AMM Math: Constant Product, StableSwap, Concentrated Liquidity

What you'll learn

• From Order Books to Curves
• Constant Product: x · y = k
• StableSwap: Curve's Hybrid Invariant
• Uniswap V3: Concentrated Liquidity and Ticks
• Uniswap V4: Hooks and the DEX Landscape
• What to Remember

Key terms

AMM (Automated Market Maker)

A smart contract that holds reserves of multiple assets and uses a deterministic pricing function (invariant) to allow swaps without an order book.

Constant-product invariant

The pricing function x · y = k used by Uniswap V1 and V2. Reserves of two assets multiply to a constant; the curve is a hyperbola.

StableSwap invariant

Curve's hybrid pricing function that behaves like constant-sum near the peg (low slippage) and transitions to constant-product as the pool becomes imbalanced. Designed for pegged or highly correlated assets.

Amplification coefficient (A)

A parameter in Curve's StableSwap that controls how flat the curve is around the peg. Higher A means lower slippage but greater vulnerability to depeg events.

Concentrated liquidity

A Uniswap V3/V4 feature where LPs allocate capital only to a specific price range. In-range capital is dramatically more efficient; out-of-range capital earns nothing.

Tick

A discrete price increment (0.01% / 1 basis point) used by Uniswap V3 and V4 to discretize the price space and make concentrated liquidity computable on-chain.

Fee tier

A configurable swap fee level (0.01%, 0.05%, 0.30%, 1.00% in standard Uniswap V3) chosen per pool. LPs and traders self-sort across tiers based on volatility and trade size.

Hook (Uniswap V4)

A smart contract attached to a V4 pool that runs custom logic at lifecycle events (before/after swap, on position change, on initialization). Enables dynamic fees, limit orders, custom MEV protection, and more.

Singleton architecture

Uniswap V4's design where one contract holds the state of all pools, dramatically reducing gas costs for multi-hop swaps and pool creation versus V3.

Curve Wars

The competitive bribing of veCRV holders (and later Convex/Aura holders) to direct CRV emissions toward specific pools. A central economic game in DeFi 2021-2024 and still relevant in muted form.