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Geometric mean market makers (G3Ms), such as Uniswap and Balancer, comprise apopular class of automated market makers (AMMs) defined by the following rule:the reserves of the AMM before and after each trade must have the same(weighted) geometric mean. This paper extends several results known forconstant-weight G3Ms to the general case of G3Ms with time-varying andpotentially stochastic weights. These results include the returns andno-arbitrage prices of liquidity pool (LP) shares that investors receive forsupplying liquidity to G3Ms. Using these expressions, we show how to createG3Ms whose LP shares replicate the payoffs of financial derivatives. Theresulting hedges are model-independent and exact for derivative contracts whosepayoff functions satisfy an elasticity constraint. These strategies allow LPshares to replicate various trading strategies and financial contracts,including standard options. G3Ms are thus shown to be capable of recreating avariety of active trading strategies through passive positions in LP shares.

Liquidity Provider Returns in Geometric Mean Markets