Automated Market Makers: A Stochastic Optimization Approach for Profitable Liquidity Concentration

Ali Sunyaev; Daniel Kirste; Niclas Kannengie{\ss}er; Paul-Niklas Ken Kandora; Simon Caspar Zeller; Steffen Rebennack

REVIEW 4 major objections 5 minor 21 references

Automated Market Makers: A Stochastic Optimization Approach for Profitable Liquidity Concentration

T0 review · 4 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper claims that on Uniswap v3's USDC/ETH pool, the optimal liquidity interval is effectively full range (α ≈ 4), and even that best case underperformed simply holding the tokens.

desk verdict A useful LP-interval formulation undercut by a bound artifact: the reported optimal α is the imposed upper bound, not a resolved trade-off. read the letter →

arxiv 2504.16542 v1 pith:JJGKPTUW submitted 2025-04-23 q-fin.TR q-fin.CPq-fin.MF

Simon Caspar Zeller , Paul-Niklas Ken Kandora , Daniel Kirste , Niclas Kannengie{ss}er , Steffen Rebennack , Ali Sunyaev This is my paper · ORCID

classification q-fin.TRq-fin.CPq-fin.MF

keywords automatedmarketmakersUniswapv3concentratedliquidityprovisionstochasticoptimizationdivergencelossreallocationcostssampleaverageapproximation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to turn an automated market maker's concentrated-liquidity interval choice into a solvable stochastic optimization problem. It models a single liquidity provider who reallocates only when the token price leaves the chosen interval, and maximizes terminal wealth over an interval-size parameter α while accounting for liquidity rewards, divergence loss, and reallocation costs. Applied to hourly on-chain data from Uniswap v3's 0.05% USDC/ETH pool for 2023–2024, the model's optimal α saturates at the upper bound 4, which the authors equate with full-range provision; they conclude that wide, full-range intervals would have been most profitable and that even the optimal interval would have earned less than passively holding the tokens. The contribution is a tractable formulation that can be adapted to other price processes, fee tiers, and cost structures, not a general claim that concentration is useless.

What carries the argument

The load-bearing object is a mixed-integer nonlinear stochastic program (equations (2) and (3)) whose single decision variable is α, the half-width of the square-root-price interval [π/α, απ]. A binary variable z_t records whether the price has left the interval and forces a reallocation; the new interval is then recentered at the current price. Liquidity rewards are computed as a constant reward rate c_f times the linearly interpolated fraction of each hourly step spent inside the interval (equation (3p)), terminal wealth is the value of the real token reserves from equation (1) plus accumulated unclaimed fees, and sample average approximation converts the expectation into an average over simulated geometric Brownian motion paths with zero drift. This machinery lets a commercial mixed-integer solver return a single number α that balances the three competing effects.

What would settle it

On the same 0.05% USDC/ETH pool, replace hourly linear price interpolation in equation (3p) with Brownian-bridge occupation-time estimates of the active fraction and re-solve the sample average approximation; if the optimal α falls well below 4, or if any finite α beats both full-range provision and the passive holding benchmark, the paper's central conclusion is falsified.

Watch

Extended reading notes

Core claim

The paper's central claim is that optimal liquidity provision on a concentrated-liquidity constant-product AMM can be stated as a finite-horizon stochastic optimization over the interval scale α, with bounds [π/α, απ] on the square-root price; the objective is expected terminal wealth, and the trade-off is driven by three competing terms: liquidity rewards grow with narrowness, divergence loss grows as the interval narrows, and reallocation costs grow with how often the price exits the interval. Solving the resulting mixed-integer nonlinear program by sample average approximation on the Uniswap v3 0.05% USDC/ETH pool (Jan 11, 2023 to Nov 30, 2024) gives an optimal α ≈ 4 across seeds and sample sizes. Because an interval of that width covers price moves from 1/16 to 16 times the starting price, the authors identify α ≈ 4 with full-range provision and claim that full-range intervals would have been the most profitable choice in retrospect. They further claim that liquidity provision even at this optimal width would have returned less than holding the initial token basket, because fee income did not compensate divergence loss.

