REVIEW 4 major objections 5 minor 2 cited by
Optimal Dynamic Fees in Automated Market Makers
T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read This paper derives closed-form optimal dynamic fees for a constant function AMM and identifies two fee regimes: one that penalizes arbitrageurs and one that attracts noise traders.
desk verdict A useful first pass at optimal dynamic AMM fees, with a clean exact result in the constant-price case and a heuristic quadratic approximation that needs an error bound before the near-optimality claim is taken literally. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is a controlled point-process model of order flow, with buy and sell intensities proportional to $\exp\left(k\left(Z_+^p(y)-(s+\zeta)\right)\Delta_+(y)\right)$ and its symmetric counterpart for sells. The exponential form, adapted from market-making intensity models, converts the Hamilton-Jacobi-Bellman equation into a linear system through the substitution $w=e^{kg}$; the tridiagonal matrix $A$ in Theorem 4.1 encodes arrival rates, exchange rates, and inventory grid, and optimal fees are read from log-ratios of $w$ across neighboring inventory states. In the stochastic-price case the same HJB equation is replaced by a quadratic expansion, yielding a Riccati ODE for the ansatz $g(t,y,s)=y^2A(t)+yB(t,s)+C(t,s)$, from which the fees are affine in $y$ and $s$.
What would settle it
Take a pool with a known fee rule and record, for each one-step inventory level, the empirical log arrival rate of buys and sells as a function of the fee-inclusive exchange rate minus the oracle price. If the log rate is not well approximated by a line with slope $k$ over the range of typical gaps, the exponential-intensity premise is violated and the optimality result does not carry over. A second check is whether the predicted revenue gap between optimal and constant fees appears in a live or simulated pool under a different order-flow model.
Extended reading notes
Core claim
On the paper's own terms, the central discovery is an explicit solution to the AMM fee-control problem. When the oracle price is constant, the value function takes the form $v(t,y,c)=c+\frac1k\log w(t,y)$ with $w(t,y)=\exp(A(T-t))\mathbf{1}$, and the optimal fees are $p^*(t,y_i)=\frac{1}{kZ_+(y_i)\Delta_+(y_i)}(1+\log(w(t,y_i)/w(t,y_{i+1})))$ and the symmetric formula for $m^*$; in the quadratic approximation these become fees that are affine in inventory $y$ and oracle price $s$. The fee schedule splits into two regimes, penalizing arbitrageurs on one side of the oracle price and subsidizing noise traders on the other. Simulations show the optimal policy outperforms constant fees by a clear margin, and the linearized version gives nearly identical revenue.
Load-bearing premise
All results rest on the assumption that buy and sell order arrivals are controlled point processes whose intensity is exponential in the gap between the pool's fee-inclusive exchange rate and the oracle price, with trades of fixed size and constant pool depth; if real order flow does not react this way, the derived fee formulas are model outputs rather than robust prescriptions.
Editorial extensions
If this is right
- The fee formulas in Theorem 4.1 can be precomputed for a given inventory grid and applied as a deterministic function of time and inventory, enabling real-time fee setting without per-trade optimization.
- Under the quadratic approximation, optimal fees are affine in inventory and oracle price and do not depend on price volatility, giving a simple implementation rule that only needs the current pool inventory and an oracle price feed.
- The two-regime structure implies that a uniform constant fee is revenue-dominated; the simulated gap between optimal and constant fees grows with arrival intensity and shrinks as traders become less price-sensitive.
- Because the linearized policy reproduces the optimal revenue in the paper's simulations, AMM designers can restrict attention to a small family of linear fee schedules rather than solving the full control problem.
Reading between the lines
- Beyond the paper, a testable prediction is that pools switching from constant to inventory-linear dynamic fees should see arbitrage trades concentrated in the high-fee direction and noise-trading volume rise in the low-fee direction, something the paper's simulations exhibit but do not validate on market data.
- The affine fee rule in the stochastic-price case could be implemented on-chain with periodic recalibration of its coefficients, since the oracle price enters linearly and no per-trade optimization is needed.
- The analysis fixes pool depth and excludes liquidity-provider entry and competition among pools; if those are endogenized, the slope of the optimal linear fee may shift, but the two-regime shape is likely to persist.
