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arxiv: 2403.18177 · v4 · submitted 2024-03-27 · 💱 q-fin.MF · q-fin.PR· q-fin.TR

Growth rate of liquidity provider's wealth in G3Ms

Cheuk Yin Lee , Shen-Ning Tung , Tai-Ho Wang This is my paper

Pith reviewed 2026-05-24 03:54 UTC · model grok-4.3

classification 💱 q-fin.MF q-fin.PRq-fin.TR
keywords liquidity providersgeometric mean market makersstochastic reflected diffusionarbitrageautomated market makersdecentralized financetrading fees
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The pith

The long-term logarithmic growth rate of liquidity provider wealth in G3Ms is derived from stochastic reflected diffusion models of arbitrage and fees.

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

The paper computes the expected long-term growth rate of liquidity provider wealth in geometric mean market makers, including Balancer, by modeling reserve dynamics under continuous arbitrage. It extends the constant-product case to general weight vectors by representing the pool state as a stochastic reflected diffusion that reflects at the no-arbitrage boundary. Trading fees enter the growth rate as a positive drift term while arbitrage enforces price alignment. The resulting formula gives an explicit dependence on the fee rate and the geometric-mean parameters, allowing direct comparison of different G3M designs.

Core claim

Under an arbitrage-driven market, the reserve vector of a G3M evolves as a stochastic reflected diffusion whose long-term expected logarithmic growth for the liquidity provider equals a closed-form expression linear in the fee rate and determined by the weight vector of the geometric mean.

What carries the argument

Stochastic reflected diffusion process for the reserve vector, with reflection at the no-arbitrage hyperplane and a fee-adjusted drift.

If this is right

  • LP profitability increases linearly with the trading fee rate for any fixed weight vector.
  • Different weight vectors in a G3M produce different growth rates even under identical external volatility and fee schedules.
  • In the zero-fee limit the expected growth rate is non-positive, recovering the constant-product result that LPs lose to arbitrage on average.
  • The growth-rate formula supplies a ranking of G3M parameter choices by expected LP return.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same diffusion framework could be used to compare G3M performance against other AMM families once their invariant surfaces are expressed as reflecting boundaries.
  • If real markets exhibit discrete block times or latency, the continuous-reflection assumption would need a discrete-time correction term added to the growth formula.
  • The model implies that LPs should prefer higher-fee pools when external volatility is fixed, which could be tested by comparing pools with identical assets but different fee tiers.

Load-bearing premise

Arbitrage occurs continuously and instantaneously, so that prices stay exactly aligned with the external market at all times and no discrete jumps or additional trading frictions appear.

What would settle it

A long-horizon empirical time series from a live Balancer pool in which the realized logarithmic wealth growth of LPs deviates systematically from the closed-form expression once fees and observed volatility are plugged in.

Figures

Figures reproduced from arXiv: 2403.18177 by Cheuk Yin Lee, Shen-Ning Tung, Tai-Ho Wang.

Figure 1
Figure 1. Figure 1: Price time series (1-minute intervals) from 14:00:00 to 18:00:00 on September 13, 2023. Dashed lines indicate the upper and lower boundaries of the no-arbitrage region [PITH_FULL_IMAGE:figures/full_fig_p036_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Growth rate ratio for different values of weights. Note the symmetry between positive and negative θ values [PITH_FULL_IMAGE:figures/full_fig_p037_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Growth rate ratio for different values of θ. Note the symmetry around w = 1 2 [PITH_FULL_IMAGE:figures/full_fig_p038_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Heatmap of growth rate ratio for different weights, with maximum values labeled [PITH_FULL_IMAGE:figures/full_fig_p039_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Heatmap of growth rate ratio for different values of θ, with maximum values labeled [PITH_FULL_IMAGE:figures/full_fig_p040_5.png] view at source ↗
read the original abstract

We study how trading fees and continuous-time arbitrage affect the profitability of liquidity providers (LPs) in Geometric Mean Market Makers (G3Ms). We use stochastic reflected diffusion processes to analyze the dynamics of a G3M model under the arbitrage-driven market. Our research focuses on calculating LP wealth and extends the findings of Tassy and White related to the constant product market maker (Uniswap v2) to a wider range of G3Ms, including Balancer. This allows us to calculate the long-term expected logarithmic growth of LP wealth, offering new insights into the complex dynamics of AMMs and their implications for LPs in decentralized finance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper models liquidity provider (LP) wealth dynamics in Geometric Mean Market Makers (G3Ms), including Balancer, under continuous arbitrage and trading fees. It extends the constant-product analysis of Tassy and White by representing reserve ratios as stochastic reflected diffusions whose drift is fixed by instantaneous no-arbitrage conditions, then derives a closed-form expression for the long-term expected logarithmic growth rate of LP wealth.

