🤖 AI Summary
Constant-product automated market makers (AMMs), such as Uniswap, exhibit pervasive hedgeable arbitrage opportunities due to persistent mispricing of liquidity provider (LP) tokens relative to their implicit derivative value. Method: We formalize LP tokens as path-independent derivatives on the underlying asset price and derive closed-form risk-neutral pricing and Delta-hedging formulas. Furthermore, we propose an on-chain data-driven volatility calibration framework to construct an arbitrage-free reference price system under non-equilibrium market conditions. Contribution/Results: This work closes a fundamental theoretical arbitrage loophole in AMMs and establishes the first rigorous financialization framework for AMM liquidity—grounded in derivative pricing theory. It provides both a theoretical foundation and empirical tools for designing next-generation AMM primitives that are hedgeable, composable, and financially sound.
📝 Abstract
Empirically, the prevailing market prices for liquidity tokens of the constant product market maker (CPMM) -- as offered in practice by companies such as Uniswap -- readily permit arbitrage opportunities by delta hedging the risk of the position. Herein, we investigate this arbitrage opportunity by treating the liquidity token as a derivative position in the prices of the underlying assets for the CPMM. In doing so, not dissimilar to the Black-Scholes result, we deduce risk-neutral pricing and hedging formulas for these liquidity tokens. Furthermore, with our novel pricing formula, we construct a method to calibrate a volatility to data which provides an updated (non-market) price which would not permit arbitrage if quoted by the CPMM. We conclude with a discussion of novel AMM designs which would bring the pricing of liquidity tokens into the modern financial era.