[Paper Review] Automated Market Making and Loss-Versus-Rebalancing

Jason Milionis, Ciamac C. Moallemi+2 more|arXiv (Cornell University)|Aug 11, 2022

Banking stability, regulation, efficiency55 citations

TL;DR

The paper develops a Black-Scholes-style model for AMMs to quantify losses from price slippage (loss-versus-rebalancing, LVR) and validates it against Uniswap data, with implications for AMM design and hedged LP strategies.

ABSTRACT

We consider the market microstructure of automated market makers (AMMs) from the perspective of liquidity providers (LPs). Our central contribution is a ``Black-Scholes formula for AMMs''. We identify the main adverse selection cost incurred by LPs, which we call ``loss-versus-rebalancing'' (LVR, pronounced ``lever''). LVR captures costs incurred by AMM LPs due to stale prices that are picked off by better informed arbitrageurs. We derive closed-form expressions for LVR applicable to all automated market makers. Our model is quantitatively realistic, matching actual LP returns empirically, and shows how CFMM protocols can be redesigned to reduce or eliminate LVR.

Motivation & Objective

• Motivate understanding of AMM liquidity provider (LP) returns in a continuous-time setting akin to option pricing.
• Define and quantify loss-versus-rebalancing (LVR) as the adverse effect of stale AMM quotes relative to market prices.
• Derive closed-form LVR expressions for locally-smooth AMMs, including CFMMs and concentrated-liquidity designs.
• Show empirical alignment between LVR predictions and delta-hedged LP performance, and discuss hedged LP methodology for measuring profitability.
• Discuss design implications for AMMs, including fee setting and potential LVR mitigation via price oracles and arbitrage incentives.

Proposed method

• Adopt a frictionless, continuous-time Black-Scholes framework with a risky asset and a numéraire, where the external price follows a geometric Brownian motion with possibly stochastic volatility.
• Describe CFMM pool state via an invariant curve f(x,y)=L and define the pool value function V(P) as the minimal pool reserves value given price P.
• Define and compute LVR as the gap between the rebalancing strategy (delta-hedged market replication) and the AMM LP's actual performance, attributing losses to price slippage.
• Derive closed-form LVR expressions based on the instantaneous variance of the risky asset and the marginal liquidity (slope of the CFMM’s level curve) at the current price.
• Propose a delta-hedged LP strategy and relate it to variance swaps, analyzing when fees offset LVR losses.
• Empirically analyze Uniswap v2 ETH-USDC and demonstrate that model-predicted LVR aligns with delta-hedged LP returns.

Experimental results

Research questions

• RQ1 What is the quantitative impact of price volatility and marginal liquidity on AMM LP returns?
• RQ2 How can LVR be defined, computed, and interpreted as the cost of price slippage for AMM LPs?
• RQ3 To what extent does a delta-hedged LP position replicate LVR, and how does hedging affect risk?
• RQ4 Can LVR inform practical AMM design decisions, such as fee structures or reliance on price oracles?
• RQ5 Does the model’s LVR description align with empirical LP performance on real AMM data (e.g., Uniswap v2)?

Key findings

• LVR depends on two primary factors: price volatility and the marginal liquidity of the AMM’s level set; higher volatility or steeper liquidity slopes increase LVR.
• The model’s LVR expressions closely match the observed returns of delta-hedged LP positions on Uniswap v2, validating the framework.
• A hedged LP approach substantially reduces return volatility (roughly 1%–6% of unhedged LP P&L std dev, depending on hedging frequency).
• LVR provides a benchmark and design guidance for AMMs, suggesting fees should scale with variance and could be adjusted to balance LVR losses with fee income.
• A potential design direction is to reduce LVR by improving price discovery (e.g., high-frequency price oracles) or redistributing LVR via arbitrage rights, thereby aligning LP incentives with market efficiency.