Skip to content

Models for the Impact of All Order Book Events

Authors: Zoltán Eisler, Jean-Philippe Bouchaud, Julien Kockelkoren Institution: Capital Fund Management, Paris Year: 2011 (arXiv July 2011; published Quantitative Finance, 2012) arXiv: 1107.3364 Categories: q-fin.TR, q-fin.ST

Plain-language abstract

Follow-up to the 2009 event-type decomposition paper. Where the earlier work established the machinery and showed it fits data, this paper proposes two competing structural models for multi-event-type impact and adjudicates between them. The TIM (Transient Impact Model) just generalises the market-order propagator: every event type \(\pi\) has its own decaying kernel \(G_\pi(\tau)\). The HDIM (History-Dependent Impact Model) is more principled: only price-changing events cause instantaneous price jumps, but the size of each jump depends on the past history of all six event types via an "influence matrix" \(\kappa_{\pi_1, \pi'}\). The influence matrix is the quantitative handle on a well-known phenomenon: aggressive market orders soften future impact of same-sign market orders and amplify opposite-sign impact — the mechanism that keeps prices diffusive despite strongly autocorrelated trade signs. Empirically the TIM fits data slightly better even though HDIM is theoretically cleaner; the authors attribute this to the Gaussian factorisation approximation required to calibrate HDIM.

Key contributions

  1. Two competing structural models for multi-event impact:
  2. TIM (Transient Impact Model): \(p_t = p_0 + \sum_{t' < t} \sum_\pi G_\pi(t-t') \cdot \epsilon_{t'} \cdot I(\pi_{t'} = \pi)\). Each event type has a bare propagator \(G_\pi(\tau)\) (derived from the 2009 paper). Theoretically inconsistent: assigns non-zero impact to non-price-changing events.
  3. HDIM (History-Dependent Impact Model): Return at \(t\) is zero unless \(\pi_t\) is a price-changing event, in which case the jump \(\Delta_t\) depends on past order flow via \(\Delta_t = \Delta^R_\pi + \sum_{\pi_1, \ell > 0} \kappa_{\pi_1, \pi'}(\ell) \cdot \epsilon_{t-\ell} \cdot I(\pi_{t-\ell} = \pi_1)\).
  4. Equivalence condition: TIM and HDIM coincide iff \(G_\pi(\ell) - G_\pi(1) = \sum_{\ell' \leq \ell} \kappa_{\pi, \pi'}(\ell')\) — a strong structural restriction not supported by data.
  5. Empirical influence matrix \(\kappa_{\pi_1, \pi'}(\ell)\) (6 × 3 = 18 kernels):
  6. \(\kappa_{\text{MO}', \text{MO}'} < 0\): a price-moving buy market order reduces the subsequent gap (on the same side) for future price-moving market orders — the book "hardens" against further aggression.
  7. \(\kappa_{\text{MO}^0, \pi'} > 0\): a small (non-price-moving) market order softens the book — gaps grow.
  8. Queue events (LO⁰, CA⁰) also harden the book for small-tick stocks.
  9. Large-tick stocks: \(\kappa \approx 0\) (gaps are always one tick).
  10. Three-process interpretation of price dynamics:
  11. Instantaneous jumps from price-changing events.
  12. Future event-rate modulation (captured by \(C_{\pi_1, \pi_2}\) correlations).
  13. Future jump-size modulation (captured by \(\kappa_{\pi_1, \pi'}\)).
  14. Verdict: TIM beats HDIM on volatility fit despite HDIM's cleaner structure. Root cause is the Gaussian factorisation approximation used to calibrate \(\kappa\) — multiplying calibrated \(\kappa\) by 3 restores most of the fit. Suggests a numerically heavier but more accurate calibration is needed.
  15. Relation to VAR: The TIM is a multi-event extension of Hasbrouck's VAR model with no direct past-return regressor. The HDIM is a VAR model with event-type-specific return equations.

Method summary

TIM (Transient Impact Model)

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

Inversion formula for \(G_\pi\):

\[R_\pi(\ell) \;=\; \sum_{\pi_1}\; P(\pi_1) \sum_{\ell' = 0}^{\ell-1} G_{\pi_1}(\ell - \ell') \cdot C_{\pi, \pi_1}(\ell')\]

Numerically better to invert in terms of increments \(\Delta G_\pi(\ell) = G_\pi(\ell+1) - G_\pi(\ell)\), which yields a stable block-Toeplitz system.

