds with the idea that a ''Fundamental'' component accounts for a large share of the volatility of financial markets.

Date: 2025-06, Revised 2025-06
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