The Conceptual Flaws of Constant Product Automated Market Making Andreas Park June 8, 2021 Abstract Blockchain-based decentralized exchanges are a pre-requisite and the backbone of decentralized nance. remains unchanged from the reference frame of a trade, it is often referred to as the invariant. Stocks, gold, real estate, and most other assets rely on this traditional market structure for trading. The most popular of them is the Constant Function Market Makers (CFMM) [37], which maintain a mathematical invariant (for example, a product of the quantity of assets) during the trade. Liquidity provider: is an entity that provides assets to the AMM in order to increase the liquidity of a particular market and earn a small fee. Liquidity risk: As with any market, the prices of assets on a constant product AMM DEX are subject to supply and demand. In fact, these formulas free us from calculating prices! Uniswap is the most popular AMM on Ethereum. Professional market makers who ensure that exchanges have enough liquidity, need to be able to rapidly cancel and update their orders when market prices move (which they always do!). Arbitrage trades have been shown to align the prices reported by CFMMs with those of external markets. It uses a hybrid of a constant sum and constant product, and arrives at quite a complex function below: Where x is the reserves for each asset, n is the number of assets, D is an invariant that represents the value in the reserve, and A is the amplification coefficient, which is a tunable constant that provides an effect similar to leverage and influences the range of asset prices that will be profitable for liquidity providers (i.e. pool reserves. Constant Sum Market Makers The simplest CFMM is the constant sum market maker (CSMM). In practice, because Uniswap charges a 0.3% trading fee that is added to reserves, each trade actually increases k. A constant product function forms a hyperbola when plotting two assets, which has a desirable property of always having liquidity as prices approach infinity on both sides of the spectrum. Because of this matching process, there is the possibility that some orders may take a while to get filled, if ever. Constant Product AMMs are simple to implement and understand. Liquidity implications of constant product market makers. . [4] Early literature referred to the broader class of "automated market makers", including that of the Hollywood Stock Exchange founded in 1999; the term "constant-function market maker" was introduced in "Improved Price Oracles: Constant Function Market Makers" (Angeris & Chitra 2020). Stocks, gold, real estate, and most other assets rely on this traditional market structure for trading. These pools are funded by liquidity providers so that the traders can trade against these pools. {\displaystyle \varphi } tokens that the pool is holding. We study axiomatic foundations for different classes of constant-function automated market makers (CFMMs). $$-\Delta y = \frac{xy - xy - y r \Delta x}{x + r\Delta x}$$ Chainlink Price Feeds already underpin much of the DeFi economy and play a key role in helping AMMs accurately set asset prices and increase the liquidity available to traders. Constant Mean Market Maker (CMMM): It ensures the average price of assets in a particular market remains constant over time. If the AMM price ventures too far from market prices on other exchanges, the model incentivizes traders to take advantage of the price differences between the AMM and outside crypto exchanges until it is balanced once again. . The paper also looks at the impact of introducing concentrated liquidity in an AMM. vAMMs use the same x*y=k constant product formula as CPMMs, but instead of relying on a liquidity pool, traders deposit collateral to a smart contract. The price of tokens are determined by the ratio of the amount of tokens in the AMM. The profit extracted by arbitrageurs is siphoned from the pockets of liquidity providers, creating a loss. For illustration, imagine there are 2 kinds of assets in the pool, A and B, with reserve amounts RA and RB , respectively. AMMs have become a primary way to trade assets in the DeFi ecosystem, and it all began with a blog post about on-chain market makers by Ethereum founder Vitalik Buterin. Please visit our Cryptopedia Site Policy to learn more. There are several different types of AMMs and they include: We need to know a number of terms that are used in DeFi: Generally AMMs use mathematical formulas to facilitate trades inDecentralized Exchange. However, the execution price is 0.666, so we get only 133.333 of token 1! To keep things simple, let's imagine our liquidity provider supplies 1 ETH and 100 DAI to the Uniswap DAI exchange, giving them 1% of a liquidity pool which contains 100 ETH and 10,000 DAI. The product k would actually be constant, if the swap fee was 0%. During periods of low volatility, Sigmadex can concentrate liquidity near the market price and increase capital efficiency, and then expand it during periods of high volatility to help protect traders from impairment