Uniswap is an automated market maker (AMM). You can think of an AMM as a primitive robotic market maker that is always willing to quote prices between two assets according to a simple pricing algorithm. For Uniswap, it prices the two assets so that the number of units it holds of each asset, multiplied together, is always equal to a fixed constant.

That’s a bit of a mouthful: if Uniswap owns some units of token `x` and some units of token `y`, it prices any trade so that the final quantities of `x` and `y` it owns, multiplied together, are equal to a fixed constant, `k`. This is formalized as the constant product equation: `x * y = k`.

This might strike you as a weird and arbitrary way to price two assets. Why would maintaining some fixed multiple between your units of inventory ensure that you quote the right price?

Uniswap by example

Let’s say we fund a Uniswap pool with 50 apples (`a`) and 50 bananas (`b`), so anyone is free to pay apples for bananas or bananas for apples. Let’s assume the exchange rate between apples and bananas is exactly 1:1 on their primary market. Because the Uniswap pool holds 50 of each fruit, the constant product rule gives us `a * b = 2500`—for any trade, Uniswap must maintain the invariant that our inventory of fruit, multiplied together, equals 2500.

So let’s say a customer comes to our Uniswap pool to buy an apple. How many bananas will she need to pay?

If she buys an apple, our pool will be left with 49 apples, but `49 * b` has to still equal `2500`. Solving for `b`, we get 51.02 total bananas. Since we already have 50 bananas in inventory, we’ll need 1.02 extra bananas for that apple (we’ll allow fractional bananas in this universe), so the price we have to quote her is 1.02 bananas / apple for 1 apple.

Note that this is close to the natural price of 1:1! Because it’s a small order, there is only a little slippage. But what if the order is larger?

You can interpret the slope at each point as the marginal exchange rate.

If she wants to buy 10 apples, Uniswap would charge her 12.5 bananas for a unit price of 1.25 bananas / apple for 10 apples.

And if she wanted a huge order of 25 apples—half of all the apples in inventory—the unit price would be 2 bananas / apple! (You can intuit this because if one side of the pool halves, the other side needs to double.)

The important thing to realize is that Uniswap cannot deviate from this pricing curve. If someone wants to buy some apples and later someone else wants to buy some bananas, Uniswap will sweep back and forth through this pricing curve, wherever demand carries it.

Uniswap sweeping back and forth through its pricing curve after a series of trades.

Now here’s the kicker: if the true exchange rate between apples and bananas is 1:1, then after the first customer purchases 10 apples, our Uniswap pool will be left with 40 apples and 62.5 bananas. If an arbitrageur then steps in and buys 12.5 bananas, returning the pool back to its original state, Uniswap would charge them a unit price of only 0.8 apples / banana.

Uniswap would underprice the bananas! It’s as though our algorithm now realizes it’s heavy on bananas, so it prices bananas cheap to attract apples and rebalance its inventory.

Uniswap is constantly performing this dance — slightly moving off the real exchange rate, then sashaying back in line thanks to arbitrageurs.

Source📚 Dragonfly Research