  Geometric mean market makers (G3Ms), 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 G3Ms to the general case of G3Ms 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 G3Ms. Using these expressions, we show how to create
G3Ms 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. G3Ms are thus shown to be capable of recreating a
variety of active trading strategies through passive positions in LP shares.
