Flow and order book data supply the coefficients optimal execution has always taken on faith — with consequences for pre-trade cost models, liquidation schedules, liquidity risk, and TCA.

The disclaimer at the heart of optimal execution

In January 1998, Robert Almgren and Neil Chriss circulated "Optimal Liquidation," the paper that gave institutional execution its organizing mathematics.

Its contribution was a frontier: for a parent order of X shares to be liquidated by time T, every schedule maps to an expected cost E and a cost variance V, and only the one-parameter family minimizing E + λV is worth considering.

Permanent impact enters through a linear coefficient γ, temporary impact through η, and the optimal trajectory — the familiar sinh curve — is governed by a single urgency parameter κ ≈ √(λσ²/η).

Nearly every implementation-shortfall scheduler, pre-trade cost model, and execution benchmark in production today descends from this construction.

The paper is candid about what it does not do: "This is not a paper about transaction cost models. We do not propose to give a method for parametrizing the transaction cost models stated in the subsequent sections." 

The numerical examples set η by assuming that trading 1% of daily volume costs one bid-ask spread, and γ by assuming that 10% of daily volume moves price by one spread — rules of thumb, offered as such.

The gap matters because the solution is not forgiving of it. Almgren and Chriss observe that η is "the most important parameter in determining the path," and κ scales as 1/√η: mis-estimate temporary impact by a factor of four and the schedule's urgency is wrong by a factor of two.

Trade too fast and the impact cost is realized; too slow and variance is carried that the frontier said was hedged away. Every quantity downstream — the frontier itself, the liquidity-adjusted VaR the paper proposes, the benchmark against which a desk is judged — inherits the parametrization.

Figure 1 — The object every execution desk still optimizes: the efficient frontier of liquidation strategies. Each point is a schedule for the same parent order; the curve is the minimum expected cost attainable at each level of cost variance, and the tangent line selects a strategy for a given risk aversion λ. The frontier’s position is computed entirely from the impact coefficients γ and η — the inputs the paper declined to parametrize. Source: Almgren & Chriss, Optimal Liquidation (1998), Figure 1.

Why the coefficients stayed unknown

For twenty-eight years the coefficients stayed hard to know for a structural reason: the consolidated tape is anonymous.

Regressions of returns on a desk's own executions have low explanatory power at the single-order level, so practitioners pool thousands of metaorders across months to extract one average coefficient — which then drifts.

The academic consensus that emerged, the square-root law, is itself an unconditional average across metaorders.

Impact is a relationship among three variables:

1. who traded

2. how much

3. and what the book could absorb

and two of the three were unobserved.

Measuring the coefficient

The measurement problem has now been attacked from both ends.

From the top down:

Gabaix and Koijen's Inelastic Markets Hypothesis, identified with granular instrumental variables on quarterly holdings data, put a number on aggregate impact: one dollar flowing into equities raises aggregate market value by roughly five dollars — a multiplier M ≈ 5, the mirror of an aggregate demand elasticity near 0.2.

Bouchaud's latent-liquidity reading reconciled that estimate with twenty years of microstructure research, predicting that M rises with volatility and falls with the fraction of market capitalization traded daily.

Figure 2 — The top-down anchor: the aggregate market multiplier M by horizon, identified with granular instrumental variables on quarterly holdings data. A flow shock moves aggregate market value roughly five dollars per dollar invested, and the effect does not decay over the following year: impact at the market level is large and permanent. Source: Gabaix & Koijen, The Inelastic Markets Hypothesis, Figure 4.

From the bottom up:

Exponential Technology's XTech Flow applies proprietary classification to the US consolidated feed, 2007 to present, labelling each trade's initiating investor type — institutional, retail, market maker — at one-minute granularity.

On this data the multiplier stops being a quarterly artifact and becomes an out-of-sample measurement.

Quarterly institutional flow into S&P 500 constituents recovers M = 7.17 with a walk-forward out-of-sample R² of 62.6% — Gabaix–Koijen's result, replicated from order flow rather than holdings.

