# Constant Product Market Maker: making Uniswap pricing intuitive.

Krishang Nadgauda / June 10, 2021

This article is a work in progress. Math is hard and DeFi can be confusing. If you find any errors, feel free to contact me at krishang@nftlabs.co with a correction.

Decentralized exchanges (DEX'es) lie at the heart of DeFi. A core piece of each DEX is the math that regulates its asset swaps. For instance, swapping a given amount of USDC for DAI on Uniswap, compared to Curve, is not the same. This article will attempt to make the constant product market maker function intuitive, the math that Uniswap uses to price its assets, without defaulting to jargon or formulae like x * y = k without explanation.

This article is not an explanation of how DEX'es or Automated Market Makers work in general. Here, I'll primarily discuss a piece of how a DEX works — its market making function. The discussion is primarily centered around Uniswap's constant product market maker function.

## Constant product market maker

If you're familiar with Uniswap, you've seen this equation x * y = k thrown around.

The equation x * y = k governs asset swaps on Uniswap, where x and y represent the quantities of two different assets in a liquidity pool, and k represents a value called the constant product invariant. The given equation governs the price of a given amount of one asset in the liquidity pool, in terms of the other asset.

Consider a liquidity pool consisting of Token A and Token B. We can calculate the above mentioned 'constant product invariant' or k as k = (Amt. of Token A in pool) * (Amt. Token B in pool).

Pre-trade: $k = Amt. A \times Amt. B$

Given the current state of the pool, we want to find out how much a given amount of Token B costs, in terms of Token A. Now, swapping Token A for Token B changes the composition of the liquidity pool. Our governing equation x * y = k dictates that the value of the invariant k must remain constant, regardless of the composition of the pool post-trade.

Post-trade: $k = (Amt. A + x) \times (Amt. B - y)$

Here, x is the amount of Token A we'd have to pay in exchange for y amount of Token B. Now, Uniswap takes a protocol fee of 0.3% for every trade. So, when you pay x amount of Token A in exchange for some amount of Token B, what you get in return is 0.997x Token A worth of Token B.

Pricing the trade: $k = (Amt. A + 0.997x) \times (Amt. B - y)$

Again, we want to know the price of a given amount of Token B in terms of Token A i.e. the amount of Token A (x) you would need to add to the liquidity pool, to receive y amount of Token B. Therefore, we want to rearrange the equation labelled as 'Pricing the trade' to be able to calculate x (how much Token A you need to pay) as a function of y (how much Token B you want).

Price of Token B in terms of Token A:

$f(y) = \frac{1}{0.997} \times (\frac{k}{Amt. B - y} - Amt. A)$

(Similarly) Price of Token A in terms of Token B:

$f(y) = \frac{1}{0.997} \times (\frac{k}{Amt. A - y} - Amt. B)$

If we are to generalize the equations further, by not fixing the fee at 0.3% like Uniswap —

Price of Token B in terms of Token A:

$f(y) = \frac{1}{1 - fee} \times (\frac{k}{Amt. B - y} - Amt. A)$

(Similarly) Price of Token A in terms of Token B:

$f(y) = \frac{1}{1 - fee} \times (\frac{k}{Amt. A - y} - Amt. B)$

We now have a general equation that we can use to find the price of one asset in a liquidity pool, in terms of the other asset in the liquidity pool. Let's work through an example.

Consider a Uniswap liquidity pool containing 100 ETH and 1000 EXAMPLE (an ERC20 token). Here, k = 100 * 1000 = 10**5. We calculate the price of 1 ETH in terms of EXAMPLE as —

Price of 1 ETH $= \frac{1}{0.997} \times (\frac{10^5}{100-1} - 1000) = 10.13$ EXAMPLE

Note that though k is supposed to be an "invariant" i.e. considered as a constant when pricing one asset in terms of the other, k actually increases a little after every trade, since we account for the 0.3% fee in our equation for pricing the assets.

In our example, the value of k pre-trade is 10**5 whereas the value of k post-trade is around 99 * 1010.13 = 100,002.87. If we would charge no fees, the price of 1 ETH in terms of EXAMPLE would be around: $(\frac{10^5}{100-1} - 1000) = 10.10$ EXAMPLE instead, maintaining k at 10**5. The increase in the size of the liquidity pool (as reflected by the increase in k) represents the payout to the pool's liquidity providers.