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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

The paper asks about the risks and returns of liquidity provision in AMMs and develops a continuous-time model to decompose LP returns. Applying the decomposition to Uniswap v2 ETH-USDC, it finds that 99.991% of pool P&L variance is driven by beta exposure to ETH prices.

  • Problem

    The paper asks what the risks and returns of LP investments in AMMs are and addresses a missing link between theoretical and empirical AMM literature.

  • Method

    The paper develops a continuous-time model in which LP returns decompose into market-risk exposure and a benchmark-dependent martingale component, with LVR as the benchmark-independent compensator.

  • Results

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

  • Takeaways & Limitations

    LP returns can be analyzed as a beta-like market-risk component alongside an alpha-like component from fees and losses to arbitrageurs.

  • Takeaways & Limitations

    The paper does not pursue the local-time and Itô-Tanaka-Meyer generalization needed for the broader running-cost process.

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

The paper develops a model-based decomposition of AMM LP returns into market-risk exposure and microstructural gains or losses. In the Uniswap v2 ETH-USDC pool, market risk explains nearly all observed P&L variance, motivating loss-versus-rebalancing as an empirical measure.

  • Contribution: LP returns decompose into a beta-like rebalancing strategy reflecting risky-asset exposure and an alpha-like component reflecting fees minus arbitrage losses.The rebalancing strategy is the market-risk component; the residual captures microstructural forces.
  • Method: The continuous-time model interprets rebalancing as the quadratic-variation-minimizing projection of LP returns onto risky-asset price movements.Loss-versus-rebalancing is the resulting projection residual.
  • Mechanism: AMM P&L reflects fees earned from pool trading and adverse-selection losses when arbitrageurs exploit stale quotes after external price moves.The loss expression depends on asset-price variance and the AMM’s marginal liquidity.
  • Empirical result: Subtracting rebalancing-strategy profits reduces LP-return variance by four orders of magnitude, with similar results for rebalancing intervals from 1 minute to 1 hour.Longer intervals produce higher volatility, consistent with residual risk approaching trading fees minus LVR.
  • Empirical result: 99.991% of Uniswap v2 ETH-USDC pool P&L variance is driven by beta exposure to ETH prices.After subtracting this component, fees minus adverse-selection costs account for 0.009% of variance.
  • Empirical implication: Using raw P&L in empirical regressions can inflate coefficient standard errors by about 108 times and require roughly 11,664 times more data for comparable precision.Even small correlations between covariates and market risk can create omitted-variable bias that overwhelms effects on AMM alpha.
  • Empirical implication: The paper argues that many-interval loss metrics approximate fees minus arbitrage costs, whereas metrics retaining market exposure mix microstructural forces with predictable asset-price exposure.Researchers are advised to justify explicitly whether market risk belongs in the LP-return measure for their question.

2. Institutional Background

AMMs provide on-chain asset exchange without centralized-exchange intermediation, while allowing individuals to supply the pool’s inventory as liquidity providers. Their accessibility and transparency create an alternative trading venue, but LPs must evaluate changing inventories, fees, and associated risks.

  • AMM structure: AMMs are smart contracts that let participants trade cryptoassets directly on a blockchain rather than through a centralized exchange.Their behavior is determined by blockchain code and trades transfer assets atomically.
  • Centralized exchanges: Centralized-exchange trading can involve custody and credit risk, account and identification requirements, asset-listing constraints, opaque priority rules, and fees or settlement delays.These costs motivate alternatives to centralized exchange trading.
  • AMM structure: AMMs reduce these barriers through open transaction access, atomic settlement without credit risk, and comparatively transparent pricing mechanisms.The Uniswap v2 contract is described as publicly available code that cannot be modified after deployment.
  • Liquidity provision: Liquidity providers contribute paired assets to an AMM inventory and receive trading fees proportionate to their pool share.They can withdraw their share, although the quantities of each asset generally differ from the original contribution as trades and fees change inventory.
  • Liquidity provision: The central LP problem is evaluating the costs and benefits of providing liquidity to AMMs.This evaluation concerns the economic consequences of changing pool inventories and collected fees.