loss. Thank you for signing up! This new method of exchanging assets embodies the ideals of Ethereum, crypto, and blockchain technology in general: no one entity controls the system, and anyone can build new solutions and participate. Adding a bid-ask spread on top of a CFMM breaks the constant-function invariant. While this function produces zero slippage, it does not provide infinite liquidity and thus is likely unfit as a standalone implementation for a decentralized exchange use-case. This has made these rules popular in prediction markets (fixed cost of . $$x + r\Delta x = \frac{xy}{y - \Delta y}$$ The formula for this model is X * Y = K. In Vitalik Buterins original post calling for automated or on-chain money markets, he emphasized that AMMs should not be the only available option for decentralized trading. What is an automated market maker? The pool stays in constant balance, where the total value of ETH in the pool will always equal the total value of BTC in the pool. An AMM uses an algorithm and the most common algorithm used by big decentralized exchanges is called a "constant-product market maker". We derive the value function for liquidity providers . Yes, I agree to receive email communications from Chainlink. This function acts as a constant sum when the portfolio is balanced and shifts towards a constant product as the portfolio becomes more imbalanced. After a trade, theres a new spot price, at a different point on the curve. For example, a liquidity pool could hold ten million dollars of ETH and ten million dollars of USDC. Follow More from Medium Jessica Doosan 5 AI Coins For the Next Crypto Trend Ren & Heinrich in DataDrivenInvestor I analyzed 200 DeFi Projects. Here Is What I Found Out. Every trade starts at the point on the curve that corresponds to the current ratio of The proposed cost functions are computationally efficient (only requires multiplication and square root calculation) and have certain advantages over widely deployed constant product cost functions. The formula is: When you trade in an AMM X and Y can vary but the result is always a constant. What he didnt foresee, however, was the development of various approaches to AMMs. . A crowdfunded CFMM is a CFMM which makes markets using assets deposited by many different users. If there is a bug in the smart contract, or if it is exploited by malicious actors, it could result in the loss of funds or other problems. (the token they want to buy). The opposite happens to the price of BTC in an ETH-BTC pool. This is true, the price is also high. Phew! A constant sum market maker is a relatively straightforward implementation of a constant function market maker, satisfying the equation: Where R_i are the reserves of each asset and k is a constant. We can always find the output amount using the $\Delta y$ formula Minting: Minting refers to the process of creating a new asset or increasing the supply of an existing asset. At this point, Users trade against the smart contract (pooled assets) as opposed to directly with a counterparty as in order book exchanges. The prices of tokens in a pool are determined by the supply of the tokens, that is by the amounts of reserves of the Because the relative price of the two pair assets can only be changed through trading, divergences between the Pact price and external market prices create arbitrage opportunities. $$(x + r\Delta x)(y - \Delta y) = xy$$ On a traditional exchange platform, buyers and sellers offer up different prices for an asset. AMMs, or Automated Market Makers, are a financial tool that allows investors to provide two different assets so that traders can trade those assets. Since increase in liquidity is equal to increase in shares: Burning: This refers to the process of removing or destroyingan asset from circulation. When they have a larger variation of the two assets they are more likely to experience that impermanent loss. Well, this is the math of Uniswap V2, and were studying Uniswap V3. $$r\Delta x = \frac{xy - x(y - \Delta y)}{y - \Delta y}$$ In this article I explain what Automated Market Makers are, and dive deep into Constant Product Market Makers. These The actual price of the trade is the slope of the line connecting the two points. A qualified professional should be consulted prior to making financial decisions. Instead, there needed to be many ways to trade tokens, since non-AMM exchanges were vital to keeping AMM prices accurate. in-game items that are hard to market make because of low liquidity). We are still very early in the evolution of constant function market makers and I am looking forward to seeing the emergence of new designs and applications over the next several years. A constant mean market maker is a generalization of a constant product market maker, allowing for more than two assets and weights outside of 50/50. Eleven sellers are also willing to sell at the same prices. This design ensures that the pool remains balanced according to its pre-set weights for each asset. An early description of a CFMM was published by economist Robin Hanson in "Logarithmic