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Automated Market Making and Loss-Versus-Rebalancing

Jason Milionis, Ciamac C. Moallemi, Tim Roughgarden, Anthony Lee Zhang

arXiv:2208.06046v6q-fin.MFmath.OCq-fin.PMq-fin.PRq-fin.TR

TL;DR

Liquidity provision in AMMs raises open questions about its risks and returns and the link between theory and evidence. The paper develops a continuous-time decomposition of LP returns and finds that 99.991% of Uniswap v2 ETH-USDC pool P&L variance reflects beta exposure to ETH prices.

  • Problem

    Evidence remains limited on AMM liquidity-provision risks and returns and on the link between theoretical and empirical AMM research.

    Source: §1 · p. 2, §8.2 · p. 44

  • Method

    The paper develops a continuous-time model decomposing LP returns into beta-like market-risk exposure and alpha-like microstructural forces.

    Source: §1 · p. 2, §11 · p. 49

  • Results

    99.991% of Uniswap v2 ETH-USDC pool P&L variance is driven by beta exposure to ETH prices.

    Source: §1 · p. 4

  • Takeaways & Limitations

    LP losses include price-slippage losses when external prices move and AMM quotes become stale.

    Source: §10 · p. 47

    The model does not cover the generalized running-cost process that can arise from the pool value process's concavity.

    Source: §6 · p. 32

Abstract

from arXiv · show

Automated Market Makers (AMMs) are both liquidity sources and investment vehicles for market participants. This paper analyzes the risks and returns of liquidity provision (LP) investments in AMMs. In a continuous-time model, we show that LP returns decompose into a beta-like component reflecting market risk exposure, and an alpha-like component reflecting microstructural forces: accrued fees minus losses to arbitrageurs. Applying our decomposition to the Uniswap v2 ETH-USDC pool, we find that over 99.991\% of LP return variance is driven by beta exposure to market risk

1. Introduction

This section studies AMM liquidity provision as an investment, developing a model-based decomposition of LP returns into market-risk exposure and microstructural effects. It applies the framework to Uniswap v2 ETH-USDC, discusses loss-versus-rebalancing as a measure of fees minus arbitrage costs, and situates the analysis within related literature and efforts to reduce LVR.

1. Introduction

  • AMMs serve simultaneously as trading venues and investment vehicles: LPs earn trading fees while bearing risks from their contributed pool capital.In the model, noise traders generate fees, whereas arbitrageurs exploit stale AMM quotes when centralized-exchange prices move.Source: §1 · pp. 2–3
  • The paper decomposes LP returns into a beta-like rebalancing strategy for underlying asset exposure and an alpha-like residual comprising fees minus adverse-selection losses.The rebalancing strategy is the quadratic-variation-minimizing projection onto risky-asset price movements, while the residual captures returns not mechanically driven by beta exposure.Source: §1 · pp. 2–3
  • The rebalancing strategy isolates directional market exposure and can be perfectly delta-hedged, whereas the alpha-like component can be positive or negative.The rebalancing component is a risk-neutral martingale; the alpha-like component reflects the difference between accrued noise-trader fees and arbitrageur losses.Source: §1 · p. 3
  • Subtracting the rebalancing strategy reduces LP-return variance by four orders of magnitude, with similar results for rebalancing frequencies from one minute to one hour.Longer rebalancing intervals increase volatility, while better approximations converge toward trading fees minus loss-versus-rebalancing rather than zero volatility.Source: §1 · p. 4
  • For the USDC pair, 99.991% of pool P&L variance is driven by beta exposure to ETH prices, while costs represent only 0.009%.Using raw P&L as the y-variable would lead to around 108 times larger coefficient standard errors, requiring around 11,664 times more data for similar coefficient standard errors.Source: §1 · p. 4
  • The framework addresses impermanent loss’s dependence on arbitrary start points by frequently updating the holdings benchmark and aggregating loss across short intervals.Impermanent loss may fail to aggregate over time and can change sign with the chosen starting point.Source: §1 · pp. 4–5
  • AMM LP positions connect to static European options, dynamic delta-hedging strategies, and generalized variance swaps weighted by marginal liquidity.The introduction frames reserves through the static payoff of a pool value function and identifies rebalancing with delta-hedging.Source: §1 · p. 5

1.1. Related Literature

  • Dynamic LP returns contain an “alpha-like” microstructural component and a “beta-like” market-risk component, whereas one-period models lack meaningful market risk.The beta-like component reflects a mechanical trading strategy rather than returns specific to the AMM as an investment.Source: §1.1 · p. 6
  • “Many-interval” LP loss measures hedged LP returns and converges to the alpha-like component, namely fees minus arbitrage costs.The paper recommends this as the generally appropriate loss measure and calls it loss versus rebalancing.Source: §1.1 · p. 6
  • “One-interval” impermanent loss combines microstructural forces with predictable risky-asset exposure, which empirical analysis suggests can be much larger than the former.The paper recommends against this metric unless researchers explicitly justify including market risk in AMM LP returns.Source: §1.1 · p. 6
  • The empirical approach is designed to measure LP returns without imposing an equilibrium model of AMM liquidity and remains usable despite stylized-model assumptions.The authors identify the empirical decomposition as their main contribution to the literature.Source: §1.1 · p. 7
  • For general constant-function market makers, LP returns decompose path by path into rebalancing-strategy returns plus noise-trader fees minus LVR.The paper interprets LVR as arbitrageur profits and uses the decomposition to measure LP losses empirically.Source: §1.1 · p. 7
  • Related work includes liquidity market makers [Elsts, 2023, CrocSwap, 2022].Source: §1.1 · p. 8
  • Industry groups are building AMMs that use the paper’s insights to attempt to reduce or eliminate LVR [Canidio and Fritsch, 2023, Cata Labs, 2023, Fenbushi Capital, 2023].Source: §1.1 · p. 8
  • Additional groups pursuing these AMM designs include Algebra Protocol, 2023, Klages-Mundt and Schuldenzucker, 2022, Aori Protocol, 2023, and Adams et al., 2024.Source: §1.1 · p. 8

2. Institutional Background

AMMs let users trade cryptoassets directly on blockchain through smart-contract-controlled inventories, avoiding many centralized-exchange costs. Liquidity providers supply that inventory for proportional trading fees, creating the paper’s central question of how to evaluate liquidity provision’s costs and benefits.

AMM trading mechanism

  • AMMs are smart contracts that enable direct cryptoasset trades on the blockchain rather than through centralized exchanges.Their inventory exchanges one asset for another in a single atomic transaction according to blockchain code.Source: §2 · pp. 8–9

AMM advantages

  • AMMs circumvent centralized-exchange costs including custody and credit risk, access and listing barriers, opaque rules, and deposit, withdrawal, or settlement frictions.Anyone can initiate a transaction, and trade completion ensures the sold asset is not lost without receiving the purchased asset.Source: §2 · p. 9

Liquidity provision

  • Any participant can become an LP by contributing both assets to an AMM inventory, receiving trading fees proportionate to their pool share.LPs can withdraw assets corresponding to their share, although inventory amounts generally differ from their initial contribution as trades occur and fees accumulate.Source: §2 · p. 10

LP economics

  • The paper’s core economic problem is evaluating the costs and benefits of providing liquidity to AMMs.Withdrawn asset quantities generally differ from initial contributions because inventory adjusts to trade requests and fees are collected.Source: §2 · p. 10

3. Model

This section develops a continuous-time model of AMM liquidity provision with passive LPs, noise traders, and arbitrageurs, using liquidity and fees as primitives. It decomposes LP returns into market-risk and microstructural components and motivates an empirical framework while acknowledging simplifying assumptions about market depth, trading frictions, and DEX market share.

3. Model

  • The CFMM pool is defined by a bonding-function level set, with the constant-product example satisfying √xy = L.A smart contract moves reserves between feasible points when an agent contributes the reserve difference.Source: §3 · p. 11
  • Arbitrageurs continuously exploit pool mispricing, moving reserves to the bonding-curve point whose slope equals −P at the external price P.The model distinguishes rebalancing arbitrage after external-price movements from reversion arbitrage after noise-trader activity, but quantifies only rebalancing-arbitrage profits.Source: §3 · pp. 11–12
  • Noise traders trade only in the CFMM for idiosyncratic reasons, while arbitrageurs immediately offset their price impact, leaving LPs with a flow of fees.Fees are paid in numéraire units and represented by the cumulative process FEE_t.Source: §3 · pp. 12–13

3.1. Discussion of Assumptions

  • The continuous-time formulation captures dynamic market risk that single-period models miss, while its basic intuitions also hold in a two-step binomial tree.The paper uses continuous time for simpler expressions and does not construct an equilibrium model; any equilibrium models generating the same L and FEE_t imply the same LP P&L.Source: §3.1 · pp. 13–14
  • LP profit and loss decomposes into a beta-like market-risk component and an alpha-like microstructural component.The framework treats liquidity level L and cumulative fee process FEE_t as primitives; together with the bonding curve and volatility process, they determine the alpha-like return component.Source: §3.1 · pp. 13–14
  • The model assumes an infinitely deep CEX, but remains a reasonable approximation when any individual DEX represents only a small share of total market liquidity.Outside-DEX liquidity includes limit order books, OTC desks, and other DEXes; otherwise, the model can still serve as a cross-sectionally consistent, arbitrage-free conceptual benchmark.Source: §3.1 · p. 14
  • The framework abstracts from trading fees, gas fees, block-based execution, noise-trader fee conventions, and LP-share minting or burning.The empirical application in Section 7 relaxes the noise-trader fee and LP-share assumptions.Source: §3.1 · p. 15
  • Ignoring arbitrageurs’ fees overstates their profits, while gas races redirect some CEX-DEX arbitrage profits toward block miners.A more careful analysis of fees is deferred to follow-on work [Milionis et al., 2023a].Source: §3.1 · p. 15
  • The stylized model prioritizes a tractable decomposition and empirical methodology over a fully realistic representation of AMM trading.Its assumptions follow the simplifying tradition of classical asset-pricing models, including CAPM [Sharpe, 1964, Lintner, 1965] and APT.Source: §3.1 · p. 15
  • LP returns decompose into market-risk and microstructural components that should be separated empirically.Researchers need not accept all model assumptions to apply the measurement framework.Source: §3.1 · p. 15

4. Market Risk “Beta” and Microstructural “Alpha”

This section decomposes AMM pool and LP performance into a replicable beta-like market-risk component and a residual alpha-like loss-versus-rebalancing component arising from arbitrage-related price slippage. It characterizes LVR and the scope of the decomposition across AMM demand curves.

Scope and generality

  • The LVR definition and characterization apply to any AMM with a locally smooth demand curve, including concentrated-liquidity AMMs such as Uniswap v3, not only CFMMs.The framework can also be extended to more general non-locally-smooth settings with additional treatment.Source: §4 · p. 17

Market-risk and LVR decomposition

  • The rebalancing strategy breaks even in expectation under the risk-neutral measure Q and earns expected returns only when the risky asset has nonzero risk premia.Its profits therefore represent the market-risk component rather than arbitrage-induced losses.Source: §4 · p. 19
  • LVR equals the CFMM’s losses to arbitrageurs relative to a rebalancing strategy that matches its risky-asset holdings but trades at CEX prices.The rebalancing strategy captures the CFMM’s market-risk exposure, while the residual is defined as loss-versus-rebalancing.Source: §4 · pp. 18–19

Loss-versus-rebalancing

  • LVR is nonnegative, non-decreasing, and equal to cumulative profits earned by rebalancing arbitrageurs.The instantaneous LVR rate is always positive, making cumulative LVR predictable and increasing over time.Source: §4 · p. 19
  • LVR is larger when price volatility is higher or when marginal liquidity |x∗′(P)| is larger, because the CFMM trades more after price movements.Marginal liquidity is defined as |x∗′(P)|, the amount of risky-asset trading induced by a given price movement.Source: §4 · p. 20

CFMM design

  • The CFMM’s local bonding-curve curvature determines LVR, so invariants with identical local convexity at a price have identical LVR there.Flatter bonding curves have higher marginal liquidity in the model.Source: §4 · p. 20

Loss-versus-rebalancing

  • LVR arises from price slippage: CFMMs execute trades at worse-than-market prices, producing losses equivalent to arbitrageur profits.A rebalancing strategy makes the same risky-asset trades at CEX prices and therefore does not lose from rebalancing itself.Source: §4 · pp. 21–22

Microstructural losses

  • CFMM arbitrage losses are driven by CEX price volatility and quantified from volatility together with the CFMM’s mechanical price-quantity schedule.Market prices reach CFMM prices through arbitrage, unlike informed trading based on knowledge of future prices.Source: §4 · p. 22

Theorem 1

  • LP P&L decomposes into a beta-like component reflecting directional market risk and an alpha-like component reflecting microstructural forces.The beta-like component is replicable by following the CFMM’s trades on a centralized exchange, whereas the alpha-like component is comparatively unique to the CFMM.Source: §4 · pp. 22–23

Microstructural alpha

  • The alpha-like component captures fee income from noise trade minus losses to informed arbitrage and can be isolated by subtracting rebalancing-strategy profits from LP P&L.The model treats LPs as fully passive; practical attempts to withdraw liquidity or snipe quotes are outside the paper’s scope.Source: §4 · pp. 22–23

5. The Rebalancing Strategy as a Projection

This section interprets rebalancing as the quadratic-variation-minimizing projection of AMM profits or pool value onto the risky asset’s price path. It develops intuition through a binomial example and formalizes the resulting beta-like strategy and alpha-like residual in a frictionless model.

5. The Rebalancing Strategy as a Projection

  • The projection analogy treats rebalancing as capturing AMM pool-value changes explained by movements in the underlying asset’s price, paralleling beta estimation.The section develops intuition with a two-step binomial tree before proving the result in continuous time.Source: §5 · p. 23

5.1. Binomial Tree

  • In the binomial tree, LVH decomposes into a risk-neutral-martingale rebalancing gap and a strictly increasing CFMM slippage gap.Removing the first gap denoises directional risky-asset exposure without changing risk-neutral expected losses; the CFMM profits less from convergence and loses more on divergence than rebalancing.Source: §5.1 · pp. 24–25

5.2. Continuous Time

  • The rebalancing strategy is essentially the unique market-price benchmark minimizing loss-process quadratic variation, yielding predictable LVR with zero quadratic variation.Any other benchmark has positive quadratic variation unless its risky-asset holdings essentially equal the rebalancing strategy.Source: §5.2 · p. 27
  • The beta-like rebalancing strategy is the quadratic-variation-minimizing projection of the AMM pool value process onto changes in P_s.This parallels factor-model residualization by removing variation attributable to market or factor risk.Source: §5.2 · p. 28
  • Subtracting the rebalancing strategy yields an alpha-like residual free of linear price exposure to the underlying asset.The decomposition concerns realized risk exposure; when benchmark and rebalancing holdings differ, their realized returns differ because of differential exposure to risky price movements.Source: §5.2 · p. 28
  • The loss-versus-benchmark process is a super-martingale whose unique Doob–Meyer decomposition separates the benchmark-independent LVR compensator from a benchmark-dependent martingale component.LVR is the compensator, while Δ(¯x) defines the martingale component.Source: §5.2 · p. 28
  • The paper presents these decompositions as conceptual P&L tools in a frictionless model, leaving frictional costs and implementation details for future accounting.The AMM delta is simply its asset holdings x∗(P_s), unlike option deltas that depend on unobserved volatility.Source: §5.2 · pp. 28–29

6. Examples

The examples apply LVR analysis to Balancer, Uniswap v2 and v3, and linear market makers. They show how LVR depends on invariant choice, liquidity concentration, and smoothness assumptions.

Example 2: Weighted Geometric Mean Market Maker / Balancer

  • For weighted geometric mean market makers, normalized instantaneous LVR is constant, uniquely characterizing them among CFMMs up to a minor caveat, and is maximized at θ = 1/2.The θ = 1/2 case corresponds to the constant product market maker.Source: §6 · p. 29

Example 3: Constant Product Market Maker / Uniswap v2

  • For Uniswap v2, instantaneous LVR per dollar of pool value is 1/8 times instantaneous variance, implying approximately 3.125 bp of daily loss when ETH-USDC volatility is 5%.This provides a directly applicable empirical formula for the constant product market maker.Source: §6 · p. 30

Example 4: Uniswap v3 Range Order

  • Uniswap v3 range orders have the same instantaneous LVR as the constant product example, but narrower ranges reduce pool value and can make LVR per reserve dollar arbitrarily high.Pool value tends to zero as the range endpoints converge to the current price.Source: §6 · p. 30

Example 5: Uniswap v3 Pool

  • In a Uniswap v3 pool, only in-range orders contribute to LVR, so aggregate in-range liquidity is a sufficient statistic for the pool’s range-order composition.The pool aggregates liquidity across multiple range orders with potentially different price ranges.Source: §6 · pp. 31–32

Example 6: Linear Market Maker / Limit Order

  • A linear market maker is statically equivalent to a resting limit order but dynamically differs because it remains in the pool after crossing its limit price.Its value function is not smooth at P = K, so the paper’s standard LVR characterization does not apply; a local-time generalization is not pursued.Source: §6 · p. 32

7. Empirical Analysis

This section empirically analyzes Uniswap v2 WETH-USDC LP returns using data from August 1, 2021 to July 31, 2022, distinguishing market-risk exposure from microstructural effects. It examines hedged LP returns, their relation to fees and adverse-selection costs, and the equivalence between discrete-time delta-hedged profits and markout profits under matched measurement conventions.

7.1. Methodology

  • The study applies its decomposition to the Uniswap v2 WETH-USDC pool over August 1, 2021–July 31, 2022, estimating LP P&L from observed reserves, prices, trades, mints, and burns.Discrete rebalancing strategies approximate the model’s market-risk component, while LP P&L is valued using observed pool holdings and CEX prices.Source: §7.1 · pp. 33–34

7.2. Empirical Results

  • 99.991% of unhedged LP-return variance reflects ETH price exposure, while fees and adverse selection account for 0.009%.One-minute hedging reduces return standard deviation to less than 1% of unhedged LP P&L; these comparisons are reported in Figure 5 and Table 1.Source: Figure 5 · p. 35, §7.2 · p. 36, Table 1 · p. 36
  • Hedged LP P&L becomes more volatile as rebalancing intervals lengthen because coarser strategies leave more market-risk variance within the hedged series.The fixed-start-point “impermanent loss” benchmark has volatility comparable to the unhedged pool because it accumulates market risk after its initial hedge.Source: §7.2 · pp. 36–37

7.3. Testing the Theory

  • More frequent rebalancing makes hedged LP P&L track fees minus LVR more closely, supporting convergence to the alpha-like microstructural residual.The gap between hedged P&L and fees minus LVR is smaller at higher rebalancing frequencies, as shown in Figure 7.Source: §7.3 · pp. 37, 39, Figure 7 · p. 38

7.4. Implications for Empirical Analysis of AMM Returns

  • Failing to remove market risk causes empirical LP-return analyses to reflect market exposure rather than microstructural effects, with standard errors roughly 108 times larger than for hedged returns.Matching hedged-return precision with unhedged returns would require around 108^2 = 11,664 observations in the stated best-case regression setting.Source: §7.4 · p. 39
  • Market-risk variance can make even weak correlations between covariates and market risk induce omitted-variable bias that overwhelms relationships with hedged LP returns.Unhedged LP returns may instead let microstructural signals—fees minus adverse-selection costs—be overwhelmed by market noise.Source: §7.4 · pp. 39–40
  • The authors recommend using hedged LP returns as regression inputs, while noting that the analysis conceptually decomposes profits rather than precisely estimating practical hedged-LP profits.Constructing the rebalancing-strategy profits requires only the CFMM LP’s risky-asset holdings and the CEX risky-asset price; the decomposition assumes frictionless rebalancing, although implementation and margin costs exist.Source: §7.4 · pp. 39–40
  • The results appear robust to rebalancing frequency, but practical delta-hedging faces spreads, trading fees, and margin costs.The stated goal is conceptual decomposition into market-risk and residual components, analogous to alpha-beta decompositions, rather than precise practical profit estimation.Source: §7.4 · p. 40

7.5. Markouts

  • Delta-hedged LPing provides a microfoundation for markout-style analysis.Industry participants use markouts to evaluate CFMM LP P&L by comparing trade prices with later prices, while the paper relates this practice to delta-hedging.Source: §7.5 · p. 40
  • Discrete-period delta-hedged LP profit is exactly equivalent to markout profit when both use the same interval frequency and CEX end-of-period prices.The approaches differ in whether marking occurs at a fixed future offset after each trade or at the interval’s ending price, and they are likely close for short offsets or intervals.Source: §7.5 · p. 40

8. “Impermanent Loss”

This section distinguishes fixed-start “impermanent loss” or loss versus holding (LVH) from loss versus rebalancing (LVR), examining how benchmarking choices affect the interpretation of liquidity-provider performance. It contrasts market-risk exposure captured by fixed-start measures with the additive, path-consistent specification of LVR and emphasizes the need to justify metric choices.

8.1. Problems with Loss-Versus-Holding

  • LVH can be positive on each individual step (LVH0,1 > 0 and LVH1,2 > 0) yet equal zero over the whole interval when the t = 2 portfolio equals the t = 0 portfolio.Thus, LVH is not monotonic, whereas LVR is monotonically non-decreasing because adverse-selection costs accumulate irreversibly over time.Source: §8.1 · pp. 41–42
  • LVH’s change can have either sign because its incremental profit depends on the arbitrarily chosen reference point, making AMM gains or losses ambiguous over an interval.The same future holdings change can make LVH increase relative to one benchmark and decrease relative to another.Source: §8.1 · p. 42
  • LVH is path-independent, whereas LVR is path-dependent and accumulates faster during high-volatility periods, matching the greater cost of adverse selection under volatility.LVH depends only on initial and final prices, even though informed-trading costs should depend on the intervening price path.Source: §8.1 · p. 42
  • LVH conflates beta-like market risk with alpha-like adverse-selection costs, while LVR absorbs market risk through rebalancing and isolates the microstructural component.The beta-like component is orders of magnitude larger than the alpha-like component in variance terms and can overwhelm microstructural signals.Source: §8.1 · p. 42
  • LVR is the time-discretized, path-consistent version of LVH because it updates the benchmark portfolio at every interval as prices and holdings evolve.Rebenchmarking against holdings at the start of each short interval guarantees additivity and yields the definition of LVR.Source: §8.1 · p. 43

8.2. “Impermanent Loss” in the Empirical AMM Literature

  • Over a hundred times greater variance distinguishes fixed-start-point “impermanent loss” from periodically updated measures, with the former mainly reflecting ETH-price movements rather than microstructural forces.The periodically updated measure converges to fees minus adverse selection costs at reasonably high update frequencies, while the fixed-start measure can be zero when the periodically updated measure is positive.Source: §8.2 · p. 44
  • Market risk still dominates microstructural forces at daily and weekly rebalancing frequencies.This finding motivates distinguishing market exposure from microstructural effects when interpreting AMM loss measures.Source: §8.2 · p. 45
  • The paper calls the updated-holdings measure “loss versus rebalancing” rather than “impermanent loss” because the measures are mathematically and economically distinct.Different measures serve different purposes, and the paper does not claim LVR is universally best.Source: §8.2 · p. 45
  • Fixed-start “impermanent loss” equals LVR plus the market-risk exposures of a well-defined trading strategy.The market-risk component may be much larger than microstructural effects in variance terms.Source: §8.2 · p. 45
  • Papers using fixed-start “impermanent loss” should justify including market risk and specify their start point and updating frequency.These decisions should be stated explicitly when implementing metrics labeled “impermanent loss.”Source: §8.2 · p. 45

9. Option Pricing

CFMM LP positions provide volatility exposure that can be understood through static European options, dynamic rebalancing, and variance swaps. The paper establishes these equivalences and explains how LVR responds to volatility and marginal liquidity.

9. Option Pricing

  • CFMM LP positions are equivalent to volatility trades through static options, dynamic rebalancing, and variance swaps, with LVR increasing when volatility is high.The paper demonstrates these equivalences in a simple two-step setting.Source: §9 · p. 45

9.1. Static Options Positions

  • Ignoring fees, an AMM LP payoff over a finite horizon can be replicated by shorting European options, while expected LVR equals the value of the options given away.LVR increases with underlying volatility and with the AMM’s marginal liquidity because more aggressive bonding curves replicate larger option portfolios; the LP-plus-options position isolates trading fees relative to option premia.Source: §9.1 · pp. 45–46

9.2. Dynamic Trading Strategies

  • AMM rebalancing sells the asset after price increases and buys after decreases, profiting from reversals but losing when prices diverge further from the initial price.For a random-walk risky-asset price, the strategy breaks even on average, while the CFMM exactly offsets convergence gains against divergence losses.Source: §9.2 · pp. 46–47

9.3. Variance Swaps

  • The instantaneous LVR expression corresponds to the floating leg of a continuously sampled generalized variance swap weighted by marginal liquidity.The variance component is the price’s instantaneous quadratic variation, while |x*′(P)| measures marginal liquidity at price P.Source: §9.3 · p. 47

10. Discussion and Implications

The discussion identifies oracle-based pricing and privileged-arbitrage designs as ways to reduce or redistribute LVR, while noting implementation risks. It also argues that bonding functions affect LVR only through marginal liquidity, motivating further AMM design research.

Oracle-based AMMs

  • An oracle-based AMM could substantially reduce or eliminate LVR by quoting near the CEX price, but this depends on oracle accuracy and remains vulnerable to manipulation.Such pricing could approach the payoff of the rebalancing strategy.Source: §10 · p. 48

Authorized participants

  • Privileged authorized participants could capture arbitrage profits and redistribute them to LPs, including through first-execution rights or zero-fee trading.APs can profit from arbitrarily small price movements, unlike non-AP wallets that must overcome trading fees.Source: §10 · p. 48
  • Periodic auctions could sell authorized-participant rights and redistribute expected arbitrage profits, including LVR, to pool LPs.In the zero-fee design, APs may also capture reversion-arbitrage profits, so LVR is a lower bound on wallet revenue.Source: §10 · p. 48

Alternative market designs

  • Batch auctions and related request-for-quotes mechanisms are additional designs intended to reduce or eliminate LVR by changing how orders enter AMMs.The passage identifies these mechanisms as existing or discussed approaches rather than reporting a measured effect.Source: §10 · p. 48

Bonding functions

  • Bonding functions affect LVR only insofar as they change marginal liquidity, so designs with equivalent marginal liquidity have equivalent infinitesimal LVR.Price-dependent marginal liquidity may nevertheless be desirable for pairs such as stablecoins; Uniswap v3 is cited as an example of a universal AMM design.Source: §10 · p. 49

11. Conclusion · Appendix

The paper frames AMM liquidity provision as an investment and develops a framework that decomposes LP returns into market-risk and microstructural components. These components can be separated empirically to inform LP-return measurement.

11. Conclusion

  • AMMs enable participants both to trade and to contribute capital to liquidity provision.Source: §11 · p. 49
  • LP positions function essentially as investments, analogous to mutual funds or ETFs in traditional financial markets.Source: §11 · p. 49
  • The paper constructs a framework for understanding returns to investments in AMM LP positions.Source: §11 · p. 49
  • LP P&L decomposes into a beta-like component reflecting market risk and an alpha-like component reflecting fee revenue versus adverse-selection losses.Source: §11 · p. 49
  • The alpha-like component captures the microstructural balance between fee revenue and adverse-selection losses.Source: §11 · p. 49
  • The beta component can be eliminated by subtracting rebalancing-strategy profits from LP P&L.Source: §11 · p. 49
  • The results have implications for measuring returns to LP positions as investments.Source: §11 · p. 49

A. Proofs · A.1. Proof of Lemma 1

The proof invokes a theorem attributed to Bertsekas (1971) and establishes the third part using the concavity of V(·), characterized as a pointwise minimum of affine functions.

A.1. Proof of Lemma 1

  • The proof invokes Danskin’s theorem.Source: §A.1 · p. 54
  • The third part follows from the concavity of V(·).Source: §A.1 · p. 54
  • V(·) is concave as a pointwise minimum of a collection of affine functions.Source: §A.1 · p. 54

A.2. Proof of Lemma 2 … B.1. LVR and Impermanent Loss in a Two-Step Binomial Tree

The appendices establish LVR as the cumulative profit of rebalancing arbitrageurs, relate marginal liquidity to CFMM curvature, and illustrate LP payoffs through a two-step binomial tree. The binomial example links price mean reversion and divergence to option-like payoff patterns.

A.2. Proof of Lemma 2

  • The continuous-time arbitrage-profit expression follows by telescoping the discrete sum, applying summation-by-parts, and passing to the Itô-integral limit under Assumption 1 Part 3.The limiting argument also uses τ_N → T, P_{τ_N} → P_T, and x*(P_{τ_N}) → x*(P_T).Source: §A.2 · p. 55
  • Cumulative profits of rebalancing arbitrageurs from time 0 to T equal LVR.The proof uses a discrete approximation with sequential arbitrageurs, then takes the vanishing-mesh limit to an Itô integral.Source: §A.2 · pp. 54–55

A.3. LVR, Marginal Liquidity, and Bonding Function Curvature

  • Flatter, more linear CFMM invariant curves imply greater marginal liquidity dx/dP.Marginal liquidity is expressed through derivatives of the bonding function and is related to the curvature of CFMM invariant curves.Source: §A.3 · pp. 55–56

A.3.1. Derivation of (32)

  • Equation (32) is derived by differentiating the expenditure-minimization first-order conditions with the implicit function theorem and solving the resulting system for dx/dP.The derivation proceeds from the Lagrangian and FOCs through their derivatives and substitutions.Source: §A.3.1 · pp. 56–58

B.1. LVR and Impermanent Loss in a Two-Step Binomial Tree

  • In the two-step binomial tree, rebalancing outperforms buy-and-hold when prices mean-revert but underperforms when prices diverge.This payoff pattern is similar to a short European straddle or strangle.Source: §B.1 · p. 59
  • Ignoring fees, a CFMM LP earns zero when prices return to their starting point and resembles giving away a straddle without collecting upfront option premia.The analogy reflects dynamic replication of European straddles and strangles by selling when prices rise and buying when prices fall.Source: §B.1 · p. 59

B.2. Weighted Geometric Mean Market Makers · B.3. Multi-Dimensional Generalization · C. Data and Measurement

The paper characterizes weighted geometric mean market makers as essentially the only CFMMs with constant instantaneous LVR per dollar of pool value, and extends its results to multiple assets. In the multi-dimensional setting, LVR is shown to be non-negative, non-decreasing, and predictable.

B.2. Weighted Geometric Mean Market Makers

  • Constant instantaneous LVR per dollar of pool value uniquely characterizes weighted geometric mean market makers among CFMMs.The quantity ℓ(σ, P)/V(P) is constant for weighted geometric mean market makers, and the theorem establishes they are essentially the only CFMMs with this property.Source: §B.2 · p. 59
  • The two degrees of freedom are intuitive, since θ and 1 −θ are exchangeable in (18).Source: §B.2 · p. 60
  • The characterization derives from an ODE whose solution combines powers P^θ and P^(1−θ), subject to concavity of V(P).The calculation uses the limit condition V(z) → 0 as z → −∞, and the quantity under the root must be non-negative because V(P) must be everywhere concave.Source: §B.2 · p. 60

B.3. Multi-Dimensional Generalization

  • The multi-dimensional generalization considers reserves in n ≥ 2 assets, with prices following geometric Brownian motion and return covariance Σ_t ∈ R^{n×n} satisfying Σ_t ⪰ 0; an optimal solution x*(P) exists for all P ∈ R^n.Source: §B.3 · pp. 60–61
  • The generalized pool value function satisfies V(P) ≥ 0 and admits the corresponding multi-dimensional value representation.These properties follow under the stated bonding-function and regularity assumptions.Source: §B.3 · p. 61
  • In multiple dimensions, loss-versus-rebalancing takes the theorem’s generalized form; with Σ = σ^2I, LVR is non-negative, non-decreasing, and predictable.The special case corresponds to identically and independently distributed assets.Source: §B.3 · pp. 61–62

C.1. Data … 12 ORDER BY 1 ASC

The appendix constructs minute-level inputs for the Uniswap v2 WETH-USDC analysis, combining Binance prices with Dune-extracted pool activity, reserve data, fees, rebalancing returns, and LVR estimates. It also documents the SQL queries used to extract mints, burns, swaps, and synchronized reserves.

C.1. Data

  • Minute-level USDC-ETH close prices come from the Binance API, while Uniswap v2 WETH-USDC pool data come from Dune Analytics.The pool data support the subsequent calculations of reserves, liquidity-provider P&L, fees, rebalancing returns, and LVR.Source: §C.1 · p. 62
  • Pool P&L values reserves at each minute’s closing price and incorporates burned assets, minted assets, and transaction fees paid directly into reserves.Pool value is defined as V_t = y_t + P_t x_t.Source: §C.1 · pp. 62–63
  • Rebalancing benchmarks hold the pool’s starting ETH quantity throughout each period, with returns computed at multiple rebalancing frequencies.For a daily frequency, the benchmark uses the pool reserves at the start of the day for every minute in that day.Source: §C.1 · p. 63
  • Fees are estimated minute by minute from gross asset flows using Uniswap v2’s fixed 30bps rate, with fee values marked to the period’s price.The same swap-based accounting also captures flash-loan revenues because flash loans are recorded as swaps.Source: §C.1 · p. 63
  • Realized daily volatility uses Binance USDC-ETH prices sampled every 60 minutes to construct a discrete one-minute approximation of LVR.The appendix combines LP P&L, rebalancing P&L, fees, and LVR to calculate cumulative returns.Source: §C.1 · pp. 63–64

C.2. Dune SQL Queries

  • The Dune SQL queries extract Uniswap v2 Pair_evt_Mint and Pair_evt_Burn events for the specified WETH-USDC contract, ordered by block number and event index.These queries provide the mint and burn records used in the data construction.Source: §C.2 · p. 64

1 SELECT

  • The swap query groups Uniswap v2 Pair_evt_Swap records by minute and computes summed amount0In, amount1In, amount0Out, and amount1Out fields.Source: §1 · pp. 64–65
  • The reserve extraction identifies the latest reserve0 and reserve1 observations from synchronized pool events grouped by day.The query filters the same WETH-USDC contract and selects reserve records using block and event ordering.Source: §1 · p. 65

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