3. Model

The model studies AMM liquidity provision in continuous time, with arbitrageurs aligning pool prices to an external market and noise traders generating fee flow. It uses this framework to separate LP returns into market-risk and microstructural components.

  • CFMM setting: The model represents a CFMM pool through reserves constrained by a bonding-function invariant.The constant product market maker is given as an example with invariant √xy = L.
  • Agents and price alignment: Arbitrageurs continuously monitor the pool and move reserves to exploit deviations from the external market price.At external price P, reserves move to the feasible-curve point whose bonding-curve slope equals −P.
  • Agents and price alignment: Noise traders trade only in the CFMM for idiosyncratic reasons, and arbitrageurs immediately offset their price impact.From the LP perspective, noise traders therefore contribute a flow of fees.
  • Arbitrage activity: The framework distinguishes rebalancing arbitrage after external-price movements from reversion arbitrage after noise-trader activity.The model quantifies the magnitude of these two forms of arbitrage activity.
  • Return decomposition: The continuous-time framework decomposes LP P&L into a beta-like market-risk component and an alpha-like microstructural component.The beta-like component reflects market exposure, while the alpha-like component reflects fees and adverse-selection losses to arbitrageurs.

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

The paper defines a rebalancing strategy that matches the CFMM’s risky-asset holdings while trading at external-market prices, isolating market exposure from execution effects. It then characterizes loss-versus-rebalancing as the nonnegative accumulation of price-slippage losses to arbitrageurs.

  • Scope of LVR: LVR applies beyond CFMMs to AMMs with locally smooth demand curves, including concentrated-liquidity AMMs such as Uniswap v3.The paper states that the characterization requires a fixed locally smooth demand curve over an instantaneous interval.
  • Return decomposition: LP P&L equals the change in pool value plus cumulative fees, while pool-value changes split into rebalancing-strategy profits and loss-versus-rebalancing.The residual term captures the difference between CFMM execution and the rebalancing benchmark.
  • Market Risk “Beta”: The rebalancing strategy makes the same net risky-asset trades as the CFMM but executes them at CEX prices.It therefore has the same exposure to risky-asset prices as the CFMM and represents the market-risk component.
  • Loss-versus-rebalancing: LVR is nonnegative, non-decreasing, and predictable, and cumulative rebalancing-arbitrage profits equal LVR.It measures losses arising because CFMM trades occur at worse prices than the rebalancing strategy.
  • Loss-versus-rebalancing: LVR increases with volatility and with the amount of risky asset traded when prices move.The amount traded is linked to the slope of the AMM demand curve and, for CFMMs, to bonding-function curvature.
  • Return decomposition: Theorem 1 decomposes LP P&L into a beta-like market-risk component and an alpha-like microstructural component.The beta-like term reflects directional exposure; the alpha-like term captures CFMM-specific microstructural tradeoffs.

5. The Rebalancing Strategy as a Projection

The paper interprets the rebalancing strategy as the beta-like projection of AMM profits onto risky-asset price movements. In continuous time, this strategy uniquely minimizes residual loss variation, leaving LVR as a predictable, price-slippage-driven component.

  • Conceptual setup: The rebalancing strategy projects AMM profits onto the risky asset’s price path, analogous to projecting stock returns onto market or factor returns.The paper first develops intuition with a two-step binomial tree and then proves the result in continuous time.
  • Binomial-tree intuition: The rebalancing strategy matches CFMM risky-asset holdings while trading at CEX prices, whereas the CFMM trades at worse AMM prices.The comparison isolates price slippage from the directional exposure generated by changing risky-asset holdings.
  • Binomial-tree intuition: The B-C payoff difference is a risk-neutral martingale, while the C-D gap is strictly increasing because the CFMM executes trades at worse prices.Thus, LVH decomposes into a market-price rebalancing component and an increasing execution-loss component.
  • Interpretation: Removing the B-C gap does not change risk-neutral expected losses; it removes directional risky-asset exposure and leaves losses driven by price slippage.This is the denoising interpretation of loss-versus-rebalancing.
  • Continuous-time result: The rebalancing strategy is essentially unique in minimizing residual loss quadratic variation, achieving zero quadratic variation and a predictable LVR process.Any benchmark whose risky-asset position differs from the CFMM position has positive quadratic variation on a sample path.
  • Continuous-time result: Subtracting rebalancing returns produces an alpha-like residual free of linear exposure to the underlying asset.The rebalancing strategy is the beta-like, quadratic-variation-minimizing projection of the AMM pool value process.

6. Examples

The examples apply the LVR framework to several market-maker designs, showing how normalized instantaneous LVR depends on bonding functions, liquidity ranges, and in-range liquidity. They also identify a smoothness limitation for linear market makers and note an unpursued generalization.

  • Weighted geometric mean market makers: For weighted geometric mean market makers, normalized instantaneous LVR per dollar of pool reserves is constant, with LVR maximized at θ = 1/2.The constant-product market maker is the θ = 1/2 case.
  • Constant-product market maker: For the constant-product market maker, loss per unit time as a fraction of mark-to-market pool value equals 1/8 times instantaneous variance.At daily ETH-USDC volatility σ = 5%, the stated implication is σ^2/8 = 3.125 bp of daily pool-value loss to LVR.
  • Range orders: For Uniswap v3 range orders, instantaneous LVR matches Example 3, but pool value falls as the liquidity range narrows around the current price.Consequently, instantaneous LVR per dollar of reserves can become arbitrarily high for sufficiently narrow ranges.
  • Uniswap v3 pools: For concentrated-liquidity pools, only in-range orders contribute to LVR, and aggregate in-range liquidity is a sufficient statistic from the range-order perspective.The pool aggregates liquidity across multiple range orders.
  • Linear market maker: The linear market maker violates the smoothness requirement because its first derivative is discontinuous at the limit price, so Lemma 2 does not directly characterize it.Its pool value remains concave and the pool value process is therefore a super-martingale, but the relevant running cost requires a local-time construction not pursued here.
  • Linear market maker: A linear market maker is statically identical to a resting limit order but dynamically differs because its order remains after crossing the limit price and reverses direction.The paper describes this as a superficial change in the default strategy after trade execution.

7. Empirical Analysis

The empirical analysis decomposes Uniswap v2 WETH-USDC LP returns into market-risk and microstructural components, using discrete rebalancing strategies and observed pool and CEX data. Market risk overwhelmingly drives return variation, while high-frequency hedged P&L approximates fees minus LVR.

  • Data and methodology: The analysis uses WETH-USDC data from Uniswap v2, combining observed pool activity with Binance prices over the study period.LP values incorporate pool holdings, trades, mints, and burns; WETH and ETH are treated as equivalent.
  • Data and methodology: Discrete rebalancing strategies approximate the continuous-time rebalancing benchmark and produce delta-hedged LP P&L.The benchmark matches AMM holdings at fixed intervals and subtracts its P&L from the LP position.
  • Empirical results: 99.991% of unhedged LP-return variance reflects ETH price exposure, while fees and adverse selection account for 0.009%.The standard deviation of one-minute hedged P&L is less than 1% of unhedged P&L’s standard deviation.
  • Empirical results: More frequent rebalancing leaves less market-risk variance in hedged P&L, whereas coarser intervals accrue more within-interval market risk.The rebalancing strategy is described as the variance-minimizing projection of LP P&L on market risk.
  • Testing the theory: At sufficiently high rebalancing frequency, hedged P&L converges toward the alpha-like component of fees minus adverse-selection costs.Figure 7 shows smaller gaps between hedged P&L and fees minus LVR at more frequent rebalancing frequencies.
  • Implications: Using unhedged LP returns in empirical analyses can let market risk overwhelm microstructural signals and induce substantial omitted-variable bias.The paper reports much larger standard errors and recommends explicitly removing or controlling for market risk.

8. “Impermanent Loss”

The paper argues that fixed-start-point “impermanent loss,” or loss versus holding, is a problematic measure because it is non-additive, path-independent, benchmark-dependent, and mixes market risk with microstructural losses. Loss versus rebalancing instead updates the benchmark over time and isolates the microstructural component.

  • Loss versus holding: Loss versus holding benchmarks AMM performance against buying and holding the asset mix at a fixed starting point.The measure is also called “impermanent loss” in industry practice.
  • Problems with loss versus holding: Loss versus holding can be positive on separate intervals yet zero over their combined interval, so it fails a basic cumulation property.Reconverging prices can make apparent losses reverse despite adverse-selection costs incurred along the path.
  • Problems with loss versus holding: The sign and magnitude of incremental loss versus holding depend on the arbitrarily chosen reference date.Different benchmark holdings can imply opposite changes over the same future period.
  • Problems with loss versus holding: Loss versus holding is path-independent, whereas loss versus rebalancing is path-dependent and accumulates faster during high-volatility periods.The latter matches the intuition that adverse selection costs increase when prices are more volatile.
  • Problems with loss versus holding: Loss versus holding conflates market risk from evolving AMM holdings with microstructural adverse-selection costs.The beta-like component is orders of magnitude larger than the alpha-like component in variance terms in the empirical analysis.
  • Loss versus rebalancing: Loss versus rebalancing updates the benchmark each interval, guarantees additivity, and equals the time-discretized definition of LVR.This construction cleanly separates market risk absorbed by the rebalancing strategy from the residual microstructural component.
  • Empirical literature: The fixed-start-point measure has over one hundred times greater variance than the periodically updated measure and mainly reflects ETH price movements.The periodically updated measure converges to fees minus adverse-selection costs at reasonably high update frequencies.
  • Implications: The paper recommends that studies using fixed-start “impermanent loss” justify including its market-risk component for their research question.The decomposition supplies guidance for choosing between fixed-start and updated-holdings benchmarks.

9. Option Pricing

The option-pricing interpretation views a CFMM LP position as exposure to volatility and as equivalent, ignoring fees, to giving away a bundle of European options. LVR therefore rises with volatility and marginal liquidity, while rebalancing and option positions provide complementary descriptions of the same exposure.

  • Volatility exposure: CFMM LPs behave like a bet on volatility because LVR is large when volatility is high.The paper relates this exposure to static options, dynamic trading strategies, and variance swaps.
  • Option replication: Ignoring fees, a CFMM LP payoff depends only on the terminal asset price and can be replicated by a bundle of European options.The pool holds x*(P_T) of the risky asset and has value V(P_T), regardless of the path to P_T.
  • Option replication: Expected LVR until time T can be interpreted as the value of the European options given away.The option analogy explains why LVR increases with the volatility of the underlying asset.
  • Comparative statics: Higher marginal liquidity increases LVR because more aggressive AMMs effectively give away larger option positions.The replicating portfolio is larger when the AMM’s marginal liquidity is greater.
  • Hedging interpretation: A long LP position combined with a short replicating option bundle removes price-movement exposure, leaving a position that profits when fees exceed option premia.This position is described as a trading fee swap.
  • Dynamic trading: The rebalancing strategy buys after price declines and sells after price increases, while the CFMM LP position loses when prices diverge from the initial price.The contrast links dynamic rebalancing profits to the option-like payoff of the CFMM.
  • Variance interpretation: LVR can be represented as the payoff of the floating leg of a continuously sampled generalized variance swap.Its instantaneous components are price variance and the marginal liquidity term |x*′(P)|.

10. Discussion and Implications

The discussion uses the decomposition to identify design and measurement implications for AMMs. It highlights mechanisms that could reduce or redistribute LVR, while noting that oracle-based approaches introduce accuracy and manipulation risks.

  • Reducing LVR: CFMM designs can reduce or eliminate LVR by improving price updating or reallocating arbitrage rights and profits to LPs.The paper discusses oracle-based pricing, authorized participants, and auctions for arbitrage access.
  • Mechanism: LP losses arise from stale CFMM quotes when CEX prices move, causing trades to execute at worse prices than the rebalancing strategy.This price slippage is the mechanism underlying LVR in the discussion.
  • Oracle-based designs: An oracle-based AMM could quote near the CEX price and approach the rebalancing strategy’s payoff, but oracle accuracy and manipulation remain risks.The design relies heavily on a high-frequency oracle for the CEX price.
  • Arbitrage-right designs: Authorized participants can profitably arbitrage smaller price movements than non-authorized wallets when they receive preferential access or zero fees.The resulting arbitrage profits could be captured by the protocol and redistributed to LPs.
  • Arbitrage-right designs: Periodic auctions could sell arbitrage rights for longer periods, with expected arbitrage profits redistributed to LPs.Potential arbitrageurs would bid the expected profits, including LVR, for authorized-participant access.
  • AMM design: Bonding functions affect LVR through their effect on marginal liquidity, and equivalent marginal liquidity implies equivalent infinitesimal LVR across AMM designs.This includes comparisons involving Uniswap v3 and other CFMM invariants.