Market Scoring Rules for Modular Combinatorial Information Aggregation" (2002). An automated market maker facilitates trades and allows digital assets to be traded on a decentralized exchange (DEX). must be monotone (intermediate value theorem), and it can be assumed WLOG that The change in $y$ is the amount of token 1 well get. By trading synthetic assets rather than the underlying asset, users can gain exposure to the price movements of a wide variety of crypto assets in a highly efficient manner. For example, one could adjust LP fees based on trailing volatility, resulting in a stochastic pricing mechanism and the added benefit of volatility sensitivity for CFMMs. AMM systems allow users to mint new assets by providing liquidity to the AMM in the form of other assets. To incentivize liquidity providers to deposit their crypto assets to the protocol, AMMs reward them with a fraction of the fees generated on the AMM, usually distributed as LP tokens. Constant Product Market Makers. The secret ingredient of AMMs is a simple mathematical formula that can take many forms. Uniswap and Constant Product Market Makers (CPMM) There are two assets, X and Y. Denote by x the volume of X and by y the volume of Y in the reserves. Theres a pool with some amount of token 0 ($x$) and some amount of token 1 ($y$). Even though Uniswap doesnt calculate trade prices, we can still see them on the curve. Traditional AMM designs require large amounts of liquidity to achieve the same level of price impact as an order book-based exchange. and decentralized finance (DeFi). {\displaystyle V} CFMMs incur large slippage costs and are thus better for smaller order sizes. plotting them on the graph. money markets, he emphasized that AMMs should not be the only available option for decentralized trading. As a new technology with a complicated interface, the number of buyers and sellers was small, which meant it was difficult to find enough people willing to trade on a regular basis. The more assets in a pool and the more liquidity the pool has, the easier trading becomes on decentralized exchanges. CFMMs provide the ability to measure the price of an asset without the use of a central third party, addressing a problem often known as the oracle problem. The most commonly used AMM is constant product AMM, but other AMM models are also deployed in decentralized finance (DeFi). Various types of AMMs are examined, including: Constant Product Market Makers; Constant Mean Market Makers; Constant Sum Market Makers; Hybrid Function Market Makers; and, Dynamic Automated Market Makers. Liquidity Pool:a liquidity pool is a collection of assets that is used to facilitate trading in an AMM.they help to ensure that there is always a sufficient supply of assets available to buy and sell in the market. Constant product formula is probably the simplest and the earliest algorithm to come into the market. From this, it is observed that when a user places an order of tokens Since Uniswap pools are separate smart contracts, tokens in a pool are priced in terms of each other. is calculated differently. Understanding this math is crucial to build a Uniswap-like DEX, but it's totally fine if you don't understand everything at this stage. V When you want to buy a big amount relative to pool reserves the price is higher than when you want to Although Automated Market Makers harness a new technology, iterations of it have already proven an essential financial instrument in the fast-evolving DeFi ecosystem and a sign of a maturing industry. Previous Multiple Fee Tiers Next StableSwap Invariant Market Maker (SIMM) Last modified 3mo ago At its core is a very For example, the Uniswap payoff curve is concave, meaning that liquidity providers are profitable within a certain price bound and will lose money in large price movements: Ideally, we want convexity when taking risk, which means having upside on both sides of the risk spectrum. Market Makers (MMs) A centralized exchange relies on professional traders or financial institutions, to create multiple bid-ask orders to match the orders of retail traders, or in other words, to provide liquidity. If a trader's bid matches the offer of the MM, the trade is executed. V Liquidity : This is the ability of an asset to be sold without affecting the price. Lets visualize the constant product function to better understand This also holds true for AMMs. This is how markets work. Copyright 2023 Gemini Trust Company, LLC. Why there are only two reserves, x and y?Each Uniswap pool can hold only two tokens. Constant Product Automated Market Maker | Solidity 0.8 - YouTube Code for constant product automated market maker.0:00 - State variables and constructor2:38: Internal functions -. This button displays the currently selected search type. CPMMs are based on the function x*y=k, which establishes a range of prices for two tokens according to the available quantities (liquidity) of each token. Excessive Trading? The product of updated reserves must still equal $k$. We should focus on what works now and assume that it might not work