Automated Market Maker

An automated market maker (AMM) is a smart contract mechanism that enables peer-to-peer token trading using liquidity pools. Instead of matching buyers and sellers through an order book, AMMs use an algorithmic price formula (typically $x \times y = k$) where prices adjust based on the ratio of tokens in the pool.

The constant product formula

Uniswap v2 uses the constant product formula:

$$x \times y = k$$

Where:

• $x$ = amount of token A in the pool
• $y$ = amount of token B in the pool
• $k$ = a constant

When a trader swaps token A for token B:

1. They deposit some amount $\Delta x$ of token A.
2. The pool must maintain $x \times y = k$, so $y$ decreases by $\Delta y$.
3. The trader receives $\Delta y$ of token B.

The price is determined by the current ratio ($x / y$) and adjusts as the pool composition changes.

Example

Imagine a liquidity pool with 1,000 ETH and 1,000,000 USDC. The constant is $k = 1,000 \times 1,000,000 = 1,000,000,000$.

A trader wants to swap 100 USDC for ETH. After the trade:

• Pool USDC: $1,000,000 + 100 = 1,000,100$
• Pool ETH: $x \times 1,000,100 = 1,000,000,000$, so $x = 999.9$ ETH
• Trader receives: $1,000 - 999.9 = 0.1$ ETH

But wait — at the current price (1 ETH = 1,000 USDC), 100 USDC should be 0.1 ETH. So the trader gets exactly the expected amount!

However, larger trades cause slippage. If a trader swaps 100,000 USDC:

• Pool USDC: $1,000,000 + 100,000 = 1,100,000$
• Pool ETH: $x \times 1,100,000 = 1,000,000,000$, so $x = 909.09$ ETH
• Trader receives: $1,000 - 909.09 = 90.91$ ETH

The trader received 90.91 ETH, but at 1,000 USDC/ETH, 100,000 USDC should be 100 ETH. The trader lost 9.09 ETH to slippage.

This is why large trades on AMMs are expensive — they move the price.

Advantages of AMM

Permissionless. Anyone can deposit liquidity and become a liquidity provider.

Automatic pricing. No order book needed; prices adjust continuously based on supply/demand.

Capital efficient (somewhat). Even small pools can facilitate trades, though with slippage.

Composable. AMM smart contracts can be combined with other contracts, enabling complex strategies.

Disadvantages of AMM

Slippage. Large trades face high slippage because prices adjust as the pool composition changes.

Impermanent loss. Liquidity providers face losses if token prices diverge, even if they earn trading fees.

Inefficiency. The constant product formula is not the most capital-efficient pricing mechanism.

Variations

Uniswap v3: Introduced concentrated liquidity, allowing LPs to specify a price range. This reduces slippage and improves capital efficiency but adds complexity.

Curve Finance: Optimised for stablecoins using a different formula (less slippage for similar-priced tokens).

Balancer: Allows pools with arbitrary token ratios (not just 50/50).

Liquidity provider economics

Liquidity providers earn trading fees (e.g., 0.3% on Uniswap) but face impermanent loss if token prices diverge significantly.

To incentivise liquidity, AMMs sometimes offer additional rewards (governance tokens, yield farming) on top of trading fees.

Security considerations

AMM smart contracts are immutable. A bug in the price formula or token transfer logic could result in lost funds.

For example, if the code accidentally divides instead of multiplies ($x / y = k$ instead of $x \times y = k$), prices would be completely wrong, and traders would be arbitraged out.

See also