Optimal Execution with Passive Market Impact

Abstract: We derive a mesoscopic model for optimal execution with limit orders that incorporates microstructural features of passive price impact. Our framework is based on two empirical observables: the approximately exponential decay of limit-order fill probabilities with distance from the midprice, and the short-term linear response of price changes to order flow imbalance. Combining these ingredients, we obtain a reduced-form passive impact rate that decays exponentially with quote distance. The model describes passive execution at a tactical level, where fills arise from a sequence of quote adjustments that balance execution probability, adverse selection, and opportunity cost. We formulate and solve an optimal liquidation problem in which the trader controls the aggressiveness of passive sell quotes. This generates a trade-off between higher fill intensity and larger accumulated impact on the one hand, and lower impact but greater non-execution risk on the other. Empirical calibration using NASDAQ equities and public FX supports the empirical foundations of the model. We also analyse extensions with heterogeneous decay rates, transient impact, and target execution schedules.

• The paper introduces a mesoscopic model where passive (limit) order execution generates systematic price impact that decays exponentially with quote distance.
• The model derives an explicit optimal quoting strategy that balances fill probability against passive impact and inventory risk.
• Empirical calibrations on NASDAQ equities and FX data demonstrate parameter sensitivities and validate the simulation-based execution framework.

Optimal Execution with Passive Market Impact: A Technical Overview

Motivation and Context

Optimal execution theory has historically concentrated on modeling the adverse price impact of large (metaorder) executions conducted via market orders, focusing on the interplay between execution costs and inventory risk as in the classic Almgren-Chriss and Bertsimas-Lo models. These market impact models often treat passive (limit order) execution as essentially costless in terms of permanent price impact, viewing it purely as a means to capture spread at the cost of fill risk. However, recent empirical findings and microscopic queue models demonstrate that passive execution strategies—through their impact on order flow imbalance (OFI)—can themselves generate significant systematic price impact, even outside the context of explicit aggressive flow.

This paper provides a rigorous, mesoscopic model of optimal execution with passive market impact, bridging microstructural empirical phenomena with tractable optimal control. The key empirical pillars for the model are:

• The exponential decay of fill probabilities with distance from the midprice, observed for limit orders.
• The linear response of short-horizon price changes to order-flow imbalance, generalizable to multi-level order-flow imbalance (MLOFI).

Both features are confirmed by calibration to NASDAQ equity and FX market data, providing broad cross-market applicability.

Passive Impact Modeling Framework

A stylized model is constructed in which a trader executes a metaorder using a tactical chain of passive (limit) orders, each set at a distance $\delta$ from the mid. Execution arises from a repeated process of quote adjustment and reposting, summarized by an exponentially decaying fill intensity:

$\Lambda(\delta) = \lambda e^{-k \delta}$

Empirical analysis reveals that price impact coefficients $\beta(\delta)$ derived from OFI (and especially MLOFI) also display exponential decay with $\delta$:

$\beta(\delta) = \xi e^{-\ell \delta}$

The product of these two rates describes a passive impact rate that aggregates the average price impact per fill as a reduced-form, continuous-time process:

$\eta e^{-m \delta}, \quad \mbox{where } \eta = \xi \lambda, \; m = k + \ell$

This summarizes the inventory, fill intensity, and average passive impact into a tractable state variable for optimal control.

Main Model: Formulation and Solution

The agent seeks to liquidate a block of $q_0$ inventory over horizon $[0,T]$ by controlling the quote distance $\delta_t$ of posted limit sell orders. Fills arrive as jumps of a counted Poisson process with intensity $\lambda e^{-k \delta_t}$, and posting at distance $\Lambda(\delta) = \lambda e^{-k \delta}$0 induces a permanent price drift at rate $\Lambda(\delta) = \lambda e^{-k \delta}$1. The unaffected midprice follows Brownian motion, with agent's total cash and remaining inventory evolving stochastically. The objective is to maximize terminal cash plus final inventory marked to the (impacted) midprice, penalized for both running and terminal inventory risk:

$\Lambda(\delta) = \lambda e^{-k \delta}$2

For the equity-calibrated case where $\Lambda(\delta) = \lambda e^{-k \delta}$3, the model admits a fully explicit solution to the associated HJB equation. The optimal quote is:

$\Lambda(\delta) = \lambda e^{-k \delta}$4

where $\Lambda(\delta) = \lambda e^{-k \delta}$5 follows a linear ODE system with known initial and boundary conditions.

Figure 1: Optimal quote depth $\Lambda(\delta) = \lambda e^{-k \delta}$6 over time for different inventory levels $\Lambda(\delta) = \lambda e^{-k \delta}$7.

The model captures a fundamental tradeoff: posting closer to the mid increases execution probability and reduces inventory risk, but raises average passive impact incurred; conversely, posting deeper reduces passive impact but heightens fill risk.

Numerical Analysis and Sensitivity

Empirical illustrations confirm the main economic features of the model:

• Urgency and Quote Aggressiveness: Higher initial inventory prompts quoting closer to the mid; toward the end of the period, inventory is marked to the impacted price, sometimes making it optimal to avoid incurring further impact and instead absorb inventory penalties.
• Parameter Sensitivities: Increasing the passive impact parameter $\Lambda(\delta) = \lambda e^{-k \delta}$8 yields less aggressive quoting and lower average liquidation, but does not increase (and may even reduce) implementation shortfall—demonstrating the opportunity cost from cautious execution.

