Constant Product Pools

Constant product pools are one of the most popular AMM models, first introduced by Uniswap. They allow two tokens to be swapped while ensuring the reserves follow a mathematical rule that prevents depletion.

How the Formula Works

The fundamental formula for constant product pools is:

$$x \cdot y = k$$

Where:

• $x$ = reserve of token X
• $y$ = reserve of token Y
• $k$ = constant product, which must remain unchanged after a swap

This relationship ensures that when a user adds some amount $\Delta x$ of token X to the pool, the output $\Delta y$ of token Y is calculated such that:

$$(x + \Delta x)(y - \Delta y) = k$$

This is often referred to as the invariant equation.

Swap Output Formula

In practice, the amount of output token a user gets from a swap is computed using a derived form of the above invariant:

$$\Delta y = \frac{y \cdot \Delta x}{x + \Delta x}$$

This shows that as more of token X is added (larger $\Delta x$), the marginal price increases, meaning slippage occurs. The pool automatically adjusts the price based on trade size.

When to Use

Constant product pools are best suited for:

• Tokens with high volatility
• General-purpose pairs like ETH/USDC or BNB/BTC
• Use cases where any-to-any swaps are needed without relying on oracles