Uniswap AMMs: The Magic Formula Decoded
Everything you need to know about Uniswap's AMM equation
Kadeem Clarke
March 12, 2023
Curious about the math behind AMMs like @Uniswap?
Here's everything you need to know about it π

1οΈβ£ The Magical Equation π§Β
Let's look at the equation behind the Uniswap AMM
X * Y = K also known as the Constant Product Formula
β’ X = Initial Quantity of Token 1 ($aTKN) in the pool
β’ Y = Initial quantity of Token 2 ($bTKN) in the pool
β’ K = Constant value
2οΈβ£ Letβs look at an example πΒ
Let's look at an example to understand this.
πΒ Suppose at the initial state the pool has:
X = 100 $aTKN
Y = 100 $bTKN
So, 100 * 100 = 10000 = K(constant)
This constant is fixed and can't be altered.
πΒ Suppose you want to buy 20 $bTKN in exchange for $aTKN
So how many $aTKN do you need to deposit into the pool?
Let's calculate...
πΒ Remember the constant product formula:
X * Y = K
i.e β $aTKN * $bTKN = K
Change (β³) = | Initial state - Final state |
πΒ At the final state of the pool,
β³ $bTKN = 100 - 80 = 20
β (100 - 20) * $aTKN = 10000
β $aTKN = 10000/80 = 125
So there needs to be 125 $aTKN in the final state of the pool to maintain the value of K
β³ aTKN = 125 - 100 = 25
So you need to deposit 25 $aTKN to get 20 $bTKN.
3οΈβ£ Final Step : Determine token price
Now let's determine the price of the tokens w.r.t each other at the final stage.
This is the graph of the above equation
And the SLOPE of this graph will give you the PRICE of the X and Y.
So what's a Slope? ποΈΒ
And as you can see in this video as you deposit more and more token X, the slope (or price) falls very sharply.
The Red line is the slope. https://t.co/W1MHdHHuHZ
— The Crypto Illuminati (@0x_illuminati)
Jul 29, 2022