Automated Market Makers (AMMs) - Overview
- Brief history
- Implementation
- Bancor - Smart token
- Constant Sum Market Maker (CSMM)
- Constant Mean Market Maker (CMMM / Balancer)
- StableSwap (Curve)
- Hybrid Models
Automated Market Makers (AMMs) are an essential piece of decentralized finance (DeFi), enabling to swap crypto assets without intermediaries.
Unlike traditional order book-based exchanges, AMMs use mathematical formulas to determine the price of assets and facilitate liquidity. These formulas govern how users trade assets and earn rewards, making them the backbone of liquidity pools.
This article delves into the most commonly used formulas in AMM design, their benefits, and their limitations.
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Brief history
The original AMMs based on constant product market makerswere envisioned in 2017 by Ethereum founder Vitalik Buterin on a reddit post to reduce the high spread (often 10% or even higher) during a trade on the plaform available at that moment (MKR market, etherdelta).
The spreads were high because market making is very expensive, as creating an order and removing an order both take gas fees, even if the orders are never “finalized”.
State channel-based solutions could theoretically resolve this, but are far from being implemented in 2017. Vitalik proposed to use instead a style of “on-chain automated market maker”
Its proposition was based on concepts already used in prediction markets and also a first proposable by Nick Johnson’s proposal here
After that, the first xy=k liquidity pool was deployed by Bancor in 2017 with BNT and GNO. See also this video by Bancor dated from March 2017.
Reference: Jennifer Albert - A historical account with Dr. Mark Richardson, Bancor Project Lead
Constant Product Market Maker (CPMM)
The constant product formula is the most popular AMM model, popularized by Uniswap and Bancor, the first AMM-based Dex.
The equation implemented and used by Uniswap V2 is the following: \(\begin{aligned} x⋅y=k \end{aligned}\) Here:
xrepresents the quantity of one token in the pool.yrepresents the quantity of another token in the pool.kis a constant.
Price
With this formula, tokens (x, y) are priced in terms of each other.
For example: in a ETH/USDC pool, ETH is priced in terms of USDC, and USDC is priced in terms of ETH.
If 1 ETH costs 1000 USDC, then 1 USDC costs 0.001 ETH.
The prices of tokens in a pool are determined by the supply of the tokens, that is by the amounts of reserves of the tokens that the pool is holding. Token prices are simply relations of reserves: \(\begin{aligned} P_x = y/x \end{aligned}\)
[\begin{aligned} P_y = x/y \end{aligned}]
Where Px and Pxare prices of tokens in terms of the other token
Graph
The result is a hyperbola where liquidity is always available but at increasingly higher prices, which approach infinity at both ends.
Reference image: Dmitriy Berenzon - DeFi’s “Zero to One” Innovation
See also Uniswap - How Uniswap works
How It Works:
- Traders interact with the pool by adding or removing liquidity, causing
xoryto adjust. - The product of the two reserves (
k) remains constant during a trade.
Advantages:
- Simplicity: Easy to implement and understand.
- Continuous Liquidity: Always provides liquidity regardless of the pool size.
- Price Discovery: Adjusts prices based on supply and demand.
Limitations:
- Price Impact: Large trades result in significant slippage, making it unsuitable for high-volume trades.
- Impermanent Loss: Liquidity providers may suffer temporary losses compared to simply holding the assets.
Implementation
Here the implementation from Uniswap V2, function swap
require(balance0Adjusted.mul(balance1Adjusted) >= uint(_reserve0).mul(_reserve1).mul(1000**2), 'UniswapV2: K');
There are four variables inside the require statement:
balance0Adjusted: Reserves of x after the trader sends tokensX to the pool minus 0.3% of the amount sent.balance1Adjusted: Reserves of y after the tokensY are sent to the trader from the pool._reserve0: Reserves of tokenxprior to the swap._reserve1: Reserves of tokenyprior to the swap.
The simplify version gives the following formula: \(balance0Adjusted * balance1Adjusted >= \_reserve0 * \_reserve1\) In equation, it gives:
[\begin{aligned} x * y = k \end{aligned}]
[\begin{aligned} x’ * y’ = k \end{aligned}]
[\begin{aligned} x * y = x’ * y’ \end{aligned}]
Where
xisreserve0andx'isbalance0Adjustedyisreserve1andy'isbalance1Adjusted
The smart contract stores also the kvalue in the public variable KLast
See also Uniswap V2 in Depth
Resources:
- Constant Function Market Makers
- Uniswap Whitepaper
- Uniswap V2 - How Uniswap works
- Chaisomsri - [DeFi Math] About Uniswap V2 Lazy Liquidity
- Kirill Naumov - Back to the Basics: Uniswap, Balancer, Curve
Bancor - Smart token
Smart tokens are standard ERC20 tokens which implements the Bancor Protocol.
A smart token holds a balance of least one other reserve token, which can be a different smart token, any ERC20 standard token or Ether.
Smart tokens are issued when purchased and destroyed when liquidiation
Bancor uses the term of CRR to design K: Constant Reserve Ratio.
This constant was set by the smart token creator, for each reserve token and uses in price calcalation. \(Price = Balance / Supply * CRR\)
- When smarts token are purchased, the payment is added to the reserve balance.
Based on the calculation price, new smart tokens are issued to the buyed.
- When smart token are liquidates, they are removed from the supply (destroed).
Based on the current price, reserve tokens are transfereed to the liquidator.
Constant Sum Market Maker (CSMM)
The constant sum formula is expressed as: \(\begin{aligned} x + y = k \end{aligned}\) This model is less common but ensures no slippage for trades until the pool is depleted.
Graph
The result graph is a function affine, which means that this model does not provide infinite liquidity.
How It Works:
- Traders exchange assets without altering the reