Geometric mean market makers (G Ms), such as Uniswap and Balancer, comprise a popular 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 for constant-weight G Ms to the general case of G Ms with time-varying and potentially stochastic weights. These results include the returns and no-arbitrage prices of liquidity pool (LP) shares that investors receive for supplying liquidity to G Ms. Using these expressions, we show how to create G Ms whose LP shares replicate the payoffs of financial derivatives. The resulting hedges are model-independent and exact for derivative contracts whose payoff functions satisfy an elasticity constraint. These strategies allow LP shares to replicate various trading strategies and financial contracts, including standard options. G Ms are thus shown to be capable of recreating a variety of active trading strategies through passive positions in LP shares. ↩ 29. Mkaouar, F., & Prigent, J.-L. ( ). Constant proportion portfolio insurance under tolerance and transaction costs.