Thread by @expctchaos

0/ #DEXes like @Uniswap are a core piece of the decentralized financial infrastructure on $ETH & beyond✨

but what happens under the hood when you swap tokens on exchanges like $UNI, $SUSHI, $JOE or $BOO?🤔

explainer-🧵 on the constant product #AMM model (0/46)⚖️👇

1/ automated market makers have been one of the most important innovations in DeFi and the overall crypto space. The concept was ideated by @VitalikButerin in 2016 and pioneered by @Bancor & @Uniswap in 2017 & 2018 respectively

2/ the #AMM space has quickly grown into the largest sector within #DeFi & today we find decentralized exchanges using the AMM model in various forms on basically every smart contract L1/L2 (or as app-chain implementations)

3/ some examples include:

- $UNI on $ETH, $OP & #Arbitrum

- $CRV across multiple EVM chains

- $SUSHI on various EVM chains

- $JOE on $AVAX

- $BOO on $FTM

- $GLINT & $STELLA on $GLMR

- $RAY on $SOL

- $OSMO or $RUNE built on #Cosmos

4/ on $ETH, DEX landscape is heavily dominated by $UNI as the 7 day volume shows:

source: @DuneAnalytics

5/ #Uniswap's market share (based on volume) on $ETH amounts to 66%, followed by $CRV with 11% and $DODO at 8.4%

4) $SNX

5) $BAL

6) $SUSHI

source: @DuneAnalytics

6/ while aggregators like $1INCH or $ZRX's @matchaxyz have introduced a layer on top of AMM #DEX protocols to improve UX, the market remains dominated by volume generated directly on AMM DEXes

source: @DuneAnalytics

7/ over $50bn in trading volume have been generated across DEXes on $ETH in the past 30days and over 5m unique addresses have already interacted with DEX protocols on $ETH

8/ but instead of taking a closer look at the AMM #DEX landscape*, today we'll dive into how the basic constant product AMM model works and what happens under the hood when you swap tokens on a DEX like #Uniswap

*) follow-up post coming

9/ let's start with liquidity pools

on an AMM #DEX, there are two liquidity pools for each tradeable pair X and Y. The tokens are deposited by liquidity providers and allow other users to trade

10/ one pool contains a quantity x of token X, the other a quantity y of token Y

x and y are hence referred to as the available liquidity for X and Y

11/ liquidity providers (or market makers) have incentives to provide liquidity. Aside liquidity mining rewards that LPs might ear, LPs earn fees that are generated on the liquidity pools they provide liquiditiy to

12/ on @uniswap for example LPs earn transaction fees paid by other traders that amount to 0.3%. These fees are paid out to market makers, proportionally to the fraction of liquidity they contributed to

13/ while market makers can withdraw liquidity at any time, providing liquidity remains risky, for example due to divergence risk (a.k.a. impermanent loss)

thread covering impermanent loss in more detail will follow soon

14/ so, let's assume Alice holds ∆x > 0 of token X and swaps them for token Y using the #Uniswap protocol

what happens under the hood is that Alice adds her ∆x to the liquidity pool for X and in exchange takes some ∆y from the liquidity pool of token Y

15/ consequently, the liquidity pools now contain:

- token X: x + ∆x

- token Y: y − ∆y

16/ but how many tokens ∆y does Alice get? Or, what is the price of token Y in units of X?

on @Uniswap, pricing is regulated by the constant product formula...

xy = k

...and depends on the amount of available liquidity x & y, as well as the quantity ∆x Alice wants swap

17/ we define the quoted price of X in units of Y as P(XY) = y/x

however, this is a theoretical exchange rate and it's important to understand that one cannot trade at this price

18/ we also define what we refer to as the transaction price of X in units of Y as T(XY) = ∆y/∆x

as opposed to the quoted price, this is the price at which you can swap (sell) ∆x units of X for ∆y, given the currently available liquidity

19/ the constant product formula imposes that the product of the two liquidity pools xy does not change after the swap

in terms of a formula, ignoring the 0.3% transaction fees, we have:

xy = (x+∆x)(y−∆y) = k, where k is a constant

20/ we can hence solve for ∆y and then find the transaction price by computing the ratio T(XY) = ∆y/∆x

21/ starting from the constant-product equation we get:

xy = (x + ∆x)(y − ∆y)

= xy + ∆x∆y − y∆x + x∆y

⇒ ∆y = ∆x*(y/(∆x+x))

We can now compute the transaction price

T(XY) = ∆y/∆x = y/(∆x+x)

22/ so, we have:

- a quoted price P(XY) = y/x

- a transaction price T(XY) = y/(∆x + x)

therefore, if Alice sells token X, the transaction price is lower than the quoted price. The difference is the price impact of the trade

23/ hence, if Alice sells a larger quantity, she gets a worse transaction price. If the size of the trade is small relative to the available liquidity on the other hand, the transaction price approaches the quoted price

24/ @Uniswap provides the price impact Π in percentage terms, depending on available liquidity & trade size

we can easily compute price impact using elementary algebra:

Π = (P(XY)−T(XY))/P(XY) = ∆x/(∆x+x)

25/ the price impact is increasing & concave in the trade size

26/ suppose we want to trade 1k ETH in exchange for wBTC. Based on Oct 2020 prices, trade size is roughly $375'000. According to its Limit Order Book, trading on Bitfinex would result in a price impact of around 0.6%, excluding commissions

27/ on #Uniswap on the other hand, the price impact would only amount to 0.14%, saving a significant 46bps (roughly $1'700)

28/ remember, each swap impacts the available liquidity of X and Y. As a consequence, the quoted price will be updated as follows:

P(XY) = y/x → Q(XY) = (y−∆y)/(x+∆x)

29/ from the requirement (x + ∆x)(y − ∆y) = xy, it follows immediately that the new price after the trade is given by

Q(XY)= xy/(x + ∆x)^2

30/ hence, the new price is lower than the transaction price:

xy/(x+∆x)^2 = (x/(x+∆x))*(y/(x+∆x)) < y/(x+∆x)

31/ the below figure shows the price change corresponding to a swap of ∆x = 100k $ETH in exchange for $UNI

the pricing curve was computed liquidity using data from October 2020 (block 11034508)

32/ trading, listing and liquidity provision on #Uniswap is performed through smart contracts.High-level, the @Uniswap router governs and records swaps between pairs of tokens. The router does not hold any token, it just routes tokens to the correct destination

33/ each trading pair is connected to a specific smart contract & is created when the pair is listed on the #DEX for the first time

remember, a smart contract is formally an account on the $ETH blockchain & can therefore hold tokens

34/ in #Uniswap's case, the contract for the pair X/Y holds the x & y tokens in the corresponding liquidity pools. When a user provides liquidity, tokens are transferred to this contract & when a user swaps, tokens are transferred to and from this contract

35/ if a user wants to list a new trading pair X/Y for the first time, the user needs to create two liquidity pools, containing token quantities x of X & y of Y

36/ the initial ratio y/x chosen by the creator determines the initial relative prices:

- P(XY) =y/x

- PYX =x/y

37/ the emergence of automated market makers has also created major #arbitrage opportunities

38/ an #arbitrage opportunity between two financial assets A and B is a situation in which it is possible to pocket a risk-free profit trading the two assets

39/ classic examples of #arbitrage are:

- different prices for the same asset quoted in different markets

- triangular arbitrage between 3 assets

- mis-pricing of derivative securities with regards to the underlying

40/ in theory, #arbitrage opportunities should quickly disappear since as soon as traders find one, they trade to profit from it and as a consequence, prices adjust making the arbitrage disappear

41/ consistency of #Uniswap prices crucially depend on arbitrage. Assume the same pair X/Y can be traded both on #Uniswap and on other (centralized) exchanges

42/ suppose P(XY), the price on @Uniswap, is different from that on another exchange P'(XY)

the wedge P(XY) − P'(XY) provides an arbitrage opportunity

43/ since prices on an AMM #DEX like #Uniswap are solely determined by the ratio between assets in the liquidity pools, arbitrageurs can exploit the opportunity by executing trades that adjust the ratios in a way that the price converges with the price on other exchanges

44/ based on trading volume in the last 24h, overall #Uniswap volume is dominated by $ETH, followed by $USDC & USDC, $wBTC and $DAI

source: @uniswap

45/ the most active pool in the last 24h was the $USDC/$ETH pool with over $300m in volume

source: @uniswap

46/ thanks for reading if you made it until the end! If you liked this thread, please support me by following & sharing the first post of the thread 🥰✨