A passive execution model that charges only for queue time will quote too close and finish too quickly. Barzykin, Boyce, Neuman and Tuschmann add the missing cost in a form that can be solved. Their correction pushes quotes deeper and leaves more inventory behind. In the authors' own simulations, mean unsold inventory climbs from 1,003 shares to 6,367 shares on a 20,000 share order as passive impact rises from zero to η = 0.01.
The mechanism combines two familiar empirical claims. Under the Avellaneda-Stoikov specification, fill intensity falls roughly exponentially with distance from the mid, giving a quote at distance δ the rate λe^(-kδ). Short-horizon midprice changes also respond linearly to order-flow imbalance across book levels, with coefficient β_d at each level. Cont, Kukanov and Stoikov established the result at the touch and outlined the deeper-level case in an appendix. Xu, Gould and Howison then formalised the multi-level version and estimated ten levels through ridge regression, calling the measure MLOFI, multi-level order-flow imbalance.
Barzykin, Boyce, Neuman and Tuschmann impose exponential decay on the coefficient itself: β(δ) = ξe^(-ℓδ). Multiplying the two relationships gives expected impact per unit time of ηe^(-mδ), where η = ξλ and m = k + ℓ.
Post closer, fill faster, and move the price against yourself faster.
The control problem covers liquidation over a finite horizon. A trader posts unit sell limit orders, while fills follow a counting process with intensity λe^(-kδ). As long as inventory remains, the midprice receives permanent drift of -ηe^(-mδ). The objective adds terminal cash to inventory marked at the impacted mid, then subtracts a running penalty ϕ∫Q² and terminal penalty αQ_T².
The authors describe ηe^(-mδ) as a mesoscopic average. It summarises submissions, cancellations, reposts and venue choices rather than the pathwise effect of one untouched child order. They call this the tactical level, and the distinction appears in the abstract.
Calibration uses two datasets. The equity sample is the LOBSTER event-by-event NASDAQ book for AMZN, TSLA, NFLX, ORCL, CSCO and MU. It covers all 252 trading days of 2016, restricted to 10:00 to 15:30. The FX sample uses LSEG L2 quote and trade data for USDMXN, GBPUSD, AUDUSD, USDTHB and USDSGD, spanning 100 business days from the start of 2026. Its hours are 06:00-18:00 UTC, aggregated into 10-second buckets.
TSLA fill decay runs 27x faster
For TSLA, k̂ = 0.5657 per tick and ℓ̂ = 0.0209. Fill probability therefore fades roughly twenty-seven times faster with distance than the MLOFI impact coefficient. Across the six stocks, k̂ ranges from 0.4807 (AMZN) to 3.7247 (CSCO), while ℓ̂ ranges from -0.1015 to +0.0209. The authors interpret the result as evidence that deep limit orders can reveal information and move prices without filling.
The same empirical relationship makes the control problem tractable. Because ℓ̂ is an order of magnitude smaller than k̂, m̂ ≈ k̂. Under that approximation, the HJB collapses. Starting from the ansatz u = x + qs + θ(t,q), the substitution ω = e^(kθ) produces a triangular linear ODE system and a matrix-exponential solution. The optimal quote is explicit: δ*(t,q) = 1/k + (1/k)log(ω(t,q)/ω(t,q-1)) + ηq/λ. Remaining inventory enters the quote directly, and the required matrix exponential is readily evaluated.
Trouble appears in the three large-tick names. Their ℓ̂ estimates are negative: ORCL -0.0650, CSCO -0.1015, MU -0.0637. Taken literally, passive impact increases as the quote moves deeper. The authors flag the issue and suggest replacing distance from the mid with queued volume ahead of the order. Their explanation is that a coarse price grid makes distance a weak proxy.
The diagnosis fits Table 4. Average spreads for AMZN, TSLA and NFLX are 36.7, 19.2 and 4.0 ticks. ORCL, CSCO and MU average 1.2, 1.1 and 1.1 ticks. For names trading inside spreads of 1.1 to 1.2 ticks, "distance" has little variation, leaving the model's exponential specification poorly fitted.
FX does not support the main approximation as strongly
The FX estimates put k̂ between 0.49 and 0.62 and ℓ̂ between 0.13 and 0.33. For GBPUSD, k̂ = 0.49, ℓ̂ = 0.27 and m̂ = 0.76. For USDTHB, the corresponding values are 0.62, 0.33 and 0.95. Across the five pairs, ℓ̂/k̂ falls between roughly a quarter and a half. Among the small-tick equities, it is near a fortieth.
The paper says the m ≈ k result supporting its main model "is therefore not replicated as strongly in the FX market, especially for some pairs". One possible reason, according to the authors, is that public venue-level information captures only part of total FX liquidity and its information set. L2 detail is also thinner than the L3 equity data, and most FX trades OTC.
They consequently solve the m ≠ k case. Theorem 6.1 expresses the quote semi-explicitly through the principal branch of the Lambert W function. Here θ satisfies a nonlinear triangular ODE system, while the second branch W_{-1} is identified as the local minimum. The theorem applies when m > k, or when k - ε < m < k for small ε.
Every pair in Table 5 has positive ℓ̂: 0.23, 0.27, 0.13, 0.33, 0.26. Thus m > k throughout the FX sample, placing those estimates in the unrestricted branch. The neighbourhood restriction matters instead for the large-tick equities, whose negative ℓ̂ estimates imply m < k.
