Market Depth and Resilience Processes

• Market Depth and Resilience Processes are key liquidity primitives that describe available trading volume and the market's ability to recover from shocks.
• The dynamic interplay between market depth and resilience is modeled in both discrete and continuous frameworks to capture transient price impact and optimal execution costs.
• Empirical studies demonstrate that resilience levels directly influence recovery curves, thereby informing adaptive execution strategies and market quality assessments.

Market depth and resilience are two fundamental liquidity primitives that determine not only the short-term evolution of limit-order books (LOBs) following liquidity shocks but also the structure of transient price impact and optimal trade execution across market models. The joint processes of market depth and resilience encapsulate the dynamic capacity of a market to absorb trading flows and recover—or, in the case of negative resilience, amplify—shocks to its liquidity landscape.

1. Definitional Framework: Market Depth and Resilience Processes

Market depth quantifies the volume available for immediate execution at prevailing prices and governs the instantaneous price impact of market orders. In discrete-time or event-based frameworks, depth is typically represented as a strictly positive, stochastic, and filtration-adapted process, denoted by $(\delta_t)$ or $(\gamma_t)$. The reciprocal $1/\delta_t$ is proportional to the incremental widening of spreads per unit volume traded (Nagy et al., 15 Nov 2025, Ackermann et al., 2021).

Resilience captures the rate and direction of healing (or amplification) of price impact generated by trades. It is formalized as a process $(r_t)$ or $(\rho_t)$ that modulates the exponential decay (if $r_t > 0$) or amplification (if $r_t < 0$) of past liquidity shocks. In event-based limit-order books, resilience is estimated by observing the recovery curve of depth or spread post-shock, often fitted by exponential models with rate $\lambda$. The canonical update law for the half-spread $\zeta^X_t$ after trading is:

$$\zeta^X_{t+1} = e^{-r_t}\,\zeta^X_t + \frac{|\Delta X_{t+1}|}{\delta_{t+1}}$$

(Nagy et al., 15 Nov 2025, Xu et al., 2016). In continuous-time models, the transient deviation $D_t$ evolves as

$$dD_t = -\rho_t D_t\,dt + \gamma_t\,dX_t + d[\gamma, X]_t$$

(Ackermann et al., 2021).

2. Empirical Characterization in Limit-Order Books

Empirical studies of LOB resiliency implement a multi-dimensional approach to market depth and resilience. Xu et al. (Xu et al., 2016) measure three primary liquidity and resiliency metrics:

• Bid-ask spread ($s(t) = a_1(t) - b_1(t)$): tightness of quotes.
• Best-quote depths ($B_1(t), A_1(t)$): top-of-book volumes, distinguished by same-side ($d_\mathrm{same}$) and opposite-side ($d_\mathrm{opp}$) after shock.
• Limit-order intensity ($\lambda(t)$): arrival rate of new limit orders of specified types.

LOB resiliency post-liquidity shock is quantified via normalized recovery curves (fractional recovery $R_X(\tau)$), with empirical recovery commonly complete within 20 top-of-book updates for spread and depth—even across diverse types of effective market orders. Resilience patterns in LOB depth and spread provide direct evidence for transient impact kernels in theoretical models (Xu et al., 2016).

3. Mathematical Modeling and Dynamics

a. Discrete-Time Transient Impact Models

In discrete time, the spread and impact dynamics are determined by the interactions between market depth $(\delta_t)$ and resilience $(r_t)$. The transient price-impact kernel is given by:

$$\rho_{j,t} := \exp\left(-\sum_{i=j}^{t-1} r_i\right)$$

with cumulative impact expressed as

$$\text{Cost}_\text{impact} = \sum_{t=1}^T |H_t| \left(\rho_{0,t}\zeta_0 + \sum_{j=1}^t \frac{\rho_{j,t}|H_j|}{\delta_j}\right)$$

(Nagy et al., 15 Nov 2025). A salient property is that under only the mild regularity conditions $\delta_t \geq \underline{\delta} > 0$ and $r_t \geq 0$, the convexity of attainable terminal payoffs may fail—economic reality no longer requires monotonicity or convexity in these processes (Nagy et al., 15 Nov 2025).

