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Cross-Impact

Definition

Cross-impact is the influence of order flow on asset \(j\) on the price of asset \(i \neq j\). It is the multi-asset generalisation of price impact — where standard price impact asks "how much does AAPL's order flow move AAPL's price?", cross-impact asks "how much does AAPL's order flow move GOOG's price?".

Two regimes

Regime Claim Evidence
Contemporaneous cross-impact Weak or absent once within-asset multi-level information is aggregated. Cont-Cucuringu-Zhang 2023 — integrated OFI leaves no room for cross-asset terms to add OOS \(R^2\).
Predictive (lagged) cross-impact Real at short horizons (\(\leq\) few minutes), decays rapidly. Same paper — lagged cross-asset OFIs add economic PnL in 1-min-ahead forecasting even with integrated OFI.

This asymmetry is the central finding of the 2023 Cont-Cucuringu-Zhang paper and reconciles earlier apparent-conflict results: Benzaquen et al. (2017) and others argued cross-impact was meaningful; Capponi-Cont (2020) argued that a common factor subsumed it. The resolution: contemporaneously Capponi-Cont are right; for forecasting Benzaquen et al. are right.

Why contemporaneous cross-impact vanishes under multi-level aggregation

Consider a multi-asset portfolio trader who splits their order across AAPL (best-level) and GOOG (deep level) simultaneously. A best-level-only OFI model for AAPL misses the GOOG leg and has to "borrow" that information via the cross-impact coefficient \(\beta_{\text{AAPL, GOOG}}\). But a multi-level integrated OFI for AAPL already captures the deep-level order directly, so the cross-impact term becomes redundant.

Mechanism in the paper's notation: information along the path \(A_j \to A_i \to \text{ofi}^3_i \to r_i\) is absorbed by integrated OFI; only \(A_j \to B_i\) (truly "other-stock-affects-this-stock") cross-routes survive, and those are rare.

Why predictive cross-impact persists

Traders do not instantly react to order flow in other stocks. There is a "flow formation period" — a few minutes between when a trade pattern appears in one stock and when related stocks' prices adjust. Cohen-Frazzini (2008) and Hou (2007) independently documented this lead-lag as "industry information diffusion" or "attention constraints". Cross-asset lagged OFIs are therefore a predictive signal, even when contemporaneous cross-impact is absent.

Decay profile (from Cont-Cucuringu-Zhang): - 1 min horizon: cross-impact doubles PnL vs own-OFI only (0.43 vs 0.21 annualised). - 3 min: advantage narrows. - 30 min: no advantage remains.

Data requirements

To estimate cross-impact on \(N\) stocks with 1-min bars over \(T\) minutes: - At minimum \(T \gg N\) to make OLS well-posed; in practice LASSO is needed even when \(T > N\) because of high cross-asset OFI correlations. - The cross-impact coefficient matrix exhibits a dominant "market mode" singular value (rank-1 structure) plus 6–8 "sector-mode" singular values.

Sector structure

Empirically (Nasdaq-100, 2017–2019), cross-impact is concentrated in three "source" sectors:

  1. Communication Services (highest out-degree).
  2. Consumer Discretionary.
  3. Information Technology.

Top individual stocks by out-degree centrality (forward-looking, 1-min): AMZN, NFLX, NVDA, GOOG, GOOGL. These tend to lead the rest of the market.

Tick-size dependency: stocks with larger tick-to-price ratio (small prices or large ticks) show stronger cross-impact — consistent with the tick-size regime effect in single-asset OFI work.

Portfolio-level implications

Even when individual-asset cross-impact is zero, portfolio returns still depend on cross-impact via the angle between the portfolio weight vector \(\vec{w}\) and the impact coefficient vector \(\vec{\beta}\):

\[ r_p = \sum_i w_i r_i = \sum_i w_i \beta_{i,i} \text{ofi}_i + \sum_{i \neq j} w_i \beta_{i,j} \text{ofi}_j + \epsilon \]

Only if \(\vec{\beta}\) and \(\vec{w}\) are parallel and all \(\beta_{i,i}\) are equal does the cross-term vanish. For eigenportfolios and equal-weighted portfolios on Nasdaq-100, cross-impact adds \(\sim\!3\%\) OOS \(R^2\) over the own-OFI-only portfolio model.

Open questions

  • How do multi-level cross-impact matrices evolve over time? Current estimates are static within 30-minute windows.
  • Can level-aware multi-level cross-impact (not collapsed via PCA) beat the integrated-OFI predictive cross-impact baseline at short horizons?
  • Does cross-impact depend on client-ID-level order flow data (i.e. knowing the same trader is behind orders in multiple assets)? Authors speculate yes, but no public dataset to test.
  • Cross-impact in futures, FX, crypto, and options markets — does the decay profile differ?
  • Beyond linear models — does deep learning on cross-asset LOB vectors improve short-horizon forecasting?

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