By Chadi EL ADNANI, November 2022
This article was part of a broader report on the following theme: “Challenges and opportunities: The future of DeFi in TradFi”. Crypto Valley Association selected it among the top 5 pieces for its 2022 Call For Papers challenge.
Following the previous article, where we provided a more qualitative analysis of DeFi and DEXs, we dive directly into price discovery mechanisms using the mythical constant function x ∗ y = k formula. We also take inspiration from the paper ”Automated Market-Making for Fiat Currencies” by Mr Alex Lipton and Mr Artur Sepp [LS21] to quantify on-chain/off-chain arbitrage opportunities. Finally, we provide some market adoption charts for selected DEXs native tokens.
1. Price discovery in Uniswap v3
Here is a brief reminder about how the constant function x∗y = k formula works to define prices and liquidity levels in AMMs: let tok1 and tok2 be two fungible tokens, x the liquidity pool reserves of tok1 and y the reserves of tok2, k a constant such as
x ∗ y = k = L2 (1)
where L represents the pool’s liquidity.
Let us consider the case where a user desires to sell ∆x units of tok1 in exchange for ∆y units of tok2.
With a 0% fee assumption, we have:
(x + ∆x) ∗ (y – ∆y) = L2 = x ∗ y (2)
The name constant product market maker comes from the fact that any trade x for y changes reserves such that x ∗ y after the trade remains equal to the constant L2. More precisely, we could face two cases:
- A user would like to sell x units of tok1, and the smart contract needs to compute y.
- A user would like to buy y units of tok2, and the smart contract needs to compute x.
Solving for x and y in each case, we have:
Uniswap v3 takes this approach to avoid having to compute square roots when executing swaps; all formulas in v3 are done on the square root of the price, with the front-end application converting from √p to p and vice versa. As we will see later, using L and √p is convenient because only one change at a time: Price changes when swapping within a tick, whereas liquidity changes when crossing a tick or when minting or burning liquidity.
As mentioned before, one of the main novelties of Uniswap v3 is concentrated liquidity. In earlier versions, liquidity was provided uniformly across the x ∗ y = k reserves curve, meaning that liquidity was provided across the entire price range [0; +∞[. This is simple to implement but meant that as much as 95% of the assets held in a pool were never used in some cases, especially in stablecoin to stablecoin pools. In the wake of these observations, Uniswap v3 introduced the concept of a position, where LPs allocate liquidity to a finite range.
Questions and comments about the rise of DEXs and DAOs can be addressed to email@example.com
- [LS21] Alex Lipton and Artur Sepp. “Automated Market-Making for Fiat Currencies”. In: (2021).
- [tea21] Uniswap team. “Uniswap v3 Core”. In: (2021).
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