- The quadratic approximation comes without formal error bounds, so the near-optimality of linear fees is a numerical finding rather than a proven theorem; a rigorous error estimate would be the direct next step.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies optimal dynamic fee setting in a constant-function market maker. Order flow is modelled by controlled doubly stochastic point processes with exponential intensities depending on the gap between the fee-inclusive AMM exchange rate and an external oracle price, on a finite inventory grid. The authors derive an HJB equation, solve it exactly in the constant-oracle-price case (Theorem 4.1) via the transformation e^{kg}=w, and in the stochastic-price case replace exponentials by quadratic polynomials and posit a quadratic value function to obtain closed-form fee formulas (Theorem 5.2). Simulations compare these formulas with constant fees and with a linearized rule and report that the two-regime structure (penalize arbitrageurs versus attract noise traders) and the linear rule are revenue-close to the optimal policies.
Significance. If valid, the paper offers a tractable framework for dynamic fee design in AMMs and a concrete linear fee rule implementable in venues such as Uniswap v4. Theorem 4.1 is an elegant exact solution of the reduced HJB system, and the public code is a strength. However, the central near-optimality claim rests on an approximate HJB with no error control and on simulations that compare only against constant and first-order policies; the significance is therefore conditional on fixing the verification and approximation issues below.
major comments (4)
- [Section 3, footnote 4] The control problem is never given a rigorous weak formulation, and no verification theorem is stated or proved showing that the solution of HJB (3.4) equals the value function of the controlled point-process system. Consequently Theorem 4.1 shows that v solves an HJB equation and gives candidate optimizers, but it does not establish that these policies are optimal for the original model. Please add a verification argument (or a formal statement with the required regularity) and a precise probability-space construction.
- [Section 5, Eq. (5.1)] The replacement of the exponentials by second-order Taylor polynomials, the constant trade sizes δ±, and the quadratic ansatz g(t,y,s)=y^2 A(t)+y B(t,s)+C(t,s) are made without any error estimate. No argument shows that the solution of (5.1) is close to the value function of (3.4), so the abstract's claim that linear-in-inventory, price-sensitive fees are a 'good approximation of the optimal fee structure' is not quantified. The authors' own observation after Figure 8 that the second-order approximation degrades as k→0 indicates the error is not uniformly small; an explicit bound or a numerical error study against the unapproximated HJB is needed.
- [Theorem 5.2] The statement asserts existence of 26 constants {ψ_i} whose full definition is relegated to a GitHub repository. As written, the theorem is not self-contained: the ODE system (ψ0+ψ1A+ψ2A²+A'=0, etc.) cannot be verified or implemented from the paper alone, and the claim of uniqueness of solutions to the ODE system is not justified. Please include the constants and the proof, or move them to an appendix.
- [Section 5.1, simulation table] The numerical comparison measures revenue under the same order-flow model used to derive the policies. The 'Linear' rule in Section 4.1 is a Taylor linearization of the optimal fee rule in that same model, so the near-identical revenue is partly by construction; no comparison to the true optimal policy of the original, unapproximated model is provided, and no confidence intervals are reported. To support the near-optimality claim, please benchmark against a numerical solution of the original HJB (or at least report the approximation error in the value function) and report standard errors.
minor comments (5)
- [Throughout] There are typographical errors such as 'assety' and 'the the grid'; please proofread the manuscript carefully.
- [Figures 1 and 7] The captions for Figures 1a/1b and 7a/7b are identical or nearly identical; please make the captions distinguish the two approximations explicitly.
- [Eq. (3.1)] In Eq. (3.1), the notation in the jump terms is inconsistent, writing both v(t, y, c, s) and v(t, c, y, s) for the same object; please use a uniform argument order.
- [Corollaries 4.2 and 5.3] Both corollaries assert limits as k→0, but the paper does not state whether these limits are uniform in t and y; please clarify the mode of convergence.
- [Section 2] The sentence 'the depth of the pool p² is fixed throughout [0,T]' is an important modelling limitation and should be highlighted in the introduction as an assumption, not only in the model section.
