Decentralized Exchange: Lehar & Parlour (2025)

What this is. The paper’s core results, the model (constant-product AMM and limit-order-book comparison), and the empirical specifications: enough to know what it found and how, without reading the full 54-page paper. To replicate or extend, read the original at doi.org/10.1111/jofi.13405.

This paper builds on Glosten (1994), the canonical model of limit-order-book efficiency under adverse selection, and extends the comparison to AMMs. It tests against Capponi and Jia (2021), who model AMM competition among arbitrageurs. It also draws on Angeris and Chitra (2020), who show constant-function market makers can reflect true prices. The empirical price-impact results complement Barbon and Ranaldo (2021), who compare DEX and Binance transaction costs for five token pairs. The analysis of HFT and strategic liquidity provision relates to Biais, Foucault, and Moinas (2015).

TL;DR

Lehar and Parlour analyze Uniswap, the largest decentralized exchange, as a new model of liquidity provision. In an automated market maker (AMM), liquidity suppliers passively post capital into pools rather than actively setting prices; price impact is determined mechanically by a bonding curve. The paper develops a model showing that equilibrium pool size trades off fee revenue against picking-off risk: pools are larger when token volatility is lower and noise trading is higher. Using 95.8 million Uniswap transactions from November 2018 to December 2022 across 105,098 liquidity pools, the paper finds that (i) pool size decreases in volatility and increases in uninformed trading, consistent with theory; (ii) AMM liquidity is substantially more stable than limit-order-book liquidity during extreme market events; (iii) price impact on Uniswap is lower than on Binance (a centralized exchange) for low-volatility, noise-dominated tokens; and (iv) Binance price impact becomes less volatile and converges toward Uniswap after integrating PancakeSwap, an AMM clone, in March 2022. The paper also shows conditions under which the AMM dominates a limit-order market and documents absence of long-lived arbitrage opportunities.

Core results

Magnitudes as reported; \*\*\* = 1%. Locators point into the source PDF.

#	Result	Locator	Magnitude
R1	Pool size decreases in volatility and increases in uninformed trading	Table II, p. 341	Std.Dev FX Rate coefficient: -0.152*** to -0.249*** across 6 specs (col 1: -0.201; col 2: -0.249; col 3: -0.204; col 4: -0.163; col 5: -0.158; col 6: -0.152); Reversals: 49.18*** (13.68); Number Trades: 6.633*** (2.088); R2 = 0.21-0.55
R2	AMM liquidity is stable during extreme market events; gas fees and fee revenue both increase, discouraging withdrawal	Table III, p. 344; Figure 9, p. 343	Only 2% of ETH-stablecoin liquidity withdrawn during 41% ETH price crash (May 19, 2021); gas fees rise 1.71 USD (10%) and fee revenue rises 17.7 USD on high-return days
R3	Uniswap price impact is lower than Binance on average; gap narrows after Binance integrates AMM clone PancakeSwap	Table IV, p. 352	rPI intercept significant and positive across cols (col 1: 4.080***; col 2: 4.956***; col 3: 3.126***; col 8: 3.963***); Pancake Swap dummy -2.497*** to -3.085*** across pool-FE specs; effect stronger at low volatility and medium trade sizes
R4	Price impact is more volatile on Binance than Uniswap; volatility gap drops after PancakeSwap integration	Table V, p. 353	Relative volatility rV intercept: col 1: 33.95***; col 2: 39.00***; col 8: 30.47***; col 9: 25.91***; Pancake Swap dummy: col 1: -13.63***; col 8: -18.16***; col 9: -7.756***; pattern holds for low exchange-rate volatility and medium trade sizes

Overall (paper’s conclusion). Uniswap’s AMM mechanism successfully provides stable, predictable liquidity at lower and less volatile price impact than a centralized limit-order book for asset pairs with lower volatility and more noise trading. The equilibrium pool size adjusts so that fee revenue compensates liquidity suppliers for adverse selection, and gas fees on the blockchain act as a commitment device that discourages strategic withdrawal during market turmoil.

Theory / model

The model has a single asset with current value $p_0$. With probability $\alpha$, an innovation occurs and the value jumps to $p_0 + \sigma$ or $p_0 - \sigma$ with equal probability; otherwise the value remains at $p_0$. Three agents interact: risk-neutral liquidity suppliers, a liquidity demander (noise trader) who trades a fixed quantity $q$, and an informed arbitrageur who trades whenever profitable.

