Constant Product Market Maker (CPMM)

The core mechanism is defined by the equation x * y = k, where x and y are the reserve quantities of two assets in a pool, and k is a constant product. This formula ensures that the product of the reserves remains unchanged by any trade, automatically determining the price based on the ratio of the reserves.

• Price Impact: The price changes non-linearly; large trades cause significant slippage as they move the ratio x/y.
• Example: In an ETH/DAI pool, swapping a large amount of ETH for DAI will drastically increase the price of ETH within the pool.

Prices are not set by an order book but are calculated algorithmically based on the pool's reserves. The marginal price is the derivative of the invariant curve. Slippage is the difference between the expected price of a trade and the executed price, which increases with trade size relative to liquidity.

• Impermanent Loss: Liquidity providers are exposed to divergence loss when the price ratio of the pooled assets changes versus holding them.

Anyone can become a liquidity provider (LP) by depositing an equivalent value of both assets in the pool. In return, they receive LP tokens representing their share. Trades incur a fee (e.g., 0.3%), which is distributed pro-rata to all LPs, providing a yield for their capital.

• Capital Efficiency: Basic CPMMs require capital across the entire price range (0, ∞), which can be inefficient compared to concentrated liquidity models.

The final state of the reserves depends only on the net amount traded, not the path taken. Swapping A→B→C yields the same result as a direct A→C swap of the equivalent net amount, assuming no fees and a single pool. This property is foundational for multi-hop routing in decentralized exchanges, which finds the optimal path across multiple liquidity pools.

While revolutionary, basic CPMMs have limitations that led to new AMM designs:

• Capital Inefficiency: Liquidity is spread thinly across all prices.
• High Slippage: For large trades in pools with low liquidity.

These limitations spurred innovations like Concentrated Liquidity (Uniswap V3), StableSwap invariant curves for stablecoin pairs, and Hybrid AMMs that combine multiple functions.

Impermanent loss is not a fee but an opportunity cost incurred by liquidity providers (LPs) in a CPMM. It occurs when the price ratio of the deposited assets changes compared to when they were deposited.

• The AMM automatically rebalances the pool, selling the appreciating asset and buying the depreciating one to maintain k.
• LPs end up with a higher quantity of the losing asset and less of the winning one.
• The loss is 'impermanent' until the LP withdraws; if prices return to the original ratio, the loss disappears.

In a CPMM, trade size directly affects price due to the constant product formula. Slippage is the difference between the expected and executed price.

• Price impact increases with trade size relative to pool liquidity (sqrt(k)). A large trade significantly moves the price along the bonding curve.
• This creates a natural limit to trade size within a single pool and is why deep liquidity (high Total Value Locked) is critical for minimizing slippage for users.

The core mathematical engine of a CPMM is the invariant x * y = k, which defines a hyperbolic bonding curve.

• The reserve of Asset X (x) multiplied by the reserve of Asset Y (y) always equals a constant (k).
• The slope of the curve at any point defines the marginal price. The curve ensures liquidity at all prices but with increasing price impact.
• This simple formula enables fully automated, permissionless trading without order books.

Impermanent Loss is the opportunity cost liquidity providers (LPs) face when the price of deposited assets changes compared to simply holding them. It occurs because the CPMM's constant product formula (x * y = k) automatically rebalances the pool, selling the appreciating asset and buying the depreciating one. The loss is 'impermanent' only if prices return to their original ratio.

• Magnitude: Losses increase exponentially with price divergence.
• Example: If ETH doubles in price relative to USDC in a pool, an LP will have less ETH and more USDC than if they had just held the assets, realizing a loss versus the holding strategy.

In a CPMM, the price for a trade is determined by the pool's current reserves, leading to slippage—the difference between expected and executed prices. Price impact is the direct effect a trade has on the pool's price.

• Large Trade Penalty: The required output of asset y is given by Δy = (k / (x + Δx)) - y. This creates exponential price movement for large orders relative to pool depth.
• Consequence: This makes CPMMs inefficient for large, block-sized trades common in institutional finance, as traders effectively pay a significant premium to move the market.

CPMMs require large amounts of capital (liquidity) to be locked to facilitate small trades with low slippage. This is because liquidity is distributed evenly across all possible prices, including those far from the current market price.

• Idle Capital: A significant portion of the pooled assets sits unused at price ranges where trading is unlikely.
• Comparison: This contrasts with order book models, where capital is concentrated around the current bid-ask spread. The inefficiency led to the development of concentrated liquidity models (e.g., Uniswap V3), where LPs can allocate capital to specific price ranges.

While CPMMs are often used as price oracles, providing a time-weighted average price (TWAP) for other smart contracts, they have specific vulnerabilities.

• Manipulation Risk: The spot price from a low-liquidity pool can be manipulated with a flash loan, executing a large trade to skew the price before a protocol reads it.
• Mitigation: Using TWAP oracles (averaging prices over a time window) is the standard defense, but this introduces latency and complexity. Oracles remain a critical attack vector in DeFi.

The x * y = k bonding curve is a one-size-fits-all model that may not suit all asset pairs.

• Stablecoin Pairs: For assets meant to be pegged (e.g., USDC/DAI), the hyperbolic curve creates unnecessary slippage near the 1:1 ratio. StableSwap (Curve Finance) and other Curve Stable Invariants were created to offer a flatter curve within a peg, drastically reducing slippage.
• Long-Tail Assets: For highly volatile or illiquid assets, the CPMM's slippage model can be prohibitively high, discouraging trading and liquidity provision.

The permissionless nature of CPMMs leads to liquidity fragmentation, where identical trading pairs exist across multiple protocols and even multiple pools within the same protocol (e.g., different fee tiers).

• Negative Effects: Fragmentation reduces overall capital efficiency, increases slippage for users who don't use aggregators, and complicates liquidity provision.
• Aggregator Dependence: This flaw created the need for and value of DEX aggregators (e.g., 1inch, Matcha) that split orders across pools to find the best effective price.