Model-Free Hedging of Impermanent Loss in Geometric Mean Market Makers with Proportional Transaction Fees

Abstract

We consider Geometric Mean Market Makers (G3Ms) – a special type of Decentralized Exchange – with two types of traders: liquidity takers and arbitrageurs. Liquidity takers use G3Ms to swap tokens and to speculate, while arbitrageurs exploit arbitrage opportunities arising from misalignments between the G3M's price and the external market price. We show that in continuous time, a G3M charging proportional transaction fees offers exchange rates that are of finite variation, and that the opportunity cost of providing liquidity relative to rebalancing a self-financing constant-weights portfolio is, in fact, a non-negative gain. Moreover, we demonstrate that Impermanent Loss can be super-hedged in continuous time by a model-free rebalancing strategy. We conclude with a numerical analysis discussing the approximative nature of our continuous-time results for trading in discrete time.