# Splitting the Atom in "Concentrated Liquidity" - Mark Richardson

- Speakers: Mark Richardson
- Channel: [ETH Warsaw](https://streameth.org/eth-warsaw)
- Date: 2025-10-07
- Duration: 28:10
- Watch: https://streameth.org/watch/yt-YL3XVhRGHhk
- YouTube: https://www.youtube.com/watch?v=YL3XVhRGHhk

## Description

Concentrated liquidity (aka "amplified" liquidity) refers to a specific transformation of the canonical constant product bonding curves first introduced in 2017. Concentrated liquidity protocols allow liquidity providers to quote prices with arbitrarily low rates of slippage compared to the constant product paradigm, with the caveat being that the available liquidity is enumerated over a specific, finite price interval as opposed to an infinite one.

Its most popular implementation is characterized by a large set of discrete, 2-dimensional price boundaries, from which liquidity providers may select a subset to contribute tokens into. Perhaps owing to its level of market saturation, it is now assumed that these design choices are synonymous with the concentrated liquidity general theory. This is not true. 

With this talk, I aim to introduce a more fundamental understanding of concentrated liquidity theory, including its 1-dimensional and n-dimensional realizations, and its utility beyond the status quo in contemporary DeFi architecture.


🧜🏻‍♀️ ETHWarsaw is a series of educational and entertaining events for an active community of blockchain builders, developers and enthusiasts with focus on Ethereum-related tech. Once a year, we organize a large conference and hackathon for the community in the center of the Polish capital with speakers from the best web3 projects and participants from all over the world.

Follow ETHWarsaw on social media for the latest updates!
X (Twitter): https://twitter.com/ETHWarsaw
LinkedIn: https://www.linkedin.com/company/ethwarsaw
Telegram chat: https://t.me/joinethwarsaw
See you all at our events in Warsaw 🙌🏻

## Transcript

thank you so much so yeah um first I want to offer uh a perspective an unpopular one uh about the way that blockchains attract user bases and how they treat those user bases because I think when you know with the uh the Bitcoin white paper kind of came this uh staple belief of don't trust verify and I'm sure that you've kind of heard this iterated before not just in Bitcoin communities but in all communities and what it means is all harm done in blockchain is self harm by definition because it assumes that every user knows how to use elliptic curve cryptography and every user has sufficient knowledge to audit every smart contract they interact with for example like this is the ethos of blockchains which is that no no one can do to you anything that you haven't done to yourself by not taking seriously your own U responsibility to understand the systems that you are interacting with and what's interesting is that that is a really bad precedent to have from a marketing perspective because these things are very very complicated and very difficult to use and if you're putting first your responsibility as a developer to educate the people that are using your products you're going to alienate like 99% of that audience and maybe that's okay and I'm here from the I don't care about Mass adoption perspective I'm here from the put Education First and the idea is to bring in as many people into your classroom and teach them about the things that you're doing so that they can actually make rational informed decisions rather than ad-based influencer based type decisions and so with that in mind this is a a slightly more technical talk than the ones you're probably used to seeing it's based on um a uh a paper that I wrote recently called defi concentrated liquidity from scratch um which you can access at archive I've just recently um updated it and there's a corrected version there um that I I put up just in uh in August of last week um and the this QR code will take you there now in order to understand um like why I did this and and kind of where it's going um we need to kind of go back to um to First principles I'm not going to I'm not going to uh excuse me I I don't want to assume that anyone is already familiar with what bankor is and where we've come from and how far we've we've moved but essentially in 2017 bankor which was a a company in Israel that was experimenting with Community currencies okay so this is pre uh pre- blockchain world uh we uh we had a um a head of Economics a gentleman by the name of Bernard leer um who was the gentleman who invented the Euro and introduced the euro to European Banks um so we have sort of a currency first um approach to things and these Community currencies were working really really well except for that when those currencies need to act or interact with economies outside of their own you need to discover an exchange rate and this was really really problematic because Community currencies are very illiquid and market makers basically refuse to use them and so what we wanted was a computer program that can abstract away the need for human to quot prices and instead quote those prices automatically and thereby make markets for different Community currencies versus each other and thus the automatic Market maker was born um and with it came things like pool tokens and bonding curves and so on and so forth and the idea is relatively simple you have um this uh cylinder shape is representing a liquidity pole and the Brain box above it is representing a smart contract it's basically saying that all you need is these components and then any incoming token immediately knows its quote price versus any other token and so you can exchange them versus each other you can also be the person who is providing liquidity to keep this thing running and so we Define other parts of the algorithm which knows how to keep account of who is owed what who has contributed liquidity to this contract what trades were made and what is owed back to the people that did it and these are called the liquidity provider tokens or liquidity pull tokens um which back in the day when known as smart tokens or relay tokens and so there's two ways that this thing interacts one is that you're contributing whatever your community currency is to the pool and it's issuing an LP token or you are giving back your LP token because you're trying to redeem the underlying Capital this is the formula that runs it okay and has been known since back in the day as the bankor formula um but you may know it more Cally as constant product Market making or cfmm and there's two main ways that this plays out the most familiar one is the two-dimensional version where you've got token balance um on both axes and a hyperbola that um quotes all valid points between them but the there was nothing in the original white paper or patent for this thing that suggested that only two Dimensions is allowed this is just the one that was made Popular by developers that I think weren't really you know grasping the the theory as it was originally written because you can very easily do this in three dimensional are actually any number of dimensions and build like bonding hypersurfaces and things now balancer did come along a little later and write a much better white paper um but the theory did come from bankor and it came from a need to create markets with Community currencies now the way that this thing generally operates is that you use this bonding curve or bonding surface or hypersurface to determine all of the allowed token balances and so what you're allowing people to do is essentially to come in and interact with that system by moving um what you know is the the current state of the pool around um thereby achieving exchange so each one of these arrows is representing a certain number of tokens either coming into or out of the pool and they're necessarily coupled together now when you abstract uh some of this into its first derivative you can actually see the price curves fall out of this quite nicely and what I wanted to do with this animation this was was prepared for a lecture series that I gave for token engineering Academy um a few months ago what I wanted to point out is actually all you need is this component this is kind of a onedimensional bonding curve and you can compi you can compose them together any way you like to achieve all kinds of different exchange derivatives the fact that the most common one is the one where these things are implicitly uh paired together to create a liquidity pool is a design choice or a marketing Choice the technology isn't contingent on that so with that out of the way I want to show you now where uh Amplified or concentrated liquidity came from what it looks like and where I'm going with this presentation in general so the Amplified bankor formula is this it looks a little bit more frightening than the last one but I show you it's not all we've done is to add this Delta term in here which basically moves the ASM toote of that hyperbola into the negative domain and this means that we can start quoting prices arbitrarily between certain bounds and you will know this from the concentrated liquidity protocols that have become very popular where you can basically nominate the two price bounds under which your liquidity will remain active the problem is that the way that it's been described in the white paper literature and the fact that the one that has the presiding dominance in the market today is the one that's been fored over and over and over again it has OB fiscated the fact that there are actually many better algebraic and trigonometric descriptions of concentrated liquidity that developers don't seem to be aware of and certainly the users don't seem to be aware of so the transformation that allows you to derive essentially every single description comes from this if you take the normal um pseudo uh unit hyperbola and twist it 45 Dees you can create an actual unit hyperbola and from there um you start to Define this angle which I'm calling fi now this angle um is not the angle that you might be used to on a circle that represents a rotation but rather the angle that represents an area underneath some curve so here thet over two uh is still representing a rotation on the circle but it's actually this um this area of the sector um that better describes theter and generalizes to the case in a hyperbola so this is still an angle but it's a hyperbolic angle not a circular angle and because of the ARX hinch function um is relatively well uh described across the reals we can actually use that to compute this angle uh f for any hyperbola and then that we can use that as a a basis from which all descriptions algebraic or trigonometric um can uh from which all descriptions of uh concentrated liquidity arise and so so starting with those principles and if you go to that archive paper what I'm showing is that the first description of concentrated liquidity from bankor in 2020 is literally identical to Unis swap v3s a year later and is also identical to our new uh concentrated liquidity description from October of 2022 now why would you need three different equations if they're all describing the same mathematical object right why are they all doing the same thing and why put so much effort into researching them if they are redundant the problem is that these shapes that we're using sure like in their abstraction algebraically they work very well but eventually you're going to have to represent these using bits in a smart contract and it depends on what your product is doing um that will influence which one of these descriptions you want to use so for example with a description like this one using the geometric mean point of this curve might be very helpful but in a system like Unis swap V3 where the thing that you are trying to parameterize is the edges of this curve right where the the price bounds are starting to exceed the liquidity that you have that's a much more important price point to sorry a much more important parameter to know with high accuracy because if you fail to calculate it correctly then what you're going to end up doing um is uh exposing exploit vectors such as what's happened in kibber Swap and other uh concentrated liquidity variants but what's interesting is that when you speak to developers who are trying to build their own concentrated liquidity system they will just use this system and shoehorn it into whatever product they're building even if that is introducing exploit vectors with it in essence this is the right concentrated liquidity description for UNIS swap V3 and for UNIS swap V3 only if you are trying to uh build something that's more sophisticated or has like a a different um data uh data structure it would actually be better to use uh an equation that takes that data structure as uh an assumption and then rearranges the algebra in order to make it work and this is what we did with carbon defi so again these uh you know these pictures they are fading in and out right the uh the the colors are changing the equations are changing but you can see that geometrically it's referring to one and the same shape it is the same thing underneath so what can you do with this right that you can't do with Unis swaps description just quickly if you want to go back over any of what I've done right I had to get through this very very quickly I'm running out of breath a little um but I have um a prepared a lecture series it's about five lecture series long each lecture is between 60 and 90 minutes so it's quite uh it's quite long um but I go through basically all of those steps in the detail required for you to fully internalize it unfortunately in a 30-minute lecture War sorry I can't do that but that's how we get from constant product to concentrated liquidity in the general case how do you get from now concentrated liquidity in its infinitely many algebraic descriptions to what I'm calling asymmetric liquidity which is the first of its kind okay so this is where we get to actually start discussing product and I'm going to start with a market back test so when I first developed the concept of asymmetric liquidity what I did was to build a market back testing toy basically in Python and this was the thing that I was using to demonstrate to let's say potential customers institutions but also my smart contract developers as to what this product is going to do what my front end Engineers were going to have to try and communicate and this is the way that I did it I have uh you know a bunch of different uh price API keys for things like coin gecko coin market cap crypto compare where I can pull any two tokens and then compare their price so let's just go with the BTC e price chart you can actually do this with any um any pair on our website right now so if you go to carbon defy doxyz there is a simulator resource now right there that will allow you to to do exactly the experiment that I'm doing here so you don't feel like you have to take my word for it if we look at traditional concentrated liquidity what it's saying