Automated market maker
Curve Finance
Curve builds AMMs for assets that should trade near each other, and a second family for assets that should not. StableSwap uses an amplification coefficient to flatten the price near the peg. CryptoSwap concentrates liquidity around a moving price scale. Stableswap-NG is the current factory for plain pools and metapools.
Curve is an AMM project with more than one invariant. The one it is known for, StableSwap, prices assets that are supposed to be worth nearly the same thing: two dollar stablecoins, or an ETH derivative and ETH. A second design, CryptoSwap, prices assets that move. A third system, crvUSD, mints a stablecoin against collateral and is not a StableSwap pool at all. Treating “Curve” as a single constant-product exchange misses all three.
The generation guide separates the factory eras. This page is the mechanism.
Why constant product is the wrong default for a peg
A constant-product pool spreads liquidity from price zero to infinity. Two assets that should both be worth a dollar do not need that. Traders want depth at one. The StableSwap whitepaper builds an invariant between a constant sum, which has no slippage and eventually runs out of one coin, and a constant product, which always has a price but wastes depth.
The derivations page writes the invariant as:
A * n^n * Σ x_i + D = A * D * n^n + D^(n+1) / (n^n * Π x_i)x_i is the balance of coin i, scaled into the pool’s math. n is the number of coins. D is the invariant, equal to the total balance when every coin is equal. A is the amplification coefficient.
When balances are near equal and A is large, the curve behaves like a constant sum: a trade barely moves the price. When one coin dominates, the product term dominates and the price can move sharply, which is what gives arbitrage a reason to return the pool toward the peg. The understanding-StableSwap page describes high A (it gives 500 to 10,000 as examples) as deep liquidity near the peg and a sharp drop if an asset leaves the peg. Lower A spreads liquidity and moves the price earlier. Those ranges are illustrations of the shape, not a recommendation for a new pool.
- Constant product: liquidity is spread from a very low price to a very high price.
- StableSwap shape: more of the liquidity sits near equal balances, then the curve steepens.
The paths are drawn to show the idea. They are not a plot of a live pool and they are not the StableSwap invariant evaluated at a particular amplification coefficient.
StableSwap is a poor fit for assets that are not meant to share a price. The same page says so and points volatile pairs at CryptoSwap. A high A on a volatile pair concentrates liquidity at a peg the assets will not keep.
What a liquidity provider actually holds
In Stableswap-NG, the integration guide and the factory overview say the pool contract and the LP token are the same contract. A deposit mints LP tokens. A withdrawal burns them. The LP token is a share of the pool’s liquidity, accounted against D, not a voucher for the exact basket deposited. If the pool is imbalanced, the coins you withdraw in a balanced removal are the current mix, and a one-coin withdrawal moves the price and pays the invariant’s slippage.
The NG factory deploys plain pools of up to eight coins and metapools. A metapool pairs one coin with the LP token of a base pool, so a swap can reach the base pool’s coins without merging their reserves into one contract.
Each coin has an asset type. The factory source documents four:
| Id | Name | Rate |
|---|---|---|
| 0 | Standard | No extra rate. The unit is the token balance. |
| 1 | Oracle | An external rate, used for tokens such as wrapped staked ETH. |
| 2 | Rebasing | The balance itself changes, as with stETH. |
| 3 | ERC-4626 | convertToAssets supplies the exchange rate. |
The pool can therefore hold a rebasing token or a vault token without pretending every ERC-20 has a static balance. That rate is an input. If the oracle or convertToAssets is wrong, the invariant will still clear trades against the wrong value. The oracle page is the general version of that dependency.
NG pools are deployed from a factory implementation index. The integration docs say a pool keeps the implementation it was deployed with when the factory later adds a new one. “The NG factory” is not a single bytecode frozen for every pool address.
Fees are pool parameters. The deploy documentation describes an off-peg fee multiplier that can raise the fee when the pool is away from balance. This page does not quote a universal admin-fee percentage. That cut is configuration.
CryptoSwap
CryptoSwap is for assets whose price should move. The whitepaper starts from a StableSwap-like invariant and replaces the constant amplification with a dynamic K:
K0 = Π x_i * n^n / D^n
K = A * K0 * γ^2 / (γ + 1 - K0)^2A still concentrates liquidity. gamma (γ) changes how the curve leaves that concentrated region. Curve’s conceptual docs and the paper do not describe gamma as a simple “higher means wider” switch that an integrator should hard-code. The understanding-CryptoSwap page also warns that CryptoSwap’s A is scaled differently from StableSwap’s: it says an A of 10,000 in CryptoSwap corresponds to an A of 1 in StableSwap. Read the paper and the pool’s stored parameters together. Do not copy a StableSwap A into a tricrypto pool.
The internal price is a price_scale the pool updates. Liquidity concentrates around that scale. A trade away from it pays more slippage, which is the point: the pool is not offering v2-style depth at every price, and it is not offering StableSwap’s peg.
crvUSD is not a pool type
The crvUSD overview describes a stablecoin minted against approved collateral. The monetary mechanism is LLAMMA, an AMM that converts collateral toward the stablecoin as the collateral price falls, instead of a single auction at a liquidation ratio. Soft liquidation is a different idea from Aave’s liquidationCall or Maker’s grab.
The same page versions the controller and AMM blueprints. V3 is described as the current blueprint on the controller factory. Existing markets keep the blueprint they were deployed with; setting a new implementation does not rewrite them. The overview lists markets on V1, V2, and V3. LlamaLend is a related lending deployment and is not the same contract family as a Stableswap-NG pool.
Savings crvUSD (scrvUSD) is described there as a yield-bearing wrapper that takes a portion of borrower interest. It is another tokenized claim, not the StableSwap LP token.
Trust that remains
A StableSwap pool does not know whether a “stablecoin” will keep its peg. A makes the pool willing to sell the depegged asset slowly at first and then very quickly. Liquidity providers own that inventory. Governance, and in NG the pool creator, chooses coins, A, fees, and any rate oracle. Gauges that emit CRV are an incentives layer on top of the invariant. They do not change D.
An integrator pricing a pegged asset from a Curve pool is reading an invariant that assumes the peg. That price is only as good as the assumption. The security assumptions page puts pool-as-oracle risk next to the other external inputs these protocols take.
Versions in this overview
- StableSwap (active) The pegged-asset invariant. Stableswap-NG is the current factory form documented for integrators.
- CryptoSwap (active) A separate invariant for volatile pairs, with amplification and gamma around an internal price scale.
- crvUSD (active) A lending-and-mint system beside the AMMs. Controller blueprints are versioned, and existing markets keep the blueprint they were deployed with.
Sources
- Understanding StableSwap Documentation · Curve
- StableSwap mathematical derivations Documentation · Curve
- StableSwap whitepaper Whitepaper · Curve
- Understanding CryptoSwap Documentation · Curve
- CryptoSwap whitepaper Whitepaper · Curve
- Integrating Stableswap-NG Documentation · Curve
- CurveStableSwapFactoryNG.vy Source code · Curve
- crvUSD overview Documentation · Curve