Load-bearing premise

The model treats the hourly reward rate as a constant and interpolates price linearly between hourly observations to decide when the position was active; if fee income is bursty or price moves nonlinearly within the hour, the optimal interval size could be different.

Editorial extensions

If this is right

If the model is right, narrower is not automatically better: the optimal interval sits where the marginal reward from concentration equals the marginal divergence loss plus expected reallocation cost, and for this pool that point is the full range.
The same formulation can be rerun for other pools, fee tiers, gas prices, or price processes by changing inputs or the simulated paths, so the method yields a pool-specific α rather than a universal rule.
Delaying reallocation until the price is a threshold distance outside the interval (γ > 0) can raise profits for narrow intervals by cutting reallocation frequency; the data does not settle whether γ helps when α is already optimal.
The model's forward-looking optimum matched the retrospectively best interval on the same data, which is the main evidence that the tractable formulation captures the reward–loss–cost trade-off well enough to guide liquidity providers.
Even at the optimum, the demonstration found liquidity provision less profitable than not providing liquidity, implying that fee rewards alone did not compensate divergence loss over a period in which ETH roughly tripled.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Correcting the paper's flagged overestimation of small-α rewards—by using Brownian-bridge occupation times instead of linear interpolation—would likely push the optimal α even closer to full range, since narrow intervals would lose their overstated reward advantage.
Re-running the model with the empirical upward drift of ETH would be a natural test: the passive basket captures that drift directly, while the LP's exposure is path-dependent, so the holding benchmark and the optimal α could both shift.
In a range-bound or mean-reverting market, narrow intervals might plausibly beat full-range provision; the paper's demonstration period has a near-tripling price path, so the full-range conclusion should be read as period-specific until other regimes are tested.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Short version: the paper adds a real piece to the LP interval-selection literature—an SAA-based MINLP that optimizes interval width α under a discrete reallocation rule with costs—but its headline result α≈4 is an artifact of the imposed upper bound, not a demonstrated trade-off. On the data shown, profits rise monotonically with α up to the bound, and the authors themselves note that full-range provision would have done better (71% vs 65%). So the central validation doesn't land.

What's genuinely new and good: unlike Cartea et al.'s continuous-time model or Fan et al.'s fixed-width SAA, this formulation makes α a decision variable inside a discrete-reallocation problem, and it accounts for reallocation costs. The appendix gives a solvable form, code is on GitHub, and the empirical work uses real on-chain USDC/ETH data. The paper is also honest about limitations: it states that LPs were better off holding, that fee interpolation can overestimate rewards, and that the zero-drift GBM assumption is a simplification. The related-work coverage looks fair.

The soft spots are not small. The upper bound α≤4 is chosen from the same data (α>1.73 implies no reallocation), and the solver returns the bound in most seeds. Fig. 3 shows terminal profit increasing across the entire feasible range, so there is no interior optimum—wider is better up to the constraint. The validation then compares the SAA solution to the hindsight-optimal choice under the same model, which is in-sample consistency, not external confirmation. A zero-drift GBM cannot probe a 23-month period where ETH nearly tripled, and the claim that the trend doesn't bias results is asserted, not tested. I'd also flag the appendix eq. (3k) as looking dimensionally off with π_t/2 instead of π_t^2, though the rendering makes this hard to confirm from the text.

That said, the empirical caution—that concentrating liquidity in this pool over this period would have lost to simply holding—is worth having, and the model structure can be built on. Who is this for? People working on LP strategies and AMM optimization; they'll want to fix the validation and test whether an interior α exists when the bound is removed.

For peer review: yes, send it out—it's a serious formulation with real data—but major revision is needed. The authors should re-run without the α≤4 cap, or at least report whether the objective turns over, and should do an out-of-sample or at least out-of-sample-period check before claiming the model recovers the hindsight optimum.

Yours,

Referee Report

4 major / 5 minor

Summary. The paper models a single Uniswap v3 liquidity provider's choice of interval width α as a stochastic optimization problem. The objective is expected terminal wealth in USD, comprising token holdings and accumulated fees, under a reallocation rule triggered when the price leaves the interval. The model accounts for liquidity rewards, divergence loss, and reallocation costs. The authors solve a sample-average approximation with Gurobi using hourly on-chain data from the 0.05% USDC/ETH pool (January 2023 to November 2024) and report an optimal α of approximately 4, which they interpret as full-range provision. The empirical conclusion is that full-range provision would have been most profitable but that liquidity provision still underperformed simply holding the tokens.

