How Markets Slowly Digest Changes in Supply and Demand¶

Authors: Jean-Philippe Bouchaud, J. Doyne Farmer, Fabrizio Lillo Institutions: Capital Fund Management (Paris); Santa Fe Institute / LUISS Guido Carli (Roma); Università di Palermo Year: 2008 (book chapter, arXiv September 2008) arXiv: 0809.0822 Categories: q-fin.TR (book chapter in Hens & Schenk-Hoppé eds., "Handbook of Financial Markets: Dynamics and Evolution")

Plain-language abstract¶

A canonical survey / manifesto of the "econophysics" view of price formation. Large orders cannot be executed instantly — they must be sliced and traded incrementally over hours to months — so the visible order flow is a strongly persistent, long-memory process. Reconciling long-memory order flow with the empirical fact that price series are close to martingales forces a specific theoretical structure: market impact must be transient, with a slowly decaying kernel, and liquidity must co-adjust. The paper reviews the propagator model (response function \(G\)), the long-memory facts, concave impact of individual trades, the square-root law of aggregate impact, spread-impact links, and empirical tests on equities. Its central claim is that most of the information processed by markets comes from supply and demand itself, not from external news.

Key concepts and formulas¶

The propagator (transient impact) model¶

See propagator-model for a dedicated method page. The mid-price return at time \(t\) is a linear superposition of the impact of all past signed trades \(\varepsilon_{t'} \in \{+1, -1\}\):

\[r_t = \sum_{t' \leq t} G(t - t') \cdot \varepsilon_{t'} + \eta_t\]

where: - \(G(\tau)\) is the response function / propagator, typically a slowly decaying power law \(G(\tau) \sim \tau^{-\beta}\) with \(\beta \approx 0.2\)–\(0.5\). - \(\eta_t\) is an uncorrelated noise term.

For prices to be diffusive despite long-memory in \(\varepsilon\), the decay of \(G\) must exactly offset the persistence of \(\varepsilon\) — a non-trivial self-consistency constraint.

Long memory of order flow¶

Sign autocorrelation of trades decays as a power law:

\[\mathbb{E}[\varepsilon_t \varepsilon_{t+\tau}] \sim \tau^{-\gamma}, \qquad 0 < \gamma < 1\]

on any liquid equity, for \(\tau\) over hours to days. Explained by order splitting of large hidden meta-orders: a few hundred large parent orders executed over long horizons generate the persistence.

Concave individual-transaction impact¶

Average price shift caused by a single trade of volume \(v\):

\[\mathbb{E}[\Delta p \mid v] \sim \log(v) \quad \text{or} \quad \mathbb{E}[\Delta p \mid v] \sim v^\alpha, \; \alpha \in [0, 0.5]\]

Concave: doubling the trade does not double the impact.

Square-root law of aggregate (metaorder) impact¶

For a metaorder of total size \(Q\):

\[I(Q) \sim \sigma \cdot \left(\frac{Q}{V}\right)^{1/2}\]

with \(\sigma\) daily volatility, \(V\) daily volume. This is Kyle's 1/2 exponent emerging empirically from the propagator dynamics + long-memory flow, not assumed a priori.

Spread and impact (Glosten-Milgrom / MRR models)¶

The bid-ask spread compensates market makers for two things: 1. Adverse selection — trading against better-informed agents. 2. Inventory carrying risk.

In the Glosten-Milgrom model, spread is exactly the asymmetric-information expected loss per trade. The paper reviews why empirical spreads sit close to this theoretical floor for liquid stocks.

Key empirical findings surveyed¶

Signed order flow is long-memory in 30+ liquid equities across exchanges and decades.
Impact is roughly concave in single-trade volume, power-law with \(\alpha \approx 0.1\)–\(0.3\).
Metaorder impact follows square-root law with coefficient proportional to volatility / \(\sqrt{\text{volume}}\).
Impact decays over time after metaorder completion: partial reversion but some permanent residual.
Spread and impact are strongly correlated — larger average spreads on days with larger realised volatility.
Spread has diurnal pattern (U-shape) matching volatility / impact.
Liquidity evaporates before price moves: the paper documents the "queue depletion" phenomenon where depth at best quotes decreases before a price tick.

Two interpretive frameworks contrasted¶

Fixed permanent impact (Kyle / MRR)¶

Each trade permanently shifts the efficient price. Requires that informed vs uninformed trades can be distinguished. Simple but struggles with: - Long memory of order flow should imply predictable returns, which is absent empirically. - Concavity of individual-trade impact is unexplained.

Transient impact (propagator) model¶

Each trade temporarily shifts prices; impact decays over time. Handles long-memory order flow cleanly: returns remain unpredictable because a persistent order flow is absorbed by a decaying response kernel. Predicts the right qualitative shape of impact. The paper argues this framework is more empirically supported.

The authors also discuss "history-dependent permanent impact" as a third unified framework that recovers both as limits.

Main theoretical claims¶

Market efficiency + long-memory order flow jointly force a transient, slowly-decaying impact kernel.
Square-root law of metaorder impact is a consequence of this kernel structure plus long-memory flow — not an empirical accident nor a derivation from rational expectations.
The "volatility puzzle" (excess volatility vs fundamental volatility) is partially resolved by noting that most order flow is uninformed (noise + order-splitting) and still causes real price movement through the propagator.
Supply–demand itself carries most of the processed information in markets, not external news. Most informed agents are at best weakly informed.

Limitations (acknowledged in the paper and since)¶

Linear propagator model — real impact has interaction terms (e.g., sign-dependent, spread-dependent).
Constant kernel \(G\) — in reality \(G\) is state-dependent (stress regimes, intraday patterns).
No distinction between limit and market order impact — later work (e.g., Eisler–Bouchaud–Kockelkoren) generalises with a multivariate propagator.
Most empirics are on equities — FX, futures, and crypto later show partial but not exact agreement.
Spread–impact link is broadly correct but the ratio is not universal across assets.

Why it matters for this wiki¶

This paper is one of the three foundational references on market impact theory, alongside Kyle (1985) and Almgren-Chriss (2001). It is cited by essentially every paper already in the wiki that talks about impact: price-impact-order-book-events, mpc-trade-execution, reality-gap-lob-simulation, order-flow-filtration.

The propagator formulation is what motivated Noble–Rosenbaum–Souilmi (reality-gap-lob-simulation) to add a power-law impact feedback kernel to their QR simulator. It also underlies the Bacry et al. multivariate-Hawkes impact decomposition referenced in forecasting-high-frequency-ofi and order-flow-filtration.

Connections¶

Definitional anchor for price-impact — propagator decomposition, square-root law, spread-impact link.
Classical comparator to price-impact-order-book-events — CKS derive a linear impact from OFI rather than a propagator-kernel on signed trades. Different inputs, complementary conclusions.
Long-memory order flow is the foundation underneath hawkes-process approaches in forecasting-high-frequency-ofi and order-flow-filtration.
Glosten-Milgrom review inside this paper provides the adverse-selection foundation for adverse-selection and market-making — despite GM itself not being on arXiv.
Kyle model review — the paper references Kyle 1985 (also not arXiv) but reconstructs its key results.