Latent Liquidity Models in Financial Markets

• Latent liquidity models are frameworks that capture both visible orders and hidden liquidity intentions, explaining market impact and stability.
• They employ methods like agent-based simulation and reaction-diffusion PDEs to model order revelation, cancellations, and diffusion processes.
• Empirical calibrations using exponential liquidity measures and imbalance indicators validate the square-root price impact law and identify critical market thresholds.

Latent liquidity models constitute a framework for understanding the dynamics of supply and demand in electronic financial markets by modeling not only the observable limit order book (LOB) but also the unobserved, or "latent," intentions of agents to provide liquidity at various prices. These models have become central to the quantitative analysis of price impact, market stability, and crisis phenomena, providing both empirical metrics for monitoring real markets and tractable theoretical structures capable of reproducing observed market regularities such as the square-root law of price impact. The explicit modeling of latent liquidity distinguishes this class from both traditional reduced-form models and purely agent-based descriptions by embedding the microstructural processes of liquidity revelation, cancellation, and order-flow interaction within analytic or PDE-based frameworks.

1. Definition and Core Principles

The latent order book is defined as the reservoir of limit order intentions that market participants would place if sufficiently incentivized by market order flow. At any time, only a small fraction of this reservoir is revealed as active (visible) limit orders in the LOB; the remainder is latent, only becoming visible when liquidity providers deem participation competitive or necessary in response to market demand (Corradi et al., 2015, Dall'Amico et al., 2018). The key assumption is that, under usual conditions, revealed orders are replenished dynamically from the latent reservoir, maintaining a near-equilibrium between aggressive order flow (market orders) and liquidity-providing flow (limit orders). The failure of this revelation mechanism, either due to endogenous risk limits or exogenous stress, is directly linked to market fragility and crisis events.

Multiple mathematical frameworks instantiate these principles. In the agent-based approach (notably the ε-intelligence model), liquidity at each price evolves via stochastic depositions and cancellations, with the latent book density vanishing linearly near the mid-price, which in turn generates the concave—specifically, square-root—impact relationship for meta-orders (Mastromatteo et al., 2013). Alternatively, reaction-diffusion PDEs represent the dynamics of both latent and revealed liquidity fields, capturing both the steady-state shape and nonequilibrium evolution of the book (Dall'Amico et al., 2018, Benzaquen et al., 2017, Salek et al., 12 Jan 2024).

2. Mathematical Formulation: Latent–Revealed Order Book Dynamics

The canonical mathematical formulation consists of coupled equations for latent and revealed order densities on each side (buy/sell), typically denoted as $\rho_{B,A}^{(\ell)}(x, t)$ (latent) and $\rho_{B,A}^{(r)}(x, t)$ (revealed). The latent densities evolve by diffusion in price (to model agents’ reassessment), cancellation, and depositions, while the conversion (“revelation”) from the latent to the revealed book is typically modulated by a probabilistic kernel, often exponentially decaying with the distance from the prevailing mid-price. The governing equations, in the reference frame defined by the mid-price $p_t$, take the form: $$\begin{aligned} \partial_t\,\rho_B^{(\ell)}(x,t) &= D_\ell \partial_{xx}\rho_B^{(\ell)} - \omega\left\{\Gamma(-k\xi)\,\rho_B^{(\ell)} - [1-\Gamma(-k\xi)]\, \mathbf{1}_{\{x < p_t\}} \phi_r \right\}, \ \partial_t\,\rho_A^{(\ell)}(x,t) &= D_\ell \partial_{xx}\rho_A^{(\ell)} - \omega\left\{\Gamma(+k\xi)\,\rho_A^{(\ell)} + [1-\Gamma(+k\xi)]\, \mathbf{1}_{\{x > p_t\}} \phi_r \right\}, \ \partial_t\,\phi_r(x,t) &= D_r \partial_{xx} \phi_r + \omega \left\{\Gamma(-k\xi)\,\rho_B^{(\ell)} - \Gamma(+k\xi)\,\rho_A^{(\ell)} - [1-\Gamma(k|\xi|)] \phi_r \right\}, \end{aligned}$$ where $\xi = x - p_t$, $\phi_r(x,t)$ is the difference in revealed densities, $D_\ell$ and $D_r$ are diffusion coefficients, $\omega$ is the revelation rate, and $\Gamma$ is the conversion probability kernel (Dall'Amico et al., 2018, Salek et al., 12 Jan 2024).

Steady-state solutions reveal that latent order densities exhibit a locally linear profile near the mid, while the revealed book possesses an exponentially decaying envelope away from the best price. The interplay between the diffusion/renewal timescale and conversion rate gives rise to a critical threshold: when the latent-to-revealed conversion parameter is below a critical value, the revealed order book collapses at the mid, signaling a market instability threshold (Dall'Amico et al., 2018).

3. Empirical Signatures and Calibration

Key empirical observables for latent liquidity models include:

• Exponential liquidity measure: Quantifies one-sided book depth using an exponentially weighted sum across ticks,

$$L^{(\text{side})}(t;\delta) = \frac{1}{\langle V_N \rangle} \sum_{\Delta=1}^N V_t^{(\text{side})}(\Delta) \exp(-\Delta/\delta),$$

where $V_t^{(\text{side})}(\Delta)$ is the volume at distance $\Delta$ ticks from the best quote (Corradi et al., 2015).

• Liquidity imbalance: Measures the normalized difference between bid and ask side exponential liquidity,

$$L_{\text{imb}}(t; \delta) = \frac{L_B(t; \delta) - L_A(t; \delta)}{L_B(t; \delta) + L_A(t; \delta)},$$

providing predictive power for short-term price moves.

