lp-mathlisted

# LP Math — AMM Liquidity Provision Mathematics Automated Market Makers (AMMs) replace traditional orderbooks with liquidity pools. Instead of matching buyers and sellers, a mathematical formula determines prices based on reserve ratios. Liquidity providers (LPs) deposit both assets into a pool and earn fees from every trade. Understanding the math behind AMMs is essential for: - Evaluating whether providing liquidity is profitable after impermanent loss - Estimating price impact before executing large trades - Comparing capital efficiency across pool types (constant product vs concentrated) - Calculating expected fee revenue for a given pool position **Related skills**: See `impermanent-loss` for IL calculations, `yield-analysis` for LP yield modeling, `liquidity-analysis` for pool depth assessment. --- ## 1. Constant Product AMM (xy = k) The foundational AMM model used by Raydium V4 and most Solana DEXes. ### Core Invariant ``` x * y = k ``` Where: - `x` = reserve amount of token X (e.g., SOL) - `y` = reserve amount of token Y (e.g., USDC) - `k` = constant product (increases over time from fees) ### Spot Price ``` P = x / y (price of Y in terms of X) P = y / x (price of X in terms of Y) ``` For a pool with 100 SOL and 10,000 USDC: price of SOL = 10,000 / 100 = 100 USDC. ### Trade Execution When a trader swaps Δx of token X into the pool: ```python # Output amount (before fees) delta_y = y * delta_x / (x + delta_x) # With fee (e.g., 0.3%) delta_y_after_