DEX

Uniswap V1 notes, math-first

Uniswap V1 is the first time I saw a pricing function you can touch. Not an orderbook. Just reserves and a curve.

State

Two reserves:

$x,\ y$

Constant product:

$x\cdot y = k$

I keep coming back to: the price is a slope of that curve.

Quote

If you add a small amount $\Delta x$ (ignore fees for a second):

$x' = x + \Delta x$ $y' = \frac{k}{x'}$ $\Delta y = y - y'$

So the output is not linear in $\Delta x$. The curve is the slippage.

Fee

V1 trade fee is 0.3%. A clean way to remember the implementation is:

$\Delta x_{eff} = 0.997\,\Delta x$

Then plug $\Delta x_{eff}$ into the same constant-product math.

Price

Spot price as a local ratio:

$p = \frac{dy}{dx} = -\frac{k}{x^2}$

You can read it as: as $x$ grows, the curve flattens, and you pay less $y$ per extra unit of $x$.

Notes

• The pool is the book.
• Liquidity is a scale factor (it stretches the curve).
• Most surprises in UX are just the curve meeting integer math.

References

• Uniswap V1 whitepaper
• Uniswap V1 core contracts