At daily frequency M ≈ 10 with out-of-sample R² of 71–75%, rising to 79.6% once flows are scaled by the volatility-per-volume ratio, exactly the adjustment latent-liquidity theory prescribes.

The input itself is validated independently: aggregated institutional flow predicts the direction of subsequent SEC 13F filings with 71% directional accuracy and a 45% information coefficient on high-confidence names, while classified retail flow shows no relationship to 13F outcomes — the placebo behaving as a placebo should.

Figure 3 — The flow input validated against ground truth: average hit rate and Pearson correlation between cumulative XTech institutional flow and subsequent SEC 13F filing changes, by GICS sector. Directional accuracy exceeds 0.60 in every sector (0.62–0.71); classified retail flow shows no comparable relationship. Source: Slade & Yan, Decoding Realtime Order Book Dynamics to Predict SEC Form 13F Filings, Figure 3.1.

For execution research the significance is not the level of M but its epistemic status. The coefficient that Almgren–Chriss assumed known — and every desk since has estimated by pooling — is observable daily, out of sample, with quantified error.

The coefficient is a state variable

Three further measurements change how the coefficient should be modeled.

It moves.

Estimated with exponential weighting, the daily multiplier ranges roughly 5 to 15 across 2010–2025, with peaks in March 2020 and April 2025. A fixed-coefficient scheduler is therefore mis-parametrized by up to a factor of three precisely in the states where large liquidations are most likely to be forced.

Figure 4 — The coefficient is a state variable. Left: the expanding-window daily multiplier is stable near 10. Right: estimated with a 21-day half-life, the same coefficient ranges roughly 5 to 15, spiking in March 2020 and April 2025 — the states in which forced liquidation is most likely. A fixed-coefficient scheduler is most wrong exactly when it matters most. Source: Slade & Yan, Decoding Real-Time Order Book Dynamics to Measure Market Inelasticity, Figure 3.2.2.

It decays.

Re-estimated over aggregation windows from one to sixty trading days, M declines from about 10 to about 7.2 — a 28% decay with a memory time of days to weeks.

This is Almgren–Chriss's temporary/permanent dichotomy measured as a curve: their model has temporary impact vanishing within one trading interval and permanent impact lasting forever; the data show a transient component relaxing over roughly a month onto a permanent floor.

Its permanence depends on the counterparty.

Hu (2014) decomposes stock order imbalance into an option-induced component — dealers hedging customer option flow — and the remainder, and finds the option-induced component predicts returns in the cross-section while the remainder's impact is transitory.

Exponential's companion futures study finds the same structure between trader groups: large speculative flows absorbed by commercial hedgers revert over the following month, while unabsorbed flows continue.

Permanent impact — γ in the Almgren–Chriss decomposition — is not a constant of the instrument. It is a function of who is on the other side.

Figure 5 — Permanence depends on the counterparty: cumulative forward return after large speculative-flow days in futures, split by whether commercial hedgers took the other side. Unabsorbed moves continue (+0.23% at 20 days); absorbed moves revert (−0.23%). Flow-size matched; 38 commodity markets, 2011–2026. Source: Exponential Technology, Measuring Market Impact and Inelasticity from Managed-Money and Producer Flow Forecast Data, Figure 14.

What this changes in Almgren–Chriss

None of this dents the 1998 framework; it activates clauses the authors wrote themselves.

First, the static schedule loses its justification — by the paper's own argument.

Almgren and Chriss prove the pre-committed trajectory remains optimal only so long as the trader's "estimation of the market parameters has not changed," and note the argument fails when price movements are serially correlated.

A daily coefficient nowcast violates the first premise; persistent, forecastable institutional flow violates the second. The sinh trajectory survives as the closed-form primitive inside a rolling re-solve: each day, in principle each interval, κ and the frontier are recomputed from the current flow state.

Second, the drift term comes alive.

Expected drift μ enters the solution through the static optimum x̄ = μ/2λσ²; Almgren–Chriss set it by assumption.