Constant-product bonding curve. For a pool with $E_0$ units of ETH and $T_0$ tokens, the bonding curve constant is (eq. 1, p. 326):

$$k := T_1 \cdot E_1 = T_0 \cdot E_0. \tag{1}$$

Any trade must stay on this curve. When a trader buys $t$ tokens by depositing $e$ ETH, the fee $\tau$ is collected, the post-trade ETH pool becomes $E' = E + (1-\tau)e$, and the post-trade token balance is (eq. 2, p. 327):

$$T' = \frac{T \cdot E}{E'} = \frac{T \cdot E}{E + (1-\tau)e}. \tag{2}$$

The token received by the trader is $t = T - T'$ (eq. 3), and the terms of trade in ETH per token are (eq. 4):

$$p^{tot} = \frac{e}{t} = \frac{e}{T} + \frac{E}{(1-\tau)T}. \tag{4}$$

In the limit as $e \to 0$, the spread relative to the fundamental $p_0$ is:

$$\lim_{e \to 0} \frac{p^{tot}}{p_0} = \frac{ET}{ET(1-\tau)} = \frac{1}{1-\tau}. \tag{5}$$

Limit-order-book equilibrium. Competing liquidity suppliers choose private investment $\gamma_i$ (e.g., co-location speed) at cost $I(\gamma) = a\gamma^2$ to become the monopolist with probability $\gamma_i(1 - \gamma_j)$. In symmetric equilibrium (Proposition 1, p. 330):

$$\gamma^* = \frac{(1-\alpha)\sigma q}{2a + \sigma q(1-\alpha)}.$$

A monopolist posts at $p_0 + \sigma$ (sell) and $p_0 - \sigma$ (buy), earning $(1-\alpha)\sigma$. With two competing suppliers, each earns zero and prices are $p_0 + \alpha\sigma$ and $p_0 - \alpha\sigma$ (Lemma 2, p. 330).

AMM equilibrium pool size. In the AMM, liquidity provision is non-rivalrous and payoffs are shared pro rata. There is no incentive for private investment. The equilibrium pool size (Proposition 2, p. 333, eq. 16) balances fee revenue from the noise trader against picking-off losses from the arbitrageur:

$$T_0 = q \left[ \sqrt{1 + \frac{(1-\alpha)^2 \tau^2 p_0^2}{\alpha^2 \omega^2}} - \frac{(1-\alpha)\tau p_0}{\alpha \omega} \right], \tag{16}$$

where $\omega = \sqrt{p_0(p_0+\sigma)(1+\tau)} + \sqrt{\frac{p_0(p_0-\sigma)}{1+\tau}} - 2p_0$. The equilibrium pool size is linear in noise-trade volume $q$, decreasing in innovation size $\sigma$, and decreasing in innovation probability $\alpha$ (Proposition 3, p. 339).

AMM vs. limit-order-book trading costs. Expected cost per unit on the limit-order book is (Proposition 4, p. 345):

$$E(c^{\text{limit}}) = \sigma((1-\eta)\alpha + \eta),$$

where $\eta = 2\gamma^*(1-\gamma^*)$ is the probability of facing a monopolist. Expected cost on the AMM is:

$$E(c^{\text{AMM}}) = p_0 \left(\frac{(1+\tau)\lambda^b - (1-\tau)\lambda^s}{2}\right),$$

where $\lambda^b(\tau, \alpha, \sigma) > 1$ and $\lambda^s(\tau, \alpha, \sigma) < 1$ are functions of equilibrium pool size. Proposition 5 shows the limit-order book does not universally dominate the AMM: there exists a critical innovation probability $\alpha^*$ below which the AMM is strictly preferred, and conditional on trade quantity, price impact is more volatile in the limit-order book.

Method

The paper combines a stylized two-market equilibrium model with reduced-form panel regressions.

Structural model. Equilibrium is derived analytically for both markets. The AMM equilibrium pool size (equation 16) is the closed-form solution to the indifference condition in equation (A10) (Appendix A, p. 358). The model builds on the amm-equilibrium-pool-size framework, which is the paper’s primary methodological contribution.

Panel regressions for pool size (R1). The estimating equation (eq. 17, p. 340) is at the pool-day level:

$$\text{pool size} = a + b_1 \sigma_{fx} + b_2 \mathbf{1}_{\text{airdrop}} + b_3 \text{noise trading}, \tag{17}$$

where pool size is daily average USD pool size, $\sigma_{fx}$ is the annualized block-by-block exchange-rate volatility (proxy for adverse selection), and noise trading is measured by three proxies: number of trades per day, daily volume, and immediate trade reversals (trades followed within 75% of the same size in the opposite direction). Standard errors are clustered by pool and by day. Robustness: specifications with and without pool-and-day fixed effects.