HDIM (History-Dependent Impact Model)

Return at \(t\):

\[r_t \;=\; \epsilon_t \cdot \Delta_t \cdot I(\pi_t = \pi')\]

where for price-changing \(\pi'\) (MO', LO', CA'):

\[\Delta_t \;=\; \Delta^R_{\pi'} \;+\; \sum_{\pi_1}\;\sum_{\ell > 0}\; \kappa_{\pi_1, \pi'}(\ell) \;\cdot\; \epsilon_{t - \ell} \;\cdot\; I(\pi_{t-\ell} = \pi_1)\]

The influence matrix \(\kappa_{\pi_1, \pi'}(\ell)\) is 6 event-types \(\times\) 3 price-changing-event-types \(\times\) lags \(\ell\). Calibrated via the return-response cross-correlation \(S_{\pi_1, \pi'}(\ell) = \langle r_{t+\ell} \cdot \epsilon_t I(\pi_t = \pi_1) \rangle\), under a Gaussian factorisation approximation for the three-point correlation.

Volatility prediction

The time-lagged price diffusion \(D(\ell) = \mathbb{E}[(p_{t+\ell} - p_t)^2]\) has explicit expressions in both models (Appendix A of the paper). This is the primary empirical test: a correctly-calibrated model should reproduce \(D(\ell)/\ell\) at all lags.

Data

Same as the 2009 paper: 14 NASDAQ stocks, 3 Mar – 19 May 2008. Large-tick and small-tick sub-samples handled separately. Excludes first 30 / last 40 minutes of the trading day.

Main results

Large-tick stocks

  • \(\kappa \approx 0\) (constant-gap limit).
  • Both TIM and HDIM reduce to the constant-impact model.
  • Fit to \(R_\pi(\ell)\) and \(D(\ell)/\ell\) is near-perfect (small residuals accounted for by gap-fluctuation correction).

Small-tick stocks

  • \(\kappa\) is non-trivial and event-type-dependent.
  • TIM fits \(D(\ell)/\ell\) very well at long lags (surprisingly accurate given the model's theoretical inconsistency).
  • HDIM overshoots \(D(\ell)\) by ~15% with the naive Gaussian-factorisation calibration. Multiplying \(\kappa\) by 3 restores the fit to within ~5%.
  • Both models miss a small high-frequency term \(D_{\text{hf}} \approx 0.04\) ticks² — attributed to limit-order placements-then-cancellations inside the gap not covered by the event-type taxonomy.

The influence matrix \(\kappa_{\text{MO}', \pi'}(\ell)\)

  • \(\kappa_{\text{MO}', \text{MO}'} < 0\): a MO' reduces the gap for future same-sign MO'. "Hard book" after aggressive trade on the same side.
  • \(\kappa_{\text{MO}', \text{CA}'}\), \(\kappa_{\text{MO}', \text{LO}'}\): generally also negative but smaller.
  • Integrated effect: \(\sum_\ell \kappa_{\pi_1, \pi'}(\ell)\) is mostly negative for price-changing events on the same side — formal confirmation of the "liquidity refill" hypothesis.
  • Small market orders (MO⁰) have positive \(\kappa\): they "soften" the book (gaps grow), indicating they're often followed by liquidity withdrawal or limit-order repositioning.

Limitations

  • Calibration approximation error. HDIM calibration via Eq. (19) uses a Gaussian factorisation of three- and four-point correlations. The factorisation error dominates residuals; a historical-simulation calibration would be more accurate but much more computationally expensive.
  • Linear ansatz for gap dynamics. Only quadratic-in-\(\epsilon\) terms are retained. Higher-order (\(\epsilon^4\)) gap responses are documented to exist but omitted.
  • L1 only. Events deeper in the book are unobserved; their effect shows up as residual dressing of the bare impacts.
  • Multi-venue. Same 2008 NASDAQ-only data as the 2009 paper; off-NASDAQ flow unobserved.
  • Volume-agnostic. MO' and MO⁰ differentiate on price-moving vs not, but within each class the volume dependence is collapsed.
  • TIM theoretical inconsistency. TIM assigns non-zero impact to non-price-changing events, which is structurally wrong. The empirical success of TIM over HDIM remains unexplained.

Connections to other wiki pages

Signal-design takeaways

For an L3 signal developer working from this paper:

  1. Influence matrix as a feature. The sign-asymmetric response to past events (MO' softens future same-sign MO' impact, hardens opposite-sign) is a real state-variable signal. If you can estimate even a crude \(\kappa_{\text{MO}', \text{MO}'}\) on your market, a signal conditional on "recent same-sign MO' count" has non-trivial edge.
  2. TIM is the pragmatic workhorse. HDIM is cleaner but harder to calibrate. For production use, a multi-event-type propagator (TIM) with per-event-type kernels is a sensible baseline — HDIM-like refinements are second-order.
  3. Large-tick simplification. If your universe is large-tick, \(\kappa = 0\) is a good first approximation — the signal reduces to a linear combination of event-type counts (essentially OFI).
  4. Event-type classification matters more than volume bucketing. The paper's binary MO / MO' split captures more of the signal than fine volume quantisation does. Similarly for LO and CA.
  5. Watch for three- and four-body effects in small-tick stocks. Higher-order terms in the gap dynamics are known to exist; if you linearise at 2nd order and fit doesn't close, this is where to look.