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The Price Impact of Order Book Events

Authors: Rama Cont, Arseniy Kukanov, Sasha Stoikov Institution: Columbia University (Cont), Cornell University (Stoikov) Year: 2010 arXiv: 1011.6402 Categories: q-fin.TR, q-fin.ST

Plain-language abstract

Every order book event — a limit order submission, a market order execution, or a cancellation — moves prices. This paper quantifies how and why. Using NYSE data for 50 stocks, the authors show that short-horizon price changes are driven almost entirely by order flow imbalance (OFI): the net pressure at the best bid and ask. The relationship is linear, with the slope set by the market depth. This framework also explains the empirical "square-root law" of price impact.

Key contributions

  1. Definition of OFI — formally defined as the difference between the rate of buy-side and sell-side order book events at the best prices. Captures supply/demand imbalance more cleanly than raw trade volume.
  2. Linear price impact model — \(\Delta p \approx \beta \cdot \text{OFI}\), where \(\beta \propto 1 / \text{depth}\).
  3. Square-root law derivation — the empirically observed \(\Delta p \propto \sqrt{Q}\) (price impact proportional to square root of trade size \(Q\)) is derived from the linear OFI model via a scaling argument.
  4. Robustness — results are stable across time scales, intraday seasonality effects, and across 50 heterogeneous NYSE stocks.
  5. OFI vs volume — OFI is a more reliable predictor of price changes than trade volume; the volume relationship is noisier.

Method summary

Data

One calendar month (April 2010) of TAQ data for 50 S&P 500 stocks chosen by random number generator, obtained via WRDS. Level 1 only (best bid/ask and queue sizes). Intraday 10-second time grid (21 trading days). The ratio of quote updates to trades is ≈ 40 : 1 — this disparity motivates the use of quote flow over trade flow.

OFI construction

Per-event signed contribution \(e_n\) to the bid queue:

  • \(P^B\) unchanged: \(e_n = q^B_n - q^B_{n-1}\) (limit add, market sell, or cancel at the best bid)
  • \(P^B\) increases: \(e_n = q^B_n\) (price-improving limit buy lifts the best)
  • \(P^B\) decreases: \(e_n = q^B_{n-1}\) (entire queue removed by market order or cancel)

Ask-side events use opposite signs. Aggregated over an interval:

\[\text{OFI}_k = \sum_{n = N(t_{k-1})+1}^{N(t_k)} e_n = L^b - C^b - M^s \;-\; L^s + C^s + M^b\]

where \(L, C, M\) count limit orders, cancels, and market orders on each side. Market sells and bid cancels are treated equivalently since they have identical effect on the bid queue.

Model

Under a stylized book with depth \(D\) at each level beyond the best, the relation

\[\Delta P_k = \frac{\delta}{D}\,\text{OFI}_k + \epsilon_k\]

holds exactly (tick size \(\delta\), depth \(D\)). Empirically the authors fit

\[\Delta P_k = \alpha_i + \beta_i \,\text{OFI}_k + \epsilon_k, \qquad \beta_i = c \cdot AD_i^{-\lambda}\]

by OLS on 273 half-hour sub-samples per stock, with \(AD_i\) the average best-quote depth in sub-sample \(i\). Standard errors are White (for the \(\beta\) regression) and Newey–West (for the \(\lambda\) regression).

Main results

  • Linear OFI fits extremely well: average \(R^2 = 65\%\) across 50 stocks × 273 sub-samples. Adding a quadratic term \(\gamma \cdot \text{OFI}_k|\text{OFI}_k|\) raises \(R^2\) to 68% but the quadratic coefficient is insignificant for most samples — the relationship is well-modelled as linear.
  • Depth exponent \(\lambda \approx 1\): the hypothesis \(\lambda = 1\) cannot be rejected for 35 out of 50 stocks (Newey–West, 5%). Depth enters approximately linearly, confirming the stylized model.
  • Tautology check: even after removing price-changing events from \(\text{OFI}_k\), average \(R^2\) stays in 35–60% — the explanatory power is not just a mechanical artefact.
  • Intraday pattern: depth at market open is ~½ of its daily average, so \(\beta\) is ~2× higher; across the full session, \(\beta\) at open is ~5× higher than at close. This explains the well-known intraday volatility U-shape using only depth and OFI — no need to invoke information asymmetry.
  • Robust to timescale: \(R^2\) rises with \(\Delta t\) but qualitative results are unchanged from sub-second up to 10 minutes.
  • Trade volume alone is noisier: restricting to trade volume gives an apparent square-root impact (via data aggregation), but it is less stable and less interpretable than the OFI linear model.

Limitations

  • Level 1 only: queue sizes at the best bid/ask only; deeper-level information is ignored. Later work (see price-impact-generalized-ofi) extends to multi-level OFI.
  • Residuals heteroscedastic and somewhat autocorrelated — White / Newey–West standard errors mitigate but do not remove this.
  • Exceptions: for wide-spread / low-depth stocks (APOL, AZO, CME in the sample) the depth regression fits poorly — the Level-1 framework breaks when hidden orders or deep-book dynamics dominate.
  • Tautology risk: price-changing events are both regressor and cause of the response; acknowledged but quantified (R² drops to 35–60% after exclusion).
  • Dataset vintage: April 2010 NYSE, post-decimalisation but pre-modern HFT saturation. Market structure has continued to evolve.

Connections