FX fill intensities are synthetic. A hypothetical passive quote is treated as filled when next-bucket aggressor trades reach or cross it. The underlying L2 feed is conflated at 5ms or more and contains no dealer flow. Confidence intervals are absent from the FX table. The authors disclose these limitations and describe the comparison as a public-data, venue-level result based only on LSEG L2 quote and trade data. The remaining question is whether the distinct-decay extension reflects FX generally or public venues specifically. Dealer-tagged fills from one bank's franchise, or the same estimator applied to a second venue, would distinguish the two.
What are these coefficients measuring?
A desk should press hardest on identification, as the authors themselves do. Their empirical findings support the model's ingredients "rather than providing a causal estimate of the impact generated" by an individual trader's passive strategy. Testing that causal claim would require trader-tagged order-level data connecting submissions, cancellations, fills and later price responses.
The equity estimate ℓ̂ does not come from a passive strategy run by any identified trader. The authors use the published ridge MLOFI coefficients in Xu et al.'s Table 8 at d̄ = 10. They map each book level d to average distance δ_d using the average half-spread and average gaps between occupied levels, then fit an exponential through ten points.
Those coefficients capture the contemporaneous response of the mid to aggregate net imbalance at each level. Everyone's placements, cancellations and market orders contribute. The model, however, needs the response generated by the trader's own quoting policy. Equality between those quantities requires the trader's flow to be informationally representative of aggregate flow at the same level, precisely the relationship a good execution desk hopes to avoid.
Converting levels into distance adds another problem. Period-average spreads and gaps reduce a full distribution to a single value. That distribution is much wider for AMZN at 36.7 ticks than for CSCO at 1.1.
Simulated economics reveal a useful non-monotonicity
Every reported performance figure comes from a Monte Carlo simulation of the model's own dynamics, using 1000 paths at 1000 steps. Parameters come from Table 1: T = 300s, q0 = 20 units of 1000 shares, λ = 1.4 units/s, k = 48 per dollar and η = 0.0028. The paper never identifies which stock or pair supplies η = 0.0028 and λ = 1.4. The value k = 48 per dollar does match AMZN's estimated k̂ = 0.4807 per tick with a $0.01 tick. We did not find a comparison with a naive quoting rule beyond Figure 11's qualitative inventory-path comparison against TWAP and Almgren-Chriss.
As η rises from 0.0 to 0.01, mean final inventory moves from 1,003 to 6,367 shares. Mean P&L drops from $320 to -$11, while mean trading time increases from 284.9s to 298.3s. The residual position is six times larger.
Implementation shortfall is non-monotonic: -$35 at η = 0, -$19 at η = 0.005 and -$61 at η = 0.01. Under the paper's sign convention, positive shortfall is the adverse outcome, making -$61 the best result among the three. The authors describe η = 0.01 as producing lower P&L "but not worse implementation shortfall", since wider quotes offset the added impact.
The comparison between η = 0 and η = 0.01 makes the issue plain. The headline metric improves even as P&L falls from $320 to -$11 and unsold inventory sextuples. Metric selection drives the interpretation, and the model produces the conflict directly.
The transient extension behaves differently. Increasing resilience from ρ = 0 to ρ = 0.02 raises mean P&L from $717 to $1,047. Trading time shifts from 273.8s to 280.9s. The paper identifies ρ = 0 in that table as the permanent-impact case. Greater resilience improves the mark on remaining inventory, reducing urgency.
We could not test any of these results on our data. Calibration requires fill intensity by quote distance and per-level order-flow imbalance, both of which need L2 or L3 event data. We have OHLCV bars without equity bid/ask observations or book state. Minute bars cannot represent queue position or a cancel-and-repost sequence, leaving no basis for calibrating k.
One extra term changes the liquidation curve
Ouazzani Chahdi, Rosenbaum and Szymanski provide the microscopic, queue-based foundation for passive impact in the contemporaneous point-process paper cited and discussed by the authors. Barzykin, Boyce, Neuman and Tuschmann place passive impact inside the control problem. On 2016 NASDAQ data, the empirical ratio between the two decay rates falls where an explicit solution becomes available.
Relative to Avellaneda-Stoikov and the Guéant-Lehalle-Fernandez-Tapia line of liquidation with limit orders, the model adds one term. The optimal quote then gains an ηq/λ inventory component. Its directional consequence matters: permanent passive impact determines the convexity of the liquidation curve, whereas permanent impact in Almgren-Chriss drops out of the trading rate entirely.
Deployment would require a number unavailable from the paper's data: η estimated from a desk's own tagged fills, with quote distance, queue position and later price response measured over a horizon long enough to separate impact from adverse selection. The model uses unit sizes, one venue and constant λ, k, η and m. None depends on spread, depth, volatility or intraday seasonality, all areas the authors list for further work.
An η fitted too high on proprietary flow sends quotes deeper. At η = 0.01, the model's 1,000 simulated paths finish the 300 second horizon with mean unsold inventory of 6,367 shares from an initial 20,000. Setting η to zero returns the model to the specification this paper seeks to correct. Across AMZN, TSLA and NFLX, the MLOFI impact coefficient decays 27 to 57 times more slowly than fill intensity. In those names, resting depth remains costly even when the order never trades.