Constant Function Market Maker

The foundational invariant used by Uniswap V2 and many AMMs. It states that the product of the quantities of two tokens in a pool (x and y) must equal a constant k after any trade. This creates hyperbolic price curves and infinite liquidity depth.

• Price Impact: Price changes are continuous and non-linear.
• Impermanent Loss: LPs are exposed to divergence loss when asset prices change.
• Example: A pool with 100 ETH (x) and 300,000 USDC (y) has k = 30,000,000. Swapping 1 ETH will add to x and subtract from y, recalculating the new price.

A simpler invariant that maintains a fixed, linear exchange rate (e.g., 1:1). It is used for stablecoin pairs or wrapped asset pools where assets are intended to maintain parity.

• Zero Slippage: Trades occur at a constant price as long as both assets are in the pool.
• Risk of Depletion: The pool can be completely drained of one asset if the external market price diverges, requiring external arbitrage to rebalance.
• Limited Use: Primarily for pegged assets like stablecoin-stablecoin pairs (e.g., USDC/DAI).

A hybrid invariant pioneered by Curve Finance that combines constant sum and constant product. It creates a "flatter" curve within a target price range (for stablecoins) and a steeper curve outside it.

• Low Slippage for Pegged Assets: Enables efficient trading of similarly priced assets (e.g., USDC, USDT, DAI).
• Amplification Coefficient (A): A tunable parameter that controls the "flatness" of the curve; a higher A makes it more like constant sum.
• Dominant for Stables: Captures the majority of stablecoin trading volume due to minimal impermanent loss near peg.

A generalization of the constant product formula used by Balancer. It allows for pools with more than two assets and customizable weightings for each asset.

• Multi-Asset Pools: Can hold up to 8 tokens in a single pool (e.g., 50% ETH, 30% WBTC, 20% LINK).
• Custom Weights (w_i): Token weights determine price sensitivity and pool composition; a 90/10 pool behaves very differently than a 50/50 pool.
• Flexible Indices: Enables the creation of automated, self-balancing token portfolios.

Introduced by Uniswap V3, this reparameterizes the constant product formula to allow liquidity providers to concentrate their capital within custom price ranges.

• Virtual Reserves: Uses the concept of liquidity (L) and a price range to calculate virtual reserves x and y.
• Capital Efficiency: LPs can achieve higher fees by providing liquidity where most trading occurs.
• Active Management: Requires LPs to actively manage their price ranges, shifting the impermanent loss profile.

All CFMM invariants share key properties that define their economic behavior and risks.

• Path Independence: The final state of the pool depends only on the net amount swapped, not the path of intermediate trades.
• Slippage: The price impact of a trade is a direct function of the invariant and pool depth.
• Arbitrage: The invariant-determined pool price must be kept in line with external market prices by arbitrageurs, which is the primary mechanism for price discovery.
• Fee Integration: Trading fees are typically added to the pool's reserves, slowly increasing the constant k and rewarding LPs.

The primary risk for liquidity providers (LPs) is impermanent loss (divergence loss), which occurs when the price of deposited assets changes relative to each other after deposit. The AMM automatically rebalances the pool, selling the appreciating asset and buying the depreciating one, resulting in a lower dollar value than simply holding the assets. This loss becomes permanent upon withdrawal.

• Magnitude: Loss increases with the size of the price divergence.
• Mitigation: LPs rely on trading fees to offset this risk, which requires high volume.

CFMMs do not have an order book; price is a function of the pool's reserves. Large trades cause significant price impact, moving the price unfavorably for the trader. Slippage is the difference between the expected and executed price.

• Consequence: Makes large trades expensive and inefficient, creating an upper bound on trade size.
• Solution: Traders use slippage tolerance settings, and protocols employ batch auctions or route trades across multiple pools.

Many DeFi protocols use CFMM pool prices as oracles for asset values. However, these are spot oracles vulnerable to manipulation via flash loans or large, one-sided trades.

• Risk: A manipulated price can trigger faulty liquidations or incorrect loan valuations in lending protocols.
• Defense: Advanced oracles use time-weighted average prices (TWAPs) calculated from CFMM reserves over a period, making manipulation cost-prohibitive.

CFMMs require large amounts of capital to be locked (Total Value Locked - TVL) to facilitate trading with low slippage. This capital is spread across the entire price range (0, ∞), much of which is rarely used.

• Problem: For stablecoin pairs (e.g., USDC/USDT), liquidity is only needed near the 1:1 price point.
• Innovation: Concentrated Liquidity (e.g., Uniswap v3) allows LPs to allocate capital to specific price ranges, dramatically improving capital efficiency.

CFMMs are money legos, freely composable within DeFi. While powerful, this creates systemic risk.

• Example: A flash loan can drain a vulnerable lending protocol, use the funds to manipulate a CFMM's price oracle, and trigger cascading liquidations across interconnected systems.
• Consideration: Security depends on the weakest link in the composed transaction.

While the Constant Product Formula (x*y=k) works for two assets, generalized CFMMs for 3+ asset pools (e.g., Balancer) use different invariant functions.

• Challenge: More complex math can lead to unexpected slippage surfaces and less intuitive LP positions.
• Trade-off: Enables efficient index-like pools but requires deeper analysis for LPs to understand their risk exposure across multiple correlated or uncorrelated assets.

The foundational reserve of token pairs that powers a CFMM. Each pool contains two or more assets, and trades are executed directly against this reserve.

• Reserve Balances: The quantities of each token in the pool determine the price via the constant product formula (x * y = k).
• Liquidity Providers (LPs): Users deposit an equal value of both assets to fund the pool, earning fees from trades.
• Examples: The USDC/ETH pool on Uniswap or the DAI/USDC pool on Curve.

A critical risk for liquidity providers in a CFMM, where the value of deposited assets changes compared to simply holding them, due to price divergence.

• Mechanism: Occurs when the price ratio of the two pooled assets changes after deposit. The AMM algorithm automatically rebalances the pool, selling the appreciating asset and buying the depreciating one.
• Mitigation: Often offset by trading fee rewards, but can result in a net loss during high volatility.
• Permanent vs. Impermanent: The loss is 'impermanent' until the LP withdraws, locking it in.

The difference between the expected price of a trade and the executed price, caused by the trade's size relative to the liquidity pool.

• Cause in CFMMs: Large trades significantly change the pool's reserve ratios, moving the price along the bonding curve.
• Slippage Tolerance: Traders set a maximum acceptable slippage percentage; transactions revert if exceeded.
• Impact: Higher slippage leads to worse exchange rates, especially in pools with low Total Value Locked (TVL).

The mathematical curve that defines the relationship between price and supply in an AMM. For a standard CFMM like Uniswap V2, this is the constant product curve.

• Formula: x * y = k, where x and y are reserve balances and k is a constant.
• Price Impact: The curve is convex, meaning price increases non-linearly as more of one asset is purchased.
• Variations: Other CFMMs use different curves (e.g., constant sum for stablecoins, hybrid curves like Curve's stableswap).