Classified flows explain idiosyncratic returns beyond standard factor models over horizons from one day to several months, so μ becomes a state-dependent forecast: a liquidation can be tilted to avoid trading against predicted post-flow reversion, and to coincide with forecast opposing flow that supplies the other side.

Third, risk measurement inherits the conditionality.

The paper's liquidity-adjusted VaR is defined as the minimum VaR over liquidation strategies — computed with constant coefficients.

Because measured M spikes in stress, constant-coefficient L-VaR systematically understates liquidation risk exactly when it binds. Conditioning the coefficients on the flow state corrects a correlation the original construction could not see.

Fourth, TCA changes reference point.

The natural post-trade benchmark becomes the conditional expected cost given the day's flow environment: yesterday's price move decomposes into the impact of aggregate investor-type flow (the measured multiplier times measured net flow), the desk's own footprint, and a residual.

Implementation shortfall judged against an unconditional cost curve conflates execution quality with the accident of trading on a high-inelasticity day; the conditional benchmark separates them.

Where Level 3 data is indispensable

There is an honest limit in the flow data, and it defines the division of labor.

At the single-stock level — where liquidation actually happens — flow-only multipliers average roughly 2 with a median out-of-sample R² near 16% (before considering the Bouchaud liquidity factor).

Inelasticity is strongest where substitution is constrained: indices and sectors, not individual names. Flow alone does not deliver stock-day coefficients of the strongest grade.

But the jump in fit from volatility-to-volume scaling shows where the missing signal lives: in the microstructure state.

That state is what full-depth Level 3 order book data measures directly — every insertion, amendment, cancellation and execution, nanosecond-timestamped and harmonized across venues.

It supplies the fixed cost ε as realized spread; the depth and replenishment dynamics that scale η; and the intraday decay path after flow events that separates transient from permanent impact — the γ/η split that daily data can only infer.

The estimator this suggests is hierarchical: tight sector- and index-level multipliers from flow, scaled to the stock-day by the Level 3 liquidity state.

It also gives synthetic metaorder research its missing conditioning variable. Synthetic metaorders constructed from L3 data trace impact curves; flow data labels the regime each curve was traced in — who was active, how one-sided the tape was, how much latent absorption was present.

Impact curves conditioned on flow state test the linear specification exactly where Almgren and Chriss said their approximation was "most doubtful" — the temporary term — and put regime-dependent error bars on the square-root law's prefactor.

From assumption to measurement

The practical consequences run in both directions of the trade lifecycle. 

Pre-trade:

the cost and risk of a parent order are priced off the current market context, today's coefficient rather than a multi-year pooled average, with uncertainty bands narrow enough to change decisions on sizing and urgency.

Let the math do its job.

Post-trade:

shortfall is attributed against the flow environment the order actually encountered. The frame is still Almgren–Chriss: an efficient frontier, an urgency parameter, a risk-cost trade-off chosen by the client.

What has changed is the epistemic status of the inputs. The coefficients are no longer hand-wavy assumptions calibrated by rule of thumb. Measured from investor-type flow and grounded in the full order book record, they are data.

References

Almgren, R. and N. Chriss (1998). Optimal Liquidation. Working paper, SSRN 53501.

Gabaix, X. and R.S.J. Koijen (2023). In Search of the Origins of Financial Fluctuations: The Inelastic Markets Hypothesis

Bouchaud, J.-P. (2022). The Inelastic Market Hypothesis: A Microstructural Interpretation. Quantitative Finance

Hu, J. (2014). Does Option Trading Convey Stock Price Information? Journal of Financial Economics.

Slade, J.M. and S. Yan (2025). Decoding Real-Time Order Book Dynamics to Measure Market Inelasticity. Exponential Technology

Slade, J.M. and S. Yan (2025). Decoding Realtime Order Book Dynamics to Predict SEC Form 13F Filings. Exponential Technology

Exponential Technology (2026). Measuring Market Impact and Inelasticity from Managed-Money and Producer Flow Forecast Data.