Circularity Check
No significant circularity: the fee formulas are derived forward from explicit intensity-model assumptions, and the linear-fee claim is an in-model approximation check rather than a fitted prediction; the only self-citation is a minor, non-load-bearing technical footnote.
full rationale
The derivation is forward. Order-flow intensities are specified exogenously; the dynamic programming principle leads to the HJB equation (3.1)-(3.4); under constant oracle price the substitution e^{kg}=w turns (4.1) into the linear ODE system (4.2), whose exact solution yields the fee formulas (4.3)-(4.4). The second approximation is explicit about its inputs: second-order Taylor expansion of the exponentials, constant trade sizes, a linearized exchange rate, and the quadratic ansatz g(t,y,s)=y^2 A(t)+y B(t,s)+C(t,s). Formula (5.2) then follows algebraically from (3.3) and that ansatz, so the linear-in-inventory and linear-in-price form is an assumption-consequence, not an independently discovered empirical fact; the paper does not claim to have derived the ansatz from the original control problem. The 'good approximation' claim rests on simulations within the same model, comparing the linearized policy to the exact first-approximation optimum and to constant fees. Because the linear rule is a Taylor linearization of the optimal policy itself, the near-identical revenue is partly by construction, but this is a self-consistency check, not a fitted parameter relabeled as prediction. No parameters are calibrated to external data, and no resulting quantity is renamed as an out-of-sample prediction. The only self-citation is footnote 4, which points to Barucci et al. (2025) for a rigorous weak formulation; that technical point is not load-bearing for the fee formulas or the numerical comparisons. The absence of an error bound between the solution of (5.1) and the true value function is a correctness and robustness gap, not a circularity.
Assumptions & free parameters
free parameters (5)
- k (intensity sensitivity) =
2, 1, 0.5, 0.25, 0.1 (used in figures and tables)
- lambda +/- (baseline order intensities) =
50, 100, 150 per side
- sigma (oracle volatility) =
0 (Section 4), 0.2 (Section 5)
- delta +/- (trade size / grid step) =
delta = 0.5 and exchange rate step 0.1
- phi (penalty weight) =
0 for main results; scanned up to 100
assumptions (6)
- domain assumption Order arrivals are controlled Cox processes with exponential intensities of Avellaneda-Stoikov form, symmetric in buy/sell with coefficient k.
- domain assumption Trade sizes are fixed to one grid step and the pool depth p^2 is constant over [0,T]; no LP entry or exit.
- domain assumption The oracle price is an arithmetic Brownian motion S_t = S_0 + sigma W_t.
- ad hoc to paper Fees p and m are allowed to be arbitrary real numbers, unbounded, negative, and above 100 percent.
- domain assumption The dynamic programming principle and HJB verification hold for the controlled point-process problem.
- ad hoc to paper Section 5 approximations: exponential terms are replaced by second-order Taylor expansions; g is a quadratic polynomial in y; delta +/- are constant; exchange rates are linearized.
Cite this review
Pith. "Pith review of Optimal Dynamic Fees in Automated Market Makers." pith.science (2026). https://pith.science/paper/EWZZWY3A
@misc{pith2026250602869,
author = {Pith},
title = {Pith review of: Optimal Dynamic Fees in Automated Market Makers},
year = {2026},
howpublished = {\url{https://pith.science/paper/EWZZWY3A}},
note = {Machine review of arXiv:2506.02869}
}
read the original abstract
Automated Market Makers (AMMs) are emerging as a popular decentralised trading platform. In this work, we determine the optimal dynamic fees in a constant function market maker. We find approximate closed-form solutions to the control problem and study the optimal fee structure. We find that there are two distinct fee regimes: one in which the AMM imposes higher fees to deter arbitrageurs, and another where fees are lowered to increase volatility and attract noise traders. Our results also show that dynamic fees that are linear in inventory and are sensitive to changes in the external price are a good approximation of the optimal fee structure and thus constitute suitable candidates when designing fees for AMMs.
Forward citations
Cited by 2 Pith papers
-
Optimal Fees for Liquidity Provision in Automated Market Makers
Optimal AMM fees sit just below all-in CEX trading costs in normal markets, rise with volatility, and become effectively infinite (halt trading) in extreme volatility.
-
Reinforcement Learning for Execution under Dynamic Fees in a Closed-Loop DEX Simulator
In a reserved 1,000-seed closed-loop DEX simulation with forced completion, a small DQN cuts implementation shortfall 13.3 bps versus tuned one-step routing under dynamic fees, with no detectable edge under constant fees.
Reference graph
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Reviewed August 7, 2026 · model on record in the stance chip above.
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