Mathematics of Constant Product Automated Market Makers

Uniswap V2 and Uniswap V3 are both examples of constant product automated market makers (CPAMMs). While Uniswap V2 is seen as the original and default CPAMM, Uniswap V3 decreased slippage by adding a price range input to concentrte liquidity inside the range or trade all to one asset outside of the range. This is called a concentrated liquidity CPAMM. For each CPAMM, we derive formulas for the value, delta, and gamma of an LP position at any time, we replicate the payout of an LP position with a portfolio of options, we derive the impermanent loss of an LP position for any price movement, and we project future impermanent loss based on Brownian motion and Black-Scholes assumptions