Order Book Resiliency: Dynamics & Recovery

Updated 15 December 2025

Order book resiliency is defined as the recovery trajectory of key liquidity metrics—bid–ask spread, depth, and order intensity—following market shocks.
Empirical studies reveal that while spread and depth typically revert to baseline within 5–10 seconds, limit order intensity may take around 30 minutes to recover.
These insights inform optimal execution strategies and liquidity stress tests, enhancing algorithm calibration and overall market quality.

Order book resiliency quantifies the speed and regularity with which a limit order book (LOB) restores its liquidity and price structure following a liquidity shock, typically induced by an effective market order. Resiliency governs the temporal decay of bid–ask spread dislocations, depth imbalances, and liquidity-providing limit order intensities after an exogenous perturbation. The concept is central to the microstructure analysis of order-driven markets, the modeling of optimal execution schedules, and the assessment of systemic liquidity risk.

1. Core Definitions and Empirical Measures

Order book resiliency is formally defined via the recovery trajectories of key liquidity metrics—bid–ask spread, best-quote depth, and limit order intensity—after a market event. For precise event studies, raw metrics are de-seasonalized using minute-by-minute Fourier-Flexible-Form regressions to remove intraday effects. The principal resiliency measures are:

Normalized Spread Resiliency:

$S(t) = 100 \frac{\langle \tilde{s}(t) / s(\tau_t) \rangle}{\langle \tilde{s}(0) / s(\tau_0) \rangle}, \quad t=-20,\ldots,+20$

with $S(0) = 100\%$ and deviations quantifying relative spread expansion or contraction.

Normalized Best-Quote Depth:

$D_{\rm bid}(t) = 100 \frac{\langle \tilde{d}_{\rm bid}(t) / d(\tau_t) \rangle}{\langle \tilde{d}_{\rm bid}(0) / d(\tau_0) \rangle}$

and likewise for $D_{\rm ask}(t)$ .

Normalized Limit Order Intensity:

$\Lambda([t]) = \bigg\langle \frac{2 \tilde{\lambda}([t])}{\lambda(\tau^-_{[t]}) + \lambda(\tau^+_{[t]})} \bigg\rangle$

with $[t]$ indexing 1-minute bins surrounding the event, omitting $t=0$ .

These indices are averaged cross-sectionally over events of identical type and are scaled to unity or 100 at $t=0$ . The event index $t=0$ denotes the update just before the market order execution (Xu et al., 2016).

2. Recovery Behavior and Quantitative Patterns

Across aggressive and passive market order events, empirical studies show a robust pattern in resiliency metrics:

Time to baseline: Both spread ( $S(t)$ ) and depth ( $D_{\rm bid}(t), D_{\rm ask}(t)$ ) metrics return to within ±5% of their pre-shock average within 20 best-limit updates (5–10 seconds in typical LOBs) following an effective market order, regardless of order aggressiveness (Xu et al., 2016). The limit order intensity typically exhibits a slower decay, relaxing to baseline over approximately 30 minutes.
Shape of recovery: Aggressive orders induce large, near-exponential spikes in $S(t)$ , with characteristic peak impact $\Delta S \sim 30$ – $50\%$ and relaxation time $\tau_s \approx 5$ –$10$ updates. Depth overshoots are observed on the opposite side, reflecting over-resiliency. Mild or “touch-only” orders often cause the spread to initially narrow before mean reversion.
Asymmetries: When the pre-shock spread is at a single tick, the stimulus to new limit orders immediately following the shock is asymmetric. Effective buy (sell) market orders generate a greater increase of same-side limit order arrivals in the spread and at the touch than opposite-side submissions, with this effect persisting for 3–5 minutes (Xu et al., 2016).

These empirical observations provide calibration targets and structural constraints for theoretical models of order book dynamics and execution cost.

3. Theoretical Models: Microscopic, Macroscopic, and Diffusion Limits

3.1 Reduced-Form and Structural Impact Models

The Obizhaeva–Wang (OW) structural model specifies a block-shaped LOB with depth $h_t$ and a resilience (refill) rate $\rho$ governing exponential decay of execution-induced dislocations:

$\dot{y}(t) = -\kappa y(t) + \eta \dot{x}(t)$

where $y(t)$ is the deviation from the unaffected midprice, $\kappa$ is identified empirically as $1/(\text{5--10 updates})$ (Xu et al., 2016), and $\dot{x}(t)$ is the trading rate.

The Almgren–Chriss (AC) model, recovered in the high-resilience limit ( $\rho \to \infty$ ), features quadratic-in-speed cost $\lambda (\dot{\varphi}_t)^2$ and permanent linear impact, with the OW–AC mapping:

$\lambda^\uparrow = \frac{1 - \alpha^\uparrow}{\rho K^\uparrow h^\uparrow}$

where $K^\uparrow$ is the standardized block width, and $\alpha^\uparrow$ is the permanent impact fraction (Kallsen et al., 2014).

3.2 Time-Varying and Stochastic Liquidity

Models incorporating time-varying depth $q_t$ and resilience $\rho_t$ , either deterministic or stochastic, reveal nontrivial execution scheduling effects. The cost functional in block-shaped LOBs with time-dependent parameters is:

$J(\varphi) = \int_{[0,T]} \left( D_t + \frac{K_t}{2} d_t \right) d_t, \quad D_t = D_0 e^{-\int_0^t \rho(u) du} + \int_{[0,t)} K_s e^{-\int_s^t \rho(u) du} d_s$

with $K_t = 1/q_t$ . High resilience enables more front-loaded execution; low resilience flattens optimal schedules (Fruth et al., 2011, Ackermann et al., 2020, Ackermann et al., 2020).