is that you can choose any two price bounds under which your liquidity will remain active but because of the way that the algorithm works you are buying and selling at every price point in here save for some very very small Delta that we call the fee but if you with me at ECC I've also proved there's no such thing as a fee in an amm right all you have is actually a price spread so you get a very very small difference here between your bid and ask but basically you have to service every single one of these price points as the market moves around inside it whereas with asymmetric liquidity we can actually separate our bids and our asks arbitrarily we can say rather than buy and sell in this specific range I only want to buy in this range right when the price of eth in terms of Bitcoin is in this point happy to buy and when uh the price is getting up here this is when I want to sell and this is a much more uh close to the Heart Way of Market making right this is how people think when someone buys ethereum at a certain price they usually have a price point in mind that they want to sell at and often they will actually just put a limit order in the book on coinbase or binance to sell when it gets to that point previously we haven't been able to do that kind of uh you know centralized limit order book Market making on chain because of the enormous data structures that are required to support it what's interesting is that when you abstract it into the integral like we have here um the memory footprint isn't just lower than something like ether Delta the memory footprint is lower even than a traditional amm right this despite the level of apparent sophistication is actually a smaller smart contract than Unis swap V3 it's also a smaller contract than Unis swap V2 and uh the gas consumption also drops commens with that okay so uh again the simulator has a couple of different uh features and we can go over that a bit later let's just have a look at the dynamic so that I can make clear exactly how it is that we've changed things this is uh the eth BTC chart that we saw before and we're enumerating prices from zero to Infinity right so this is the normal sort of bankor V1 um constant product price uh Market making uh algorithm and what I want you to see here is that essentially as the price is moving up and down here that translates to a left and right motion here and you can see that the liquidity on the amm is distributed much the same way you would see it distributed on a book right that is the idea behind the amm but this is maybe the more familiar representation so what I've done here is to basically just draw that part of the hyperbola that we're interacting on and so we're now transla the price point here to the first derivative of this curve and so what we're looking at is basically as the um the token balances are moving back and forth commensurate with this how is the position of our uh how is the value of our position improving versus hodle and you can see and uh you know maybe this is coming no surprise to people who have been around defi for a little while but the value tends to go down right and this wasn't necessarily a super volatile pair the fees that we're accumulating right or the really the K growth of this curve is small compared to our exposure so uh at its peak we have a minus 1.5% drop which is I guess pretty significant but over time it still Trends up right you should be able to see if I if you were to just uh draw a straight line from here to here there's still a very slight positive trend on it and this is generally where amm performance should come from very very very long well sustained uh uptrends but for tokens that have a high volatility uh response with respect to each other but Trend sideways so we can summarize these kinds of um these kinds of positions like this where the blue line here is representing the um the portfolio value in terms of the amm and the Gold Line is representing the Huddle position had you not decided to contribute liquidity with it so whenever the blue blue line is above the Gold Line This is your impermanent gain if you want to think of it like that and whenever the blue line is beneath the Gold Line like it is over here that's your impermanent loss now concentrated liquidity was originally marketed by Unis swap as being like a decent answer to impermanent loss we're going to find out if that's true so we're using our same um the same methods that we were to interrogate the the constant product case and what you should be able to see is that the price is kind of Leaping around and you can see it leaving a very small Gap right that's that effective fee level that um that you charge on a Unis swap pool um and we're letting those fees accumulate over here now importantly as our asking prices are being taken it's still being filled with bid prices immediately behind it you can see that represented here as well and when the price comes back down you'll see that those bidding prices are replaced by asking prices but once the price moves above or below that band we've effectively run out of liquidity and the thing basically stalls now we can still draw that hypera um except now it's a much more zoomed up version because on the constant product case it asint topically approaches zero in either axis but this case it actually strict zero now you can see here and I want you to pay attention to this on on this animation cuz we're about to uh have a look at as metric liquidity but what I've done is to label the eth balance on the Y and the BTC balance on the X um on the left chart and on the right chart I've got the BTC balance on the Y and the eth balance on the X now when this uh price starts falling back through this region what I want you to just pay attention to is that they respond symmetrically right and that shouldn't be too uh surprising because this is just the transpose um but they are responding the same way right they are one in the same uh price quoting algorithm now a lot more noise here so you can see that with concentrated liquidity it's a lot more sensitive to things that are happening inside it so for example we have a very large amount of impermanent loss way back here when the price is moving outside of our range but overall it trended quite high and uh this position actually is more profitable than the um standard constant product position but it didn't come without cost because this um this uh draw down that we had early on um in this simulation is quite significant okay what about asymmetric liquidity so this