• Calibration methodology: Empirical revealed book profiles $\phi_r^{\text{emp}}(\xi)$ are fit using steady-state analytical forms parameterized by latent liquidity $\mathcal{L}$, inverse book depth $k$, and conversion-related length scales $\ell_\ell, \ell_r$. A fit is considered valid only if the critical stability condition $k\ell_\ell < \zeta_c$ is satisfied (Dall'Amico et al., 2018). Example values for large-cap US equities (bid-ask symmetric) are $k\ell_\ell\sim0.1$–$0.5\ll\zeta_c$; for Euro-Stoxx futures $\mathcal{L}\approx 4600$ shares\%$^{-1}$, $k\approx 2.1$ %$^{-1}$ (Dall'Amico et al., 2018).
• Auction dynamics: In equity auctions, both submission and cancellation rates exhibit an accelerated, time-to-deadline dependence—specifically, a $1/(T-t)$ "deadline kernel"—leading to rapid buildup of depth at the indicative price and sub-diffusive dynamics for the indicative price itself ($H\approx0.35$–$0.45$), with price predictability remaining absent (Salek et al., 12 Jan 2024).

4. Impact Laws and Market Stability

All latent liquidity models reproduce, under suitable conditions, the empirically robust square-root law for price impact: $I(Q) \propto Q^{1/2}$ for meta-orders of volume $Q$ executed on a timescale shorter than the latent-book renewal time (Mastromatteo et al., 2013, Benzaquen et al., 2017, Dall'Amico et al., 2018). Analytically, this emerges from the linear vanishing of latent liquidity near the mid and integrating volume traversed by price impact. When the meta-order execution spills into the timescale of liquidity renewal, impact becomes a crossover between linear (in depleted books) and concave (in replenishing books) behavior.

The existence of a critical (market instability) threshold is a central feature. The conversion rate from latent to revealed orders must exceed a critical value; otherwise, the revealed book vanishes at the mid-price and the market becomes fragile, with a propensity to large price jumps (Dall'Amico et al., 2018). This threshold is characterized by a critical overlap parameter $\zeta_c$ determined by model parameters, with explicit values depending on the balance of diffusion and conversion terms in the dynamics.

The permanent impact of an executed meta-order is universally linear in traded volume and independent of execution rate, reflecting no-arbitrage constraints (Benzaquen et al., 2017). Multi-timescale latent liquidity models, where different classes of agents possess different cancellation/renewal rates, resolve the price-diffusivity puzzle: when order flow exhibits long-memory autocorrelations, suitably distributed liquidity memory can maintain exact price diffusion; specifically, with memory exponent $\gamma$ in order flow, diffusion requires an agent population with cancellation age parameter $\alpha = \gamma/2 < 1/2$ (Benzaquen et al., 2017).

5. Extensions: Agent-Based and Parametric Reduced-Form Models

Agent-based models (e.g., ε-intelligence) assert that the latent liquidity density is the emergent result of stochastic depositions (rate $\lambda$), cancellations (rate $\nu$), and market order arrivals (rate $\mu$). The stationary solution for the latent density $\psi_\infty(x)$ is V-shaped: $$\psi_\infty(x) = \frac{\lambda}{\nu}\left(1 - e^{-|x|/p^*}\right), \quad p^*=\sqrt{\frac{D}{2\nu}},$$ vanishing linearly at the mid. This structural property, i.e., zero density of latent liquidity at the prevailing price, robustly yields the square-root impact law. Variants incorporating reactive limit-order refill or alternate execution protocols (market vs. limit order) confirm the universality of the result, provided only that the linear vanishing condition and price diffusion are preserved (Mastromatteo et al., 2013).

Reduced-form, parametric models such as (Malo et al., 2010) model the LOB using a small number of underlying risk factors—typically the logarithms of liquidity-depth parameters on each side and the mid-price—evolving as a multivariate Ornstein-Uhlenbeck process. The factor dependence introduces mean-reversion in liquidity and enables explicit computation of the order cost function and response to liquidity shocks. "Crowding out," in this context, is mathematically encoded via cross-dependencies between side-specific liquidity and the mid-price drift, and the model admits fast calibration on real data (Malo et al., 2010).

6. Crisis Dynamics, Predictability, and Broader Implications

The models explicitly quantify that liquidity crises, characterized by large returns or price gaps, are associated with two phenomena: (1) a breakdown in the normal linear relationship between aggressive market order flow and compensating limit order revelation (at long timescales, e.g. 15-minute windows), and (2) a static depletion of visible book depth near the best quote on a short timescale (30-second windows) (Corradi et al., 2015). Critically, the exponential liquidity measure and liquidity imbalance serve as early warning metrics for such events.

Despite the sub-diffusive scaling for indicative prices in deadline-driven auctions, indicative prices remain unpredictable; central limit theorem scaling (Joseph effect), lack of autocorrelation, and efficient market property hold except at the very end of auctions when the price locks (Salek et al., 12 Jan 2024). Continuous-time markets can be modeled with the same latent–revealed PDEs but with transaction terms included, and periodic batch auctions exhibit the same accelerated liquidity dynamics and affiliated sub-diffusive signatures.

Barricading risk in modern markets thus becomes a function of monitoring latent liquidity conversion rates and critical stability parameters in real time—a fact which the theoretical and empirical structure of these modeling frameworks appears to support. Extensions to heterogeneous agent classes, nonlocal price adjustments, and agent-based coupling are ongoing research directions intended to bring these frameworks into further alignment with high-frequency data and additional market microstructure stylized facts.