in the future. The constant product market maker protocol is a form of the much known automated market maker (AMM) model. Automated market makers (AMMs) are algorithmic agents that perform those functions and, as a result, provide liquidity in electronic markets. For example, Bancor 3 has integrated Chainlink Automation to help support its auto-compounding feature. Liquidity sensitivity for todays CFMMs is limited to price (i.e. Since AMMs usually have a fee, the product of the reserves is not really a constant in practice. On a. , buyers and sellers offer up different prices for an asset. In order to understand a constant product AMM, we first need to understand what is a market maker. For example, Synthetix was able to use Uniswap to bootstrap liquidity for its sETH liquidity pool, giving users an easier way to begin trading on the exchange. This is evident in both traditional markets and centralized crypto exchanges, where asset prices are influenced by factors like order book depth, buy-side or sell-side liquidity, trading history, and private information. {\displaystyle V} If 1 ETH costs 1000 USDC, then 1 USDC In this paper, we focus on the analysis of a very large class of automated market makers, called constant function market makers (or CFMMs) which includes existing popular market makers such as Uniswap, Balancer, and Curve, whose yearly transaction volume totals to billions of dollars. Shell Protocol has similar goals but takes a different approach. The constant product formula . Decentralized exchanges (DEXes) are an essential component of the nascent decentralized finance (DeFi) ecosystem. Product-market fit is a moving target. However, AMMs have a different approach to trading assets. At its core, a liquidity pool is a shared pot of tokens. This property implies that market makers should adjust the elasticity of their pricing response based on the volume of activity in the market. AMMs democratized cryptocurrency trading by doing away with order books and institutional market makers. In this constant state of balance, buying one ETH brings the price of ETH up slightly along the curve, and selling one ETH brings the price of ETH down slightly along the curve. Visually, the prices of tokens in an AMM pool follow a curve determined by the formula. From Bancor to Sigmadex to DODO and beyond, innovative AMMs powered by Chainlink trust-minimized services are providing new models for accessing immediate liquidity for any digital asset. $21. we want to buy a known amount of tokens). We focus particularly on separability and on different invariance properties under scaling. You need to enable Javascript to view this site properly. On this Wikipedia the language links are at the top of the page across from the article title. Uniswap works. These CFMMs will have price functions that best reflect the characteristics of their respective assets, resulting in less slippage and more efficient exchange. When plotted, the constant product function is a quadratic hyperbola: Where axes are the pool reserves. Only when new liquidity providers join in will the pool expand in size. If an AMM doesnt have a sufficient liquidity pool, it can create a large price impact when traders buy and sell assets on the DeFi AMM, leading to capital inefficiency and impermanent loss. Perpetual Protocol's vAMM uses the same x*y=k constant product formula as Uniswap. When other users find a listed price to be acceptable, they execute a trade and that price becomes the assets market price. Curve (a.k.a. In effect, the function looks like a zoomed-in hyperbola. The only constant in life (and business) is Change. For example, If you want to sell token A and buy token B in the Constant product AMM then the formula will be, dx = Change in the amount of token A (there will be an in increase in token A in the AMM), dy =Change in the amount of token B (there will be a decrease in token B in the AMM), Before the trade the formula was : XY = K. After the trade the formula will be (X+dy)(Y-dy) = K. From the above graph you can tell that K is constant. CFMMs are the first class of AMMs to be specifically applied to real-world financial markets. To calculate the output amount, we need to find a new point on the curve, which has the $x$ coordinate of $x+\Delta x$, i.e. First introduced by Balancer, constant mean markets satisfy the following equation in the absence of fees: where R is the reserves of each asset, W is the weights of each asset, and k is the constant. Minting: Minting refers to the process of creating a new asset or increasing the supply of an existing asset. Anyone with an internet connection and in possession of any type of, can become a liquidity provider by supplying tokens to an AMMs liquidity pool. For example, the function for an equal-weighted portfolio of three assets would be (x*y*z)^(1/3) = k. There are several projects which use hybrid functions to achieve desired properties based on the characteristics of the assets being traded. Simple question: does it pay to split an order? With the Constant Product Market Maker (CPMM) capability, pairs act as automated market makers, ready to accept one token for the other as long as the constant product formula is preserved. As the legend goes, Uniswap was invented in Desmos. The price of tokens in the AMM before adding the liquidity = (X + dx) / (Y + dy): From the above equation we can find both the amount of token A added (dx) given the amount of token B added (dy) i.e what is dy given dx ? Proposition: For \(x>x^*\), constant product provides "higher" risk compensation than what market competition would yield, for \(x<x^*\) it is the reverse. This offers two important benefits: Slippage refers to the tendency of prices to move against a traders actions as the trader absorbs liquidity the larger the trade, the greater the slippage. Well be focusing on and They do this by using a process called "liquidity provision," in which they act as both the buyer and the seller of an asset. and this is a desirable property! These trades impose costs on Liquidity Providers (LPs) who supply reserves to CFMMs. Not only do AMMs powered by Chainlink help create price action in previously illiquid markets, but they do so in a highly secure, globally accessible, and non-custodial manner. What Are Automated Market Makers (AMMs)? Delta neutral market makers also have a difficult task at hand if they have to find a way to hedge assets off their books since it is often not possible if a natural buyer or seller does not exist. However, the CFMM + spread will never underperform the CFMM without a spread (the latter of which will never compensate for opportunity cost). However, users holding an open position in a synthetic asset are at risk of having their collateral liquidated if the price moves against them.. in a permissionless system. Heres how you can derive the above formulas from the trade function: Constant function market makers (CFMMs), such as constant product market makers, constant sum market makers, and constant mean market makers, are a class of first-generation AMMs made popular by protocols like Bancor, Curve, and Uniswap. Trading any amount of either asset must change the reserves in such a way that, when the fee is zero, the product R_*R_ remains equal to the constant k. This is often simplified in the form of x*y=k, where x and y are the reserves of each asset. Dont be scared by the long name! Using formulas derived from the constant product market maker formula (x times y equals k), we can calculate the amount they can purchase before ETH value in the liquidity pool reaches $550 as well. is a "consistent payoff function",[8] that is, a payoff function which is concave, nonnegative, nondecreasing, and 1-homogenous, it is possible to construct a trading function which achieves In contrast to regular market makers, AMMs function by using self-executing computer programs, also known as smart contracts. two USD-denominated stablecoins) then you could reduce the amount of slippage in the function. One alternative approach could be to increase the LP fee at lower levels of liquidity to incentivize LPs to deposit their assets (e.g. A constant sum function forms a straight line when plotting two assets, resulting in the equation x+y=k. Market makers do this by buying and selling assets from their own accounts with the goal of making a profit, often from the spreadthe gap between the highest buy offer and lowest sell offer. Since AMMs dont automatically adjust their exchange rates, they require an arbitrageur to buy the underpriced assets or sell the overpriced assets until the prices offered by the AMM match the market-wide price of external markets. of Uniswap V3 is different. To build a better intuition of how it works, try making up different scenarios and Their trading activity creates liquidity, lowering the price impact of larger trades. Automated market makers (AMMs) are decentralized exchanges that use algorithmic money robots to provide liquidity for traders buying and selling crypto assets. Visually, the prices of tokens in an AMM pool follow a curve determined by the formula. How do we calculate the prices of tokens in a pool? CFMMs are largely path-independent (assuming minimal fees), which means that the price of any two quantities depends only on those quantities and not on the path between them. Token prices are simply relations of reserves: $$P_x = \frac{y}{x}, \quad P_y=\frac{x}{y}$$. In non-custodial AMMs, user deposits for trading pairs are pooled within a smart contract that any trader can use for token swap liquidity. costs 0.001 ETH. As a liquidity provider you just need . Amm, but other AMM models are also willing to sell at the impact of introducing concentrated liquidity an! That some orders may take a while to get filled, if the swap fee was 0 % to understand. Trader & # x27 ; s bid matches the offer of the reserves not! And that price becomes the assets market price prices, we can still see on. Implement and understand also deployed in decentralized finance ( DeFi ) ecosystem consulted prior to making financial.... 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