Endogeneity Near Criticality in Self-Exciting Mid-Price Dynamics

Hardiman, Bercot, and Bouchaud model the sequence of mid-price changes in E-mini S&P 500 futures as a Hawkes point process, parameterizing the self-excitation kernel as a power law φ(τ) ∝ τ^{−(1+β)} with exponent β ∈ (0, 1), and estimating the branching ratio n = ∫_0^∞ φ(τ) dτ from data spanning 1998 through 2011. The central finding is that n remains persistently close to unity — in the range 0.9 to 1.0 — across the entire sample period, independent of market regimes, volatility levels, or significant structural changes in market microstructure over that decade. A branching ratio near one means that each exogenous event — a genuine news arrival or an externally motivated order — generates on average nearly one additional endogenously triggered event via the self-excitation mechanism, and the cascade of endogenous events then generates further cascades, producing a near-critical amplification of the original shock by a factor of approximately 1/(1−n). Near criticality, this amplification diverges, and the system's response becomes scale-free: the distribution of endogenous cascade sizes follows a power law with exponent related to β, there is no characteristic time scale for the decay of the self-excitation, and the market operates in a state that maximizes information transmission but also maximizes susceptibility to large endogenous fluctuations. The scale-free nature of the endogenous cascade under power-law kernels is the key structural feature that connects this paper to the rough volatility literature: the same power-law tail that generates near-criticality in the Hawkes branching structure is the kernel that, in the El Euch–Rosenbaum limit, produces the fractional Brownian motion driving rough volatility. The temporal structure of the cascades — with cross-event correlations decaying as τ^{−β} — maps directly onto the Hurst index H = 1 − β/2 of the macroscopic variance process, providing a microstructure interpretation of the roughness parameter. For regime interpretation at the desk level, the persistent proximity of n to unity suggests that the market operates in a state of permanent near-criticality rather than transitioning between clearly subcritical and clearly supercritical regimes, and this has implications for how the desk should interpret apparent momentum signals: much of what appears as directional momentum at short horizons is in fact the tail of an endogenous Hawkes cascade triggered by a small exogenous shock, and conditioning on the cascade's estimated duration is more informative than conditioning on the direction of the triggering move alone.

The univariate Hawkes model specifies the conditional intensity as λ_t = μ + ∫_0^t φ(t − s) dN_s where μ is the exogenous baseline intensity, N_t is the count of mid-price change events up to time t, and φ(τ) = n · β · τ_0^β · (τ + τ_0)^{−(1+β)} is the power-law excitation kernel normalized so that ∫_0^∞ φ(τ) dτ = n. The branching ratio n ∈ [0, 1) is the key parameter: n = 0 means no self-excitation (Poisson process), n → 1 is the critical point. The exogenous fraction of events in any time window is (1 − n), and the average cluster size — the expected number of endogenously generated events per exogenous arrival — is 1/(1 − n). In the nearly-critical regime the distribution of cluster sizes follows P(cluster = k) ∝ k^{−(1+1/β)} for large k, making large cascades power-law distributed with no finite mean for β ≤ 1. The Bartlett spectrum of the process (the spectral density of the increment process) is S(ω) = μ / |1 − φ̂(ω)|² where φ̂ denotes the Fourier transform of φ; at low frequencies near-criticality produces a divergence in S(ω) as |1 − φ̂(0)| = 1 − n → 0, encoding the long memory and the scale-free response of the system.

The estimation methodology for the branching ratio n is non-trivial and the paper devotes considerable attention to establishing the robustness of n ≈ 1 against estimation artifacts. The key difficulty is that maximum likelihood estimation of Hawkes parameters is sensitive to the choice of kernel truncation — the maximum lag beyond which φ(τ) is set to zero in the likelihood — and for power-law kernels with slowly decaying tails, this truncation can produce substantial downward bias in n if set too short. The paper uses a spectral estimation approach based on the Bartlett spectrum as a cross-check, finding consistent estimates across methods and confirming that n ≈ 1 is not a truncation artifact. The Yule–Sismondi cascade interpretation — named for the branching-process genealogy in which each event is either an immigrant (exogenous) or a descendant of a previous event (endogenous) — provides the conceptual framework for thinking about what n ≈ 1 means for market dynamics. In the subcritical regime, n < 1, cascades die out in finite expected time and the market has a well-defined mean-reversion timescale set by (1 − n)^{−1} times the kernel scale. Near criticality, n → 1, the expected cascade duration diverges and the market's self-excitation operates on all time scales simultaneously, which is precisely the mechanism that generates the multi-scale temporal correlations of volatility documented in rough-vol papers. The temporal acceleration of endogeneity — periods where n is closer to unity than usual — is empirically associated with periods of high observed volatility, suggesting a feedback between the level of volatility and the degree of self-excitation, though the Hawkes framework treats n as a fixed parameter and does not model this feedback dynamically. Comparison with realized variance cascade interpretations reveals a consistent picture: the heterogeneous time scales of volatility clustering, the long-range correlations of realized variance, and the near-unit branching ratio of mid-price moves all point to the same underlying mechanism of scale-free temporal self-amplification, and the rough-volatility literature can be understood as the continuous-time macroscopic limit of this microstructure cascade picture.

The high estimated branching ratio is a direct regime signal for mechanically-supported apparent momentum: when n is near unity, a significant fraction of observed directional price moves are endogenously generated continuations of prior cascades rather than responses to genuine new information, and any directional signal extracted from price action during such periods has correspondingly reduced information content about future price moves once the cascade exhausts itself. The desk should maintain a branching-ratio tracker alongside its standard momentum signals, and during periods of estimated high n, apply a haircut to position sizing on signals that rely on return-path autocorrelation. Asymmetric impact of genuine exogenous shocks in the near-critical regime is a second implication. When n is near one, the amplification factor 1/(1 − n) is large, so a small genuine information arrival can produce a price move far exceeding what its direct informational content would justify, and the cascade-amplified price move will subsequently revert as the endogenous excitation decays. The desk's conditioning on estimated cascade age — how many periods have elapsed since the last identified exogenous event trigger — is motivated by this asymmetry, and risk management protocols during identified cascade episodes should reflect the elevated probability of mean reversion relative to continuation.

The branching ratio estimation from event-time data faces several well-documented biases. Self-trade filtering — the removal of trades that are matched against the same participant's own orders — affects the event count in ways that depend on venue and participant mix, and its omission biases n upward or downward depending on the correlation between self-trades and market-moving events. Event-time aggregation at the millisecond or second level, required to make the Hawkes model computationally tractable, discards the intra-aggregate event ordering and conflates simultaneous events, introducing a systematic downward bias in n because clustered events are compressed into single counts. The power-law kernel assumption is both the source of the model's elegance — it provides the connection to the rough-volatility limit — and its most questionable feature. Empirical kernel estimates from nonparametric methods show deviations from pure power law at short lags (where market microstructure effects dominate) and at long lags (where the kernel may truncate faster than any power law due to the finite memory of market participants). The structural instability of power-law fits across regimes, venues, and sampling procedures means that the H = α − 1/2 identification — which connects the Hawkes tail exponent to the volatility Hurst index — should be treated as an asymptotic theoretical relationship rather than a precise empirical one, and the uncertainty in n translates into material uncertainty in any downstream roughness-based conditioning.