11. Conclusion

The paper frames AMM liquidity provision as an investment and develops a decomposition of LP P&L into market-risk and microstructural components. These components can be separated empirically by subtracting rebalancing-strategy profits from LP P&L.

  • AMMs let participants trade and contribute capital to liquidity provision, making LP positions investment-like.
  • The framework decomposes LP P&L into a beta-like component reflecting market risk and an alpha-like component reflecting fees versus adverse-selection losses.
  • The beta component can be eliminated empirically by subtracting the profits of a rebalancing strategy from LP P&L.
  • The decomposition has implications for measuring returns on LP positions as investments.

A. Proofs

The proof uses the concavity of the pool value function, established by viewing it as a pointwise minimum of affine functions.

  • The pool value function is concave because it is the pointwise minimum of a collection of affine functions.

A.2. Proof of Lemma 2

The proof identifies cumulative rebalancing-arbitrage profits with loss-versus-rebalancing by approximating trades discretely and passing to continuous time. The section also connects rebalancing and LP payoffs to short-option-like exposure.

  • Applying Itô’s lemma to the pool value function shows that cumulative rebalancing-arbitrage profits equal LVR.
  • The proof begins with a discrete sequence of arbitrageurs who observe prices and rebalance the pool.
  • Arbitrageurs purchase the pool’s price-adjusted asset quantity and sell it externally, earning profits from price differences.
  • As the partition mesh shrinks, the discrete profit sum converges to an Itô integral under the stated smoothness assumptions.
  • Relationships to option strategies: An LP position ignoring fees has payoff patterns similar to a short European straddle or strangle: positive when prices mean-revert and negative when they diverge.
  • Relationships to option strategies: The rebalancing-minus-buy-and-hold payoff is positive when prices mean-revert and negative when prices diverge, while the LP payoff is zero when prices end where they started.

B.2. Weighted Geometric Mean Market Makers

Weighted geometric mean market makers are characterized by constant instantaneous LVR per dollar of pool value. The results extend the LVR framework to multiple assets under geometric-Brownian price dynamics.

  • Weighted geometric mean market makers have constant instantaneous LVR per dollar of pool value.
  • The paper states that these are essentially the only CFMMs with this constant-LVR-per-dollar property.
  • The characterized pool is the sum of weighted geometric mean market makers with weights θ and 1 − θ.
  • Multi-dimensional generalization: The multidimensional generalization considers n asset reserves, a vector of prices, and geometric Brownian price dynamics with covariance matrix Σ_t.
  • Multi-dimensional generalization: In the multidimensional setting, LVR is non-negative, non-decreasing, and predictable.

C.1. Data

The data procedure combines minute-level Binance prices with Uniswap v2 WETH-USDC pool activity to measure pool P&L, rebalancing returns, fees, and LVR. It uses Dune Analytics queries and discrete minute-level calculations to evaluate the return decomposition.

  • Price data: Minute-level USDC-ETH close prices are downloaded from the Binance API for the price inputs.
  • Pool data: Uniswap v2 WETH-USDC pool data are downloaded from Dune Analytics, which aggregates Ethereum blockchain data into SQL databases.
  • Pool P&L: Pool value is computed as V_t = y_t + P_t x_t, and pool P&L tracks reserve values, mints, burns, and transaction fees.Minted and burned assets are valued at the time-t closing price; Uniswap v2 fees enter pool reserves directly.
  • Rebalancing strategy: Rebalancing returns are computed at different frequencies by holding the pool's starting-period ETH amount throughout each period.For daily rebalancing, the held ETH amount is set to the pool reserves at the start of each day.
  • LVR and decomposition: LVR is approximated discretely using realized daily volatility from 60-minute Binance samples and a one-minute interval of 1/(24 × 60).Cumulative returns are assembled from the LP P&L, rebalancing P&L, fee, and LVR increments.
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