Automated market making (AMM) protocols such as Uniswap have recently emerged as an alternative to the most common market structure for electronic trading, the central limit order book. Relative to limit order books, AMMs are both more computationally efficient and do not require the participation of active market making intermediaries such as high frequency traders. As such, AMMs have emerged as the dominant market mechanism for trust-less decentralized exchanges (DEXs) implemented through smart contracts on programmable blockchain platforms such as Ethereum. In cryptocurrency markets, the aggregate trading volume on the Uniswap DEX exceeds that of the much better known Coinbase centralized exchange.
We develop a model the underlying economics of AMMs from the perspective of their passive liquidity providers (LPs). Our central contribution is a "Black-Scholes formula for AMMs". Like the Black-Scholes formula, we consider the return to LPs once market risk has been hedged. 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. In a continuous time Black-Scholes setting, we are able to derive closed-form expressions for this adverse selection cost. Qualitatively, we highlight the main forces that drive AMM LP returns, including asset characteristics (volatility), AMM characteristics (curvature / marginal liquidity, fee structure), and blockchain characteristics (block rate). Quantitatively, we illustrate how our model's expressions for LP returns match actual LP returns for the Uniswap v2 WETH-USDC trading pair. Our model provides tradable insight into both the ex ante and ex post assessment of AMM LP investment decisions. LVR can also inform the design of the next generation of DEX market mechanisms - in fact, in the short time since our work has been released, "LVR mitigation" has already emerged as the dominant challenge among practitioners in the AMM protocol designer community.