Price impact comparison (R3, R4). Relative price impact is defined as (eq. 18, p. 350):

$$rPI = \frac{PI_{\text{Binance}}}{PI_{\text{Uniswap}}} - 1, \tag{18}$$

estimated at the daily level for 43 token pairs cross-listed on both venues. The estimating equation (eq. 19, p. 350) is:

$$rPI = a + b_1 \mathbf{1}_{\text{pancake}} + b_2 \sigma_{FX} + b_3 \text{trade size} + b_4 \text{trade size}^2 + b_5 \text{noise trading} + \epsilon. \tag{19}$$

The PancakeSwap dummy $\mathbf{1}_{\text{pancake}}$ equals one after March 25, 2022 (when Binance integrated a Uniswap clone). Relative volatility of price impact is defined analogously (eq. 20, p. 351):

$$rV = \frac{V_{\text{Binance}}}{V_{\text{Uniswap}}} - 1, \tag{20}$$

using the same specification. Standard errors are clustered by pool and by day.

Empirical specifications

All regressions are panel (pool-day or pool observations) with standard errors clustered by pool and by day unless noted.

R1: Pool size and volatility/noise trading. Pool-day level on 1,525 pools (997,507 observations). Dependent variable: daily average pool size in million USD. Key regressors: exchange-rate volatility (Std.Dev FX Rate, annualized daily sd of block-by-block price changes), airdrop dummy, and three alternative noise-trading proxies (daily USD volume, number of trades, and reversals). Six columns spanning no-FE and pool-and-day FE; R2 = 0.21-0.55 (Table II, p. 341).

R2: Stability during extreme events. Analyzed via Figures 9 and Table III. Gas fees and fee revenue are regressed on High Return dummy (absolute daily price change > 10%) and absolute return. Pool fixed effects. 1,303,869 observations. Gas fees rise 1.71 USD on high-return days; fee revenue rises 17.73 USD (Table III, p. 344).

R3/R4: Relative price impact AMM vs. Binance. 43 cross-listed token pairs; 24,963 pool-day observations in col (1) of Table IV (pool FE only, full sample without pancake dummy); 24,224 in cols with pancake dummy included; 21,409 in Table V col (1). Specifications with pool FE only and pool-and-day FE. Pre/post-PancakeSwap subsamples in columns (8) and (9) (Table IV, p. 352; Table V, p. 353).

Datasets used

Dataset	Role in paper	Wiki page
Uniswap V1 and V2 blockchain data (Ethereum)	Primary data: 95.8 million interactions across 105,098 pools; liquidity injections, withdrawals, token swaps Nov 2018-Dec 2022	Uniswap on-chain
Binance minute-by-minute price data	Price benchmark for USD volume conversion and price impact comparison; 43 cross-listed token pairs	No page yet
Ethereum blockchain gas-price data	Gas cost estimation for AMM withdrawal transactions	No page yet

Sample: November 2, 2018 to December 21, 2022 (Uniswap V1 launch through sample end); reduced econometric sample: 1,525 pools with at least 30 trading days and 100 ETH average balance (59,606,977 observations).

When to read the full paper

Read the original if you are: building models of AMM or decentralized exchange mechanisms; studying how blockchain-specific costs (gas fees) affect market design and liquidity stability; comparing trading costs across centralized and decentralized venues; analyzing the market-microstructure implications of DeFi protocols; or extending the model to concentrated-liquidity AMMs (Uniswap V3) or multi-pool settings. Exact tables are at pages 341 (pool size), 344 (stability), and 352-353 (price impact comparison).

Attribution and rights

Source: peer-reviewed, The Journal of Finance 80(1), February 2025, pp. 321-374. Copyright 2024 the American Finance Association. This distillation was extracted by an LLM on 2026-06-06 and is not human-verified or independently reproduced. The paper is paywalled; only text extracts are reproduced here under fair-use principles.

Lehar, Alfred, and Christine A. Parlour. “Decentralized Exchange: The Uniswap Automated Market Maker.” The Journal of Finance 80, no. 1 (February 2025): 321-374. DOI: 10.1111/jofi.13405.