b. Stochastic Order Book Models and Negative Resilience

Recent models allow for stochastic depth ($\gamma_t$) and resilience ($\rho_t$), with resilience possibly negative. When $\rho_t < 0$, past liquidity shocks are amplified rather than healed—modeling self-exciting dynamics. The transient impact kernel is

$$K(s, t) = \gamma_s\,\exp\left(-\int_s^t \rho_u\,du\right)$$

Negative resilience leads to sharply non-monotonic and possibly sign-changing optimal trading strategies, including “overjumping zero” and “premature closure” phenomena (Ackermann et al., 2021).

c. High-Resilience Asymptotics

In block-shaped order book models, sending resilience ($\kappa$) to infinity yields instantaneous impact recovery, reducing wealth dynamics to a quadratic-cost Almgren–Chriss formulation:

$dW_t = X_t\,dS_t - \lambda\,\dot{X}_t^2\,dt$

with $\lambda \sim \frac{1}{\kappa h}$ (Kallsen et al., 2014). The optimal control structure simplifies, allowing explicit leading-order ODE solutions for trading trajectories.

4. Empirical Recovery and Predictive Modelling

The empirical literature confirms that depth and resilience processes directly govern the short-term restoration of market quality post-shock. Both microscopic and meso-scale studies utilize exponential recovery models for depth:

$D(t) = D_\infty - (D_\infty - D_0) e^{-\lambda t}$

and define normalized resilience curves:

$R(t) = 1 - \exp(-\lambda t)$

where $\lambda$ is estimated via least-squares on depth time-series (Bechler et al., 2017). Meso-scale econometric frameworks further identify the predictive power of limit-order flows and depth metrics for subsequent price changes and liquidity scarcity; top predictors include net limit flows, deeper-book impact metrics, and cancellation rates (Bechler et al., 2017).

5. Qualitative Effects of Depth–Resilience Structure

The interaction of depth and resilience dictates the market's response to order flow:

• Low resilience (scarce liquidity): Depth recovers slowly, price impact persists, and prediction accuracy of ensuing large price movements rises.
• High resilience: Rapid healing, low permanent impact, and execution costs decrease.
• Negative resilience: Self-exciting dynamics, optimal strategies may reverse direction or close prematurely to exploit momentum induced by one's own trades (Ackermann et al., 2021).

Empirical results document that execution algorithms must adaptively calibrate execution rates in response to real-time recovery metrics; trading into the tightest possible spread can provoke short-term price continuation via herding in same-side limit order submissions (Xu et al., 2016).

6. Utility Maximization and Theoretical Integration

In discrete-time transient impact models, market depth and resilience directly shape the cost functional in utility maximization:

$$\sup_{X \in \mathcal{A}_0} \mathbb{E}[u(z + \xi^X_T - B)]$$

Existence results are established under very mild process assumptions: depth strictly positive, resilience nonnegative (not necessarily monotonic), and utility bounded above (Nagy et al., 15 Nov 2025). Non-convexity of attainable payoffs does not preclude optimality, but requires measurable-selection and dynamic-programming arguments.

Portfolio optimization in highly-resilient models reduces to Almgren–Chriss-style problems with quadratic costs, due to rapid spread recovery; classic risk-tolerance based ODEs become asymptotically optimal (Kallsen et al., 2014).

7. Connections and Implications for Market Quality

The structure and empirical characteristics of market depth and resilience processes underpin both transient and permanent impact, execution efficiency, and the statistical properties of post-shock recovery. Their interplay modulates:

• Price reversals versus continuation depending on order aggressiveness and pre-shock tightness (Xu et al., 2016).
• Predictability of scarce liquidity events via limit-order flow and depth profiles (Bechler et al., 2017).
• Validity and calibration of optimal execution models, with direct mapping from observed healing times/resilience rates to control-theoretic kernels (Kallsen et al., 2014, Nagy et al., 15 Nov 2025).

A plausible implication is that real-time estimation and stochastic modeling of these liquidity primitives is essential for designing execution algorithms, market-making strategies, and empirical studies of market quality across trading venues and regimes.