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The Price Impact of Order Book Events: Market Orders, Limit Orders and Cancellations

Authors: Zoltán Eisler, Jean-Philippe Bouchaud, Julien Kockelkoren Institution: Capital Fund Management, Paris Year: 2009 (arXiv April 2009; published Quantitative Finance 12(9), 2012) arXiv: 0904.0900 Categories: q-fin.TR, q-fin.ST

Plain-language abstract

The standard impact-model story treats "a trade" as the object that moves prices. This paper says no — on modern electronic markets every order-book event (market order, limit order arrival, cancellation, at or inside the spread) contributes, and you should measure them all. By tracking six event types (MO⁰, MO', LO⁰, LO', CA⁰, CA' — with ' meaning "changes the best price") on 14 NASDAQ stocks in 2008, the authors extract the "bare" impact of each event type — what the event would contribute if it happened in isolation, stripped of correlations with surrounding flow. Two headline findings: (i) for large-tick stocks, bare impacts are permanent and non-fluctuating — a simple constant-impact model fits; (ii) for small-tick stocks, bare impacts are history-dependent — the gap behind the best quote fluctuates and carries a non-trivial memory of past flow, accurately modelled by an autoregressive correction. The decomposition framework generalises the earlier market-order-only propagator model and connects to Hasbrouck's VAR.

Key contributions

  1. Six-event-type taxonomy — MO⁰, MO', LO⁰, LO', CA⁰, CA' (0 = no price change at best; ' = changes best price). Each event carries a sign \(\epsilon\) (direction of expected price effect) and a gap \(\Delta\) (size of price move if it moves the price). This taxonomy is now standard in empirical microstructure.
  2. Bare vs dressed impact distinction — The market-order propagator \(G(\tau)\) measured in trades-only data is "dressed" by unobserved limit-order / cancellation flow. The authors derive the bare-impact extraction:
\[R_\pi(\ell) = \sum_{\pi_1} \sum_{\ell' = 0}^{\ell - 1} G_{\pi_1}(\ell - \ell') P(\pi_1) C_{\pi, \pi_1}(\ell')\]

where \(R_\pi\) is the observed response to event type \(\pi\), \(C_{\pi, \pi'}\) is the signed event-event correlation, and \(G_\pi\) is the unobserved bare impact. Inverting this linear system recovers \(G_\pi\). 3. Constant-impact model works for large-tick stocks — Assuming \(G_\pi(\ell) = \text{const}\) for each event type and using constant realised gaps \(\Delta^R_\pi\), the predicted response functions and price diffusion match data nearly perfectly. Large-tick stocks have negligible gap fluctuations, so the propagator is truly permanent. 4. Small-tick stocks need gap dynamics — Gap fluctuations behind the best quote carry history dependence. A linear AR model on past order flow \(\Delta^R_\pi\) captures this:

\[\Delta_{\pi, \epsilon, t} = \Delta^R_\pi + \sum_{\pi', \ell > 0} K_{\pi, \pi'}(\ell) \cdot \epsilon_{t-\ell} \cdot I(\pi_{t-\ell} = \pi')\]

The kernels \(K\) are fitted via OLS from empirical cross-correlations and restore the fit to data. 5. Limit-order impact is real — Contrary to earlier trade-only work, limit orders have measurable (though smaller) impact than market orders. Implies that a cancellation inside the spread is nearly as potent as a market sell at the bid, and that "market-order impact" in earlier studies conflates the bare impact with induced limit-order/cancellation flow. 6. Three-way impact decomposition — The price change triggered by an event splits into: (a) the instantaneous jump, (b) the modification of future event-type rates (induced compensating flow), (c) the modification of future gap sizes. All three observable within this framework. 7. Hasbrouck VAR relation — The propagator model is a structural restriction of the VAR framework: \(B_{rr}(\ell) = 0\) (past returns cannot directly affect current returns) plus a specific causal interpretation. 8. Spread dynamics — Appendix extends the framework to the bid-ask spread, which is expressed as a linear function of past signed events — a bonus by-product.

Method summary

The six event types

Event Description Gap \(\Delta\)
MO⁰ market order, volume \(<\) outstanding at best 0 (no price move)
MO' market order, volume \(\geq\) outstanding at best half the gap to the second-best
CA⁰ partial cancellation at the best 0
LO⁰ limit order at the current best 0
CA' complete cancellation of the best half the gap to the second-best
LO' limit order inside the spread half distance from the previous best

Event sign \(\epsilon = +1\) for buy-side (buy MO, cancelled sell LO, incoming buy LO), \(-1\) for sell-side. Side \(s\) distinguishes bid vs ask — different from \(\epsilon\) for LO events.

The response function

For event type \(\pi\) with sign \(\epsilon\) at time \(t\), the mid-price moves by \(p_{t+\ell} - p_t\) on average:

\[R_\pi(\ell) = \mathbb{E}[(p_{t+\ell} - p_t) \cdot \epsilon_t \cdot I(\pi_t = \pi)] / P(\pi)\]

Six response functions measured directly from data.

Correlation structure

Signed event-event correlations \(C_{\pi_1, \pi_2}(\ell)\): 36 of them, measured. Unsigned \(\Pi_{\pi_1, \pi_2}(\ell)\) captures clustering. Key empirical finding: the sign of events overall (\(\epsilon_t\)) is short-range correlated (auto-corr dies at ~100 events), but the side of events (\(s_t\)) is long-range correlated with decay exponent \(\gamma \approx 0.7\). The mixing of limit orders and market orders compensates the market-order persistence to keep prices near-diffusive.