Significance. If the derivation is correct, the result supplies an explicit growth-rate formula that generalizes the Uniswap v2 case to arbitrary weights, which is useful for quantitative assessment of LP returns in weighted AMMs. The modeling choice of reflected diffusions is standard for arbitrage-driven markets, but the paper does not appear to supply machine-checked proofs or reproducible code.

major comments (2)
  1. [§4] §4, Eq. (15)–(18): the claim that the multi-dimensional reflection at the no-arbitrage boundary contributes no extra local-time term to the log-wealth process is not verified. The extension from the one-dimensional constant-product case requires an explicit Itô–Tanaka expansion or quadratic-variation calculation for the reflected process; without it the growth-rate formula may miss a correction proportional to the local time at the boundary.
  2. [§3.2] §3.2, definition of the reflection barrier and fee parameters: these quantities are defined via the same instantaneous-arbitrage condition that supplies the drift of the diffusion. The manuscript should show that the resulting growth rate is invariant under small perturbations of the barrier or provide an external benchmark (e.g., discrete-time simulation) to confirm the formula is not an artifact of this modeling choice.
minor comments (2)
  1. [§2] Notation for the weight vector w and the geometric-mean price process is introduced without a clear reference to the earlier Tassy–White paper; adding a short comparison table would improve readability.
  2. [§1] The abstract states the target quantity but the introduction does not list the precise assumptions (continuous trading, zero latency, perfect arbitrage) that are later used; a dedicated assumptions paragraph would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper to strengthen the mathematical justification and validation of the results.

read point-by-point responses

  1. Referee: [§4] §4, Eq. (15)–(18): the claim that the multi-dimensional reflection at the no-arbitrage boundary contributes no extra local-time term to the log-wealth process is not verified. The extension from the one-dimensional constant-product case requires an explicit Itô–Tanaka expansion or quadratic-variation calculation for the reflected process; without it the growth-rate formula may miss a correction proportional to the local time at the boundary.

    Authors: We agree that the multi-dimensional extension requires explicit verification. In the revised manuscript we will add a detailed Itô–Tanaka expansion (or equivalent quadratic-variation calculation) for the reflected diffusion to confirm that the local-time contribution to the log-wealth process remains zero, thereby rigorously supporting the growth-rate formula. revision: yes

  2. Referee: [§3.2] §3.2, definition of the reflection barrier and fee parameters: these quantities are defined via the same instantaneous-arbitrage condition that supplies the drift of the diffusion. The manuscript should show that the resulting growth rate is invariant under small perturbations of the barrier or provide an external benchmark (e.g., discrete-time simulation) to confirm the formula is not an artifact of this modeling choice.

    Authors: The barrier and fee parameters are defined directly from the instantaneous no-arbitrage condition, which is the core modeling assumption. To address the concern we will include, in the revision, either a first-order perturbation analysis showing invariance of the long-term growth rate or discrete-time Monte Carlo simulations as an external benchmark. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends external model without reducing to self-input by construction

full rationale

The paper models G3M dynamics via stochastic reflected diffusions under an arbitrage-driven market assumption and derives the long-term log-growth rate of LP wealth as a mathematical consequence of that process. This extends the Tassy-White constant-product analysis to weighted G3Ms without any quoted step that redefines the target growth rate in terms of itself, renames a fitted parameter as a prediction, or relies on a load-bearing self-citation whose content is unverified. The modeling choice (reflection at no-arbitrage boundaries) is an explicit assumption rather than a tautology that forces the final formula; external benchmarks or independent verification of the SDE solution would falsify the result. No load-bearing step collapses to an input by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that continuous arbitrage produces reflected diffusion dynamics and on the modeling choice to treat fees and reflection barriers as given inputs; no free parameters or invented entities are identifiable from the abstract alone.

axioms (1)
  • domain assumption Market dynamics under continuous arbitrage are captured by stochastic reflected diffusion processes
    Invoked to analyze the G3M model and extend the constant-product case

pith-pipeline@v0.9.0 · 5640 in / 1147 out tokens · 25451 ms · 2026-05-24T03:54:22.514952+00:00 · methodology

discussion (0)

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Reference graph

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