3.3 Diffusion and PDE Formulations

In the heavy-traffic limit, the LOB queue sizes $Q^b(t), Q^a(t)$ evolve as coupled reflected diffusions with drift $\mu_i$ and volatility $\sigma_{ii}$ , yielding closed-form mean refill times:

$E[\tau^i_h] = \frac{h-x}{\mu_i}, \quad \operatorname{Var}[\tau^i_h] = \frac{\sigma_{ii}^2 (h-x)}{\mu_i^3}$

This allows for real-time monitoring of resiliency via rolling window drift and volatility estimates (Cont et al., 2012).

At the macroscopic scale, nonlinear PDEs of thin-film type

$u_t = (u^2)_{xx}$

(where $u(x,t)$ is local volume density) predict universal $t^{1/3}$ scaling for the recovery front following aggressive liquidity-taking, both in simulations and empirically. This provides a robust phenomenological law for recovery distance and peak depth rescaling (Rosenzweig, 2019).

4. Temporal and Scale-Dependent Aspects: Micro, Meso, and Macro

LOB resiliency operates across multiple time and volume scales:

Microscopic: Event-level dynamics (e.g., 1–20 updates after a shock) characterize the true time scale of liquidity replenishment. Metrics include immediate returns of $S(t)$ , $D_{\rm bid/ask}(t)$ , and $\Lambda([t])$ (Xu et al., 2016).
Meso-scale: Volume-bucketed analysis (e.g., 0.25–2% of ADV per bucket) highlights the importance of limit order flows and cancellation–addition rates as predictors of recovery and price moves. Net liquidity, constructed as a weighted sum of trade and limit flows, achieves a nearly linear relation with price impact and tracks resiliency more accurately than trade imbalance alone (Bechler et al., 2017).
Macroscopic: Burst-driven “excitation–relaxation” patterns dominate, with fast spikes in execution flow $I = dv/dt$ and best-level depth, followed by slow multi-scale decays. Power-law tails in relaxation times and spike intervals confirm the absence of stationarity, identifying resiliency with the distributional properties of recovery times rather than a single exponential constant (Malyshkin et al., 2016).

5. Role in Optimal Execution and Market Quality

Resiliency directly informs optimal trade scheduling. Execution algorithms leverage empirical recovery speeds and asymmetries. For example, after a single-tick spread shock, it is optimal to exploit elevated same-side limit order intensity for 3–5 minutes with passive orders, thus reducing adverse selection (Xu et al., 2016). In stochastic and time-varying liquidity models, feedback-adaptive trading adjusts to evolving $\rho_t$ and $q_t$ , sometimes delivering block trades when resilience jumps or even permitting “self-exciting” regimes where negative resilience accelerates or reverses executions (Ackermann et al., 2020, Ackermann et al., 2020, Ackermann et al., 2021).

In the high-resilience regime, the complexity of structural models reduces to the tractable AC framework, with quadratic trading penalties and parameters calibrated from observable $h$ and $\rho$ (Kallsen et al., 2014). In moderate or low-resilience conditions, richer singular-control and barrier strategies emerge, explicitly adapting to transient market liquidity (Fruth et al., 2011).

6. Asymmetries, Nonstationarity, and Regime Effects

Empirical studies highlight several nuanced implications:

Asymmetries: Single-tick spreads generate asymmetric limit order replenishment, breaking the symmetry observed for wider spreads (Xu et al., 2016, Bechler et al., 2017).
Regime sensitivity: Recovery laws (e.g., $t^{1/3}$ , mean queue-refill times) break down in thin or volatile periods, providing real-time risk signals (Rosenzweig, 2019, Cont et al., 2012).
Nonstationarity: No persistent regime dominates; resilience varies dynamically in response to market states, order flow, and diurnal patterns. Parameters must be recalibrated in rolling windows or at event times to preserve model accuracy (Malyshkin et al., 2016).
Negative Resilience/Self-Excitation: In some stochastic models, negative resilience ( $\rho_t < 0$ ) leads to self-exciting dynamics, with positive feedback in price impact and path-dependent, potentially nonmonotonic execution patterns (Ackermann et al., 2021).

7. Practical Applications and Model Calibration

Practitioners use resiliency metrics for:

Real-time risk monitoring: Monitoring the mean and volatility of queue recovery times or the distribution of relaxation times to detect and adapt to transient liquidity droughts (Cont et al., 2012, Malyshkin et al., 2016).
Execution algorithm calibration: Setting the speed and strategy of algorithms in response to recovery time distributions, observed intensity spikes, and predicted relaxation rates (Xu et al., 2016, Fruth et al., 2011).
Liquidity stress testing: Backtesting resiliency metrics against historical shocks to determine thresholds for risk controls and liquidity provisioning (Rosenzweig, 2019).

Calibrating structural models to empirical resilience enables theoretically optimal yet practically robust trading strategies, reduces execution costs in electronic markets, and contributes to overall market quality.

The concept of order book resiliency integrates empirical regularities, microstructural theory, stochastic control, and macroscopic scaling laws to provide a rigorous framework for understanding transient liquidity, impact decay, and optimal adaptation in order-driven financial markets.