is the one that comes with bangor's new product that we call carbon this is not a mistake it's an anagram of the word bankor um it's meant to represent kind of A New Perspective on Market making so let's have a look at those EXA same animations now because our bids and our asks they're still linked together but they're not necessarily dependent on each other when the price comes down through our bids rather than the asking prices continuing to chase the market down it stays where it is and so again this is a design Choice as our asking prices are taken we're still converting our base asset to cash asset but the asking prices do not continue to chase the market up which is another way of saying that this protocol allows you to wait for better prices it allows you to wait for cheaper prices to buy and it allows you to wait for higher prices before selling now um on the invariant function graph we can see unlike before these are not the transpose of each other the eth balance has its very own bonding curve and the BTC balance has its very own bonding curve and I've labeled the X um axis on both of these as a fictional x coordinate um because this is really just the shadow of performing an integration across the price curve now the performance versus hudle is much much better right this is actually giving you and we can see in the next uh chart here um a dramatically improved uh uh portfolio performance profile versus either concentrated liquidity or constant product in the simulator quantifies it right if you want the code for this I have it in Python so if you want to audit this yourself I'm very happy to do that um but these these results really do speak for themselves so if we're using constant product from 2017 this that exact heads up comparison gives a 1% profit um for the yeah the 2017 technology 14% profit for 2020 and for carbon defi which is from 2022 um about a 37% profit which is pretty interesting now you might think that I've like cherry-picked this but actually I chose this because I think it's the one that you might find more believable because of this ability to set arbitrary bids and asks you can create ridiculously high Returns on this so you didn't want to you know I could very easily have chosen a back testing period that would produce stupid apys like the ones we're used to seeing in defi but what I've tried to do is to choose one that I think has a pretty decent heads up comparison so yeah that's the product is carbon defied is the new um the new generation of uh of concentrated liquidity protocols if you want to know more about this um this is the website where you can access it this is the the simulator and if you want to um you can reach me at any of these um at any of these destinations so thank you so much for your time I've tried my best to get through it um without uh contributing too much to the uh how much we're running over um but I'd be very happy to accept any questions and thank you for your attention massive Round of Applause for Mark guys yes we do have time for questions and thank you for going through so swiftly by the way and I had no idea carbon was an anagram for a banker and I've been using it please question guys who's up for the first question for Mark here uh thank you for your presentation my question uh very quickly do you think the API will be decreasing over the years as just the 2017 and the other numbers that she showed us well because it's a really so the question is do I think that the API using this product is going to Trend down over the years um yes and no as volatility between various cryptocurrencies continues to like tighten then yes the ability to trade against that volatility and earn a profit is going to tighten with it that's another way of saying that the margins are going to become diminishing over time but it still comes down to the skill of the trader so for example for things like Ethan BTC let's say 20 years from now there might just be like free floating you know currencies and they have like a very very minor um volatility like bitcoin's basically just approached the ASM toote and settled on whatever price it settled on ether settled on whatever price it's settled on and now trading between them you're trading like basis points you're not going to be able to get you know a return like that but that doesn't mean that someone very cleverly has released a new meme token on salana or something and then created a competing meme token on ethereum and is like bridging them to create more volatility or something like that the point is is that the product is only as good as the person is using it you know you need to make these decisions these are the prices that I'm selling and buying at the protocol doesn't prescribe that for you so I think my perspective is that people will actually get better at using these kinds of tools over time and so I think the apys will will increase slowly but they should stabilize and then eventually Trend down as volatility Trends down any any other questions there guys what time for one more I love it straight in so um I'm sorry because I wasn't here since the beginning but so you said this protocol is as good as a Trader so there is no option to really like automatically set up the most optimized I I would say way to do it because I I assume also if everybody would do it then they will lose on the effectivity too right yeah correct there's I mean there there's no automated way to do it at the contract level because this would require things like Oracle price feeds and other like third party dependencies that add like a lot of let's say uh attack surface area that we're not really happy with so the contracts are bare Burns having said that that doesn't mean that you can't use your own like offchain API or something to modify your position over time according to whatever signals and things that you want to use so for example um one of the institutions that I was speaking with is using a like momentum indicator to trade ethereum and they like this product a lot because they don't like spot trading on chain because of Sandwich attacks and other things and this is sandwich attack immune and so what you can do is say wait for the momentum indicated a hit and then quote a price on chain and wait for it to be taken now that price quoting component doesn't know about why you quoted that price so the automation is on the user right if you want to automate It Go right ahead but it's not built into the product and it never will be amazing another massive run of Clos from Mark there guys