Volatile curve (x·y = k)

Volatile pools keep the product of reserves constant (it only grows, from fees).

$$x \cdot y = k$$

Output for an exact input

With reserves $x$ (in) and $y$ (out), fee $f$ and input $\Delta x$:

$$\Delta x' = \Delta x\,(1-f) \qquad \Delta y = \frac{\Delta x' \cdot y}{x + \Delta x'}$$

Worked example

Pool: 100 ETH and 300,000 USDC, fee 0.30%. Swap 1 ETH.

• $\Delta x' = 1 \times 0.997 = 0.997$
• $\Delta y = 0.997 \times 300{,}000 / 100.997 \approx 2{,}961.5$ USDC
• Spot price was 3,000 → you pay ~1.28% (0.30% fee + 0.98% price impact).

Marginal price

$$P = \frac{y}{x}$$

The curve never runs out: as $x$ grows, $y$ approaches zero but never reaches it, so the pool can always quote a price.