Temporary impact model (generalised)

\[p_t = p_0 + \sum_{t' < t} \sum_\pi G_\pi(t - t') \cdot \epsilon_{t'} \cdot I(\pi_{t'} = \pi)\]

Recover \(G_\pi\) by inverting the response equation. Diagnostic: compute the predicted price variance \(D(\ell)/\ell\) and compare to data. Large-tick: poor fit. Small-tick: decent but not great.

Constant-impact model

Replace \(G_\pi(\ell) \to G_\pi\) and \(\Delta_{\pi, \epsilon, t} \to \Delta^R_\pi\). Both conditions ("permanent impact" + "no gap fluctuations"). Works well for large-tick. For small-tick, extend with the linear AR model on gap fluctuations.

Data

  • 14 NASDAQ stocks, 3 Mar – 19 May 2008 (53 trading days).
  • ~10⁶–10⁷ events per stock.
  • Split large-tick (spread ~1 tick: AMAT, CMCSA, CSCO, DELL, INTC, MSFT, ORCL) vs small-tick (spread ~3–4 ticks: AAPL, AMZN, APOL, COST, ESRX, GILD).

Main results

Large-tick stocks

  • Event-type frequencies: LO⁰ ≈ CA⁰ ≈ 40% each, MO⁰ ≈ 5%, price-changing events total ~3%.
  • Constant-impact model: near-perfect fit to both \(R_\pi(\ell)\) and \(D(\ell)/\ell\).
  • Bare impacts: permanent and non-fluctuating. Limit-order impact ~ 60–70% of market-order impact.

Small-tick stocks

  • Event-type frequencies: LO⁰ ~ 33%, CA⁰ ~ 26%, MO⁰ ~ 5%, price-changing events ~35–40%.
  • Realised gaps much larger than average gaps (AAPL: \(2\Delta^R_{\text{MO'}} = 1.31\) ticks vs \(2\langle\Delta_{\text{MO'}}\rangle = 1.14\)). The act of moving the price is correlated with opening a larger gap.
  • Constant-impact model noticeably off. History-dependent AR model restores the fit.

Stylised facts reconfirmed / refined

  • Market-order signs \(\epsilon_{\text{MO}}\) are long-range correlated (\(\gamma \approx 0.7\)). All-event signs \(\epsilon\) are not — the reverting limit-order flow compensates.
  • MO, CA, LO cluster in time — aggressive orders induce more aggressive orders.
  • "Stimulated refill" of liquidity after a price-changing market order (LO' arrives quickly to rebuild the book).
  • Time-reversal symmetry holds for some event pairs (MO⁰/CA⁰, MO'/CA') but not all (MO'/LO' is asymmetric).

Limitations

  • Level-1 only. Events deeper in the book are treated as unobserved; their influence appears as residual "dressing" of the bare impact. A stronger version would observe all levels.
  • Multi-venue liquidity. Stocks trade across many platforms; events on other venues are unobserved and also "dress" the impacts measured on NASDAQ.
  • Short data window. 53 days in 2008 only — a turbulent year. Stability across regimes not tested.
  • Signless volume. Volume dependence is weak and collapsed into a dichotomy (moves-price vs does-not-move-price). A richer volume treatment might uncover non-linearities.
  • Linearity assumption. The superposition-of-bare-impacts model is linear. Large events or stressed regimes likely break linearity.
  • Gaps only between levels 1 and 2. The AR gap model covers only the first hidden gap — deeper gap dynamics are lumped in the residual.

Connections to other wiki pages

  • Extends: price-impact-order-book-events (Cont-Kukanov-Stoikov 2014, same data idea at a different resolution — OFI vs event-type decomposition); bouchaud-farmer-lillo-propagator (the single-event market-order propagator).
  • Directly extended by: models-for-all-order-book-events (Eisler-Bouchaud-Kockelkoren 2011) — elaborates the history-dependent model for small-tick stocks with explicit dynamic gap modelling.
  • Cross-connection: cross-impact-ofi-equity-markets — which side of the "bare impact" story does integrated OFI capture? Almost all of it for large-tick stocks; small-tick behaviour still partially non-linear via gap dynamics.
  • Method it defines: event-type-impact-decomposition — the per-event-type bare-impact extraction framework.
  • Relates to: propagator-model — this paper is its multi-event-type generalisation.
  • Companion: order-flow-imbalance — OFI as defined by Cont-Kukanov-Stoikov is a specific linear combination of the event-type counts (\(L^b - C^b - M^s - L^a + C^a + M^b\)); this paper dissects those ingredients individually.

Signal-design takeaways

For someone building an L3 signal from order-book events:

  1. Treat event types separately. Collapsing everything into "buy volume – sell volume" throws away structure that matters — especially the MO vs LO vs CA distinction.
  2. Tick size is the key regime. Large-tick: a simple linear-event-weighted signal works (think OFI). Small-tick: need the gap-fluctuation correction — your signal must depend on the current book shape.
  3. Limit-order cancellations count. A cancelled best limit order on the bid is nearly a sell market order in directional content. If you filter for executions only, you miss a third of the signal.
  4. Signed correlations tell you which events cluster with which. Useful for designing state-conditional signals (e.g. "a MO' at the ask attracts LO' inside the spread within ~10 events" = a refill signal).
  5. Bare vs dressed matters. If you fit a market-order-only propagator on your own data without limit-order awareness, the estimated \(G\) decays in a time-scale that reflects average limit-order compensation, not true impact decay. Mis-calibrated executions follow.