  Figure 2: Effect of the passive impact parameter $\Lambda(\delta) = \lambda e^{-k \delta}$9 on the optimal quote depth $\beta(\delta)$0.

  

  Figure 3: Simulated unaffected midprice (blue), impacted midprice (red), quoted price (purple), and fill times (green vertical lines) under the baseline strategy.

  

  Figure 4: Monte Carlo diagnostics for final inventory, P&L, implementation shortfall, and trading time for baseline parameters.

Strong numerical evidence is provided in simulation, with performance and risk metrics tracked over thousands of Monte Carlo paths. Results confirm that aggressive quoting under high passive impact leads to partial rather than full liquidation and impact-aware quoting policies.

Empirical Grounding: Cross-Asset Calibration

Calibrations for both US equities and G10/emerging FX pairs demonstrate high generality:

• Equities: The exponential decay rates for fill intensity $\beta(\delta)$1 are much steeper than those for impact $\beta(\delta)$2, with $\beta(\delta)$3. Empirical fits (Figure 5, Figure 6) support the primary modeling hypothesis.
• FX: The gap between decay rates is smaller, requiring the more general case $\beta(\delta)$4. Still, exponential decay in both fill probability and impact is evident (Figure 7, Figure 8).

  Figure 5: Fill intensity estimation for TSLA, showing empirical fill probabilities and exponential fits.

  

  Figure 6: Passive impact estimation for TSLA, showing MLOFI impact coefficients and exponential fit.

  

  Figure 7: Fill intensity estimation for GBPUSD, with 10-second bucketed fill intensity and exponential fit.

  

  Figure 8: Passive impact estimation for GBPUSD, showing MLOFI price impact coefficients and exponential fit.

These calibrations support both the parametric structure and the adequacy of the mesoscopic execution model across market microstructures: equities (price-time priority LOBs) and FX (fragmented, bilateral, or quote-driven venues).

Model Extensions and Generalizations

Heterogeneous Decay Rates

In FX and other contexts where $\beta(\delta)$5, the optimal quote can still be written semi-explicitly, but now involves the principal branch of the Lambert W function due to the misalignment in fill and impact decay rates. This broader class of models accommodates empirically observed differences in market structure and information environments.

Figure 9: Effect of the passive impact decay parameter $\beta(\delta)$6 on the optimal quote depth $\beta(\delta)$7.

Transient Impact

Moving beyond permanent impact, the model admits a specification in which passive impact decays at resilience rate $\beta(\delta)$8. The associated control problem increases in state dimension, and the optimal quoting rule adapts dynamically to the current level of transient (recovering) impact.

Target Execution Schedules

Risk penalties can penalize deviation from deterministic target inventory paths (e.g., TWAP trajectories), yielding optimal strategies that interpolate between classical market order–driven execution and stochastic fill-driven paths.

Figure 10: Optimal quote depth $\beta(\delta)$9 under various model extensions and target schedules.

Figure 11: The median path of the inventory $\delta$0 across the baseline, transient impact, and TWAP-target extensions, shading quantiles.

Theoretical Insights and Implications

A distinctive theoretical result is that—contrary to the Almgren-Chriss/MO setting—permanent passive impact does affect trading rates due to nonlinearity in fill probability as a function of quote distance. The optimal path exhibits characteristic convexity, mediated by the joint effects of inventory risk and passive impact. Extensions to transient impact confirm that the optimal profile approaches TWAP as resilience increases.

The explicit, tractable solution in the $\delta$1 case provides analytical benchmarks for both theoretical work and empirical calibration. The modeling architecture supports immediate inclusion of more complex features (state-dependent parameters, adverse selection, cross-asset portfolios). Importantly, the empirical methodology is constructed to be implementable on publicly available L2/L3 market data, enhancing reproducibility.

Limitations and Future Directions

While the reduced-form, tactical model summarizes key economic forces, it bypasses explicit treatment of adverse selection, dynamic order size, multi-venue posting, and asymmetric information. Additionally, fill and impact parameters are held static in the baseline model, whereas in real-world execution they may depend on local microstructure features and market conditions.

The model forms a foundation for both further theoretical extension (such as cross-impact, state-dependent kinetics, and learning-based execution strategies) and practical applications in algorithmic liquidity provision and risk management. Precise calibration to trader-tagged order data and integration with modern reinforcement learning approaches remain open areas.

Conclusion

This paper rigorously formulates and solves an optimal execution problem recognizing that passive trading strategies generate their own systematic price impact at the mesoscopic level. The model unites empirical microstructure regularities with tractable optimal control, showing that optimal limit order execution must balance inventory risk, execution probability, and endogenous market impact. The framework supports a diverse spectrum of market environments and sits naturally between microscopic order-book modeling and coarse empirical calibration, offering both robust theoretical insights and practical implementational guidance for sophisticated algorithmic execution strategies.