Significance. If the formulation were correct and the empirical result robust, the paper would offer a useful decision-support tool for LPs and a surprising counterpoint to the common view that concentrated ranges are the most profitable. The authors provide an explicit mixed-integer nonlinear formulation, an SAA scheme, a real dataset, and open-source code, which are notable strengths. However, the central numerical claim is weakened by a dimensional inconsistency in the appendix formulation and by the fact that α≈4 is the solver's imposed upper bound, not an interior trade-off point. The validation is in-sample and compares the widest allowed range with the widest possible range. Consequently, the paper's main claim—that the optimization resolves the rewards-versus-divergence-loss-versus-reallocation-cost trade-off—is not currently demonstrated.

major comments (4)

[Appendix, eq. (3k)] Equation (3k) is dimensionally inconsistent with the wealth definition used in the objective (3a). The objective values terminal wealth as π_T^2 \tilde{x}^r_T + \tilde{y}^r_T + Σ^f_T, but (3k) writes the wealth balance at reallocation times using π_t/2 x^r_t + y^r_t and π_t/2 \tilde{x}^r_t + \tilde{y}^r_t. Since π_t is the square root of the price, π_t/2 has units of sqrt(price), not price, so the equality cannot hold as a USD wealth conservation law; the rebalancing cost term ctr |y_t^r − \tilde{y}_t^r| also lacks the appropriate price conversion. This is not a notational slip: the SAA results in Table II are obtained by solving this formulation, so the numerical results must be recomputed with a corrected eq. (3k).
[Section IV-A, IV-B, Fig. 3, Table II] The reported optimum α≈4 is the imposed upper bound, not an interior solution. Section IV-A states that the upper bound was set to 4 because values above 1.73 result in no reallocation and that α=4 can be interpreted as going full range. Figure 3 shows terminal profit increasing monotonically for α≥1.73 on the realized path, and the text states that full-range provision gives 71% profit versus 65% at α=4, so the hindsight optimum lies outside the feasible set. In Table II, all medians are 4.00 but minima range from 1.01 to 1.04 with standard deviations up to 0.95, indicating a flat or regime-dependent SAA objective rather than a robust trade-off surface. The validation in Section IV-B therefore compares the widest allowed interval with the widest possible interval and does not establish that the optimization balances rewards, divergence loss, and reallocation costs.
[Section V-C and eq. (3p)] The activity-fraction interpolation used in eq. (3p) can materially overestimate expected liquidity rewards for small α, as the authors themselves acknowledge in Section V-C. Since the objective is precisely what determines the optimal α, and since the finding that α=4 is optimal depends on the relative magnitude of rewards, the paper should quantify the bias (for example, with Brownian bridge estimates or intra-hour price data) or show that the optimum is insensitive to this modeling choice. Without such an analysis, the empirical demonstration is not robust.
[Section III-C and Section IV-A] The price process is assumed to be GBM with zero drift, but the sample period has ETH rising from approximately $1,350 to $3,725. The claim in Section IV-B that this trend does not substantially bias the results is not adequately supported: both terminal wealth and the comparison with holding assets depend directly on the realized appreciation, while the zero-drift SAA paths make such appreciation unlikely. The authors should either fit a drift, allow time-dependent α_t, or provide a concrete counterfactual or robustness check for the zero-drift assumption before the empirical conclusion can be accepted.

minor comments (5)

[Section III-A, eq. (1)] The second case of eq. (1) states the condition as π_l ≤ π ≤ π_l, which appears to be a typo for π_l ≤ π ≤ π_u.
[Section III-C and Appendix] The main text refers to 'eq. (3) in section VI-A' for the tractable formulation, but the formulation is in the Appendix, not in Section VI-A; the cross-reference should be corrected.
[Section IV-A] The GitHub URL contains spaces ('AMM Liquidity Concentration Optimization') and should be given as a proper hyperlink or with URL encoding.
[Section IV-B] The text says α=4 'can be interpreted as going full range' but then reports a distinct profit of 71% for going full range versus 65% for α=4; this internal inconsistency should be clarified.
[Section III-C] The wealth variable Y_t is used in formulation (2) but is not explicitly defined in the main text; the definition appears only in the appendix objective (3a), and it would help to define Y_t in Section III-C.

Circularity Check

2 steps flagged · score 6.0 of 10

Reported optimal α≈4 is the imposed upper bound, and the 'hindsight' validation compares constrained solutions computed from the same fitted data, making the demonstration's central result partially circular.

other [Section IV-A, Methods (upper bound on α); Section IV-B, Demonstration Results]
"We set the upper bound to 4, as values of α above 1.73 result in no reallocation taking place over the entire time frame of the used data. ... Reallocation costs therefore do not influence terminal wealth and the solution α = 4 can be interpreted as going full range."

The reported optimum α≈4 is not an interior solution where liquidity rewards, divergence loss, and reallocation costs balance: it is the solver's upper bound, chosen from the same dataset to represent 'going full range'. Figure 3 is monotonically increasing in α over the feasible range (71% at full range vs. 65% at α=4), so any optimizer restricted to α≤4 returns the boundary. Thus the central 'optimal α' result is imposed by the constraint plus monotonicity rather than derived from the modeled trade-off.
fitted input called prediction [Section IV-B, Demonstration Results; Section IV-A, Methods (parameter estimation)]
"The values in Table II show that the optimal solution was consistently set to ≈ 4; equivalent to the value for α that an LP should have chosen in retrospect."

The 'retrospective optimum' is not an independent benchmark: it is computed with the same model, the same fitted inputs (σ estimated from the same log returns, cf from the same fee data, same cg and ctr), and the same α≤4 upper bound. The SAA result and the hindsight result are therefore the same constrained maximization output on the same period. Their agreement is in-sample consistency by construction, not an external validation or out-of-sample prediction.

full rationale

The optimization formulation itself is self-contained and does not rest on a load-bearing self-citation chain; parameters are stated and estimated from data, and the GBM/ULRA modeling choices are attributed to external prior work. The circularity is localized to the demonstration's headline quantitative claim. The solver selects the upper bound α=4 because the modeled terminal profit increases with α over the tested range; the paper even notes that full-range liquidity provision (beyond the bound) would have yielded 71% versus 65% at α=4. Consequently, the reported 'optimal α≈4' is a boundary artifact, and the validation against 'the value an LP should have chosen in retrospect' compares two outputs of the same constrained, same-data optimization. This makes the central empirical result partially circular (the result reduces to the imposed bound and in-sample fit), although the model formulation retains independent content and the honest finding that liquidity provision underperforms holding is an empirical statement rather than a definitional one.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The model's outputs depend on a price-process specification, a reward-accrual approximation, an inherited reallocation rule, and several parameters fitted to the same dataset used for the backtest. No new entities are introduced.

free parameters (6)

GBM volatility σ = ≈ 0.006 hourly
Estimated as the standard deviation of log returns of hourly price data (Section IV-A). Drives the SAA price paths.
Constant reward rate cf = historical median of cft
Section IV-A: 'cf are set to the median of cft, and consequently assumed to be constant.' Used in the optimization; validation uses the actual time-varying cft.
Transaction cost per reallocation cg = $109.80
Section IV-A: '$84.8 as of [6] plus an additional estimated $25 to claim liquidity rewards'. Hand-estimated.
Reallocation trigger threshold γ = 0 (base), -6 and +6 for sensitivity
Section IV-A. γ=0 gives the base ULRA rule; the paper tests fixed alternatives but does not optimize γ in the main problem.
Bounds on interval width α = [1.01, 4]
Section IV-A: lower bound 1.01 because α below ≈1.02 gives -100% profits; upper bound 4 because α>1.73 implies no reallocation and α=4 covers 1/16 to 16 times the initial price. The reported optimum sits at this upper bound.
Pool fee rate ctr = 0.05%
Set to the fee tier of the analyzed Uniswap v3 USDC/ETH pool (Section IV-A). A pool parameter, not fitted.

assumptions (5)

domain assumption Marginal price π_t^2 follows a geometric Brownian motion with constant volatility σ and zero drift
Section III-C: 'we model the marginal price π_t^2 as a GBM process with constant volatility σ but without drift because forecasting price trends is beyond the scope of this work.' This underpins the SAA input paths; the actual data had a strong upward drift.
ad hoc to paper Liquidity rewards for a partially active position scale linearly with the interpolated fraction of time the price is in range
Section III-C and eq (3p). The paper acknowledges this can overestimate rewards for small α (Section V-C).
domain assumption The LP reallocates only when the price exits the current interval (ULRA-style rule)
Section III-C: 'we assume the LP to reallocate their liquidity at time t when the liquidity position becomes inactive.' Inherited from Fan et al. (2023); the paper tests fixed γ adjustments but does not optimize the rule.
domain assumption The LP is small enough that its actions do not affect pool prices or total liquidity
Section V-C lists this as a limitation; the model treats fee income per unit liquidity (fg) and price as exogenous.
domain assumption The optimization uses a constant reward rate cf equal to the historical median, and constant volatility σ estimated from the same period
Section IV-A: 'The maximum liquidity rewards per liquidity cf are set to the median of cft, and consequently assumed to be constant.' The validation uses time-varying rewards, so the optimization input differs from the evaluation input.

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Cite this review

Pith. "Pith review of Automated Market Makers: A Stochastic Optimization Approach for Profitable Liquidity Concentration." pith.science (2026). https://pith.science/paper/JJGKPTUW

@misc{pith2026250416542,
  author       = {Pith},
  title        = {Pith review of: Automated Market Makers: A Stochastic Optimization Approach for Profitable Liquidity Concentration},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JJGKPTUW}},
  note         = {Machine review of arXiv:2504.16542}
}

read the original abstract

Concentrated liquidity automated market makers (AMMs), such as Uniswap v3, enable liquidity providers (LPs) to earn liquidity rewards by depositing tokens into liquidity pools. However, LPs often face significant financial losses driven by poorly selected liquidity provision intervals and high costs associated with frequent liquidity reallocation. To support LPs in achieving more profitable liquidity concentration, we developed a tractable stochastic optimization problem that can be used to compute optimal liquidity provision intervals for profitable liquidity provision. The developed problem accounts for the relationships between liquidity rewards, divergence loss, and reallocation costs. By formalizing optimal liquidity provision as a tractable stochastic optimization problem, we support a better understanding of the relationship between liquidity rewards, divergence loss, and reallocation costs. Moreover, the stochastic optimization problem offers a foundation for more profitable liquidity concentration.

Figures

Figures reproduced from arXiv: 2504.16542 by the authors.

**Figure 1.** Illustration of the evolution of an LP’s liquidity provision interval, [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 5.** Profits of an LP with initial wealth $ 100,000 for different sizes [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Liquidity rewards per time step of an LP with initial wealth $ 100,000 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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How automated market makers approach the thin market problem in cryptoeocnomic systems,

D. Kirste, N. Kannengießer, R. Lamberty, and A. Sunyaev, “How automated market makers approach the thin market problem in cryptoeocnomic systems,” 2023. [Online]. Available: https://arxiv.org/ abs/2309.12818

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J. Xu, K. Paruch, S. Cousaert, and Y . Feng, “Sok: Decentralized exchanges (dex) with automated market maker (amm) protocols,” ACM Comput. Surv. , vol. 55, no. 11, Feb. 2023. [Online]. Available: https://doi.org/10.1145/3570639

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J. Milionis, C. C. Moallemi, T. Roughgarden, and A. L. Zhang, “Automated Market Making and Loss-Versus-Rebalancing,” Nov. 2023. [Online]. Available: http://arxiv.org/abs/2208.06046

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Thorough mathematical modelling and analysis of Uniswap v3,

M. Echenim, E. Gobet, and A.-C. Maurice, “Thorough mathematical modelling and analysis of Uniswap v3,” Sep. 2023. [Online]. Available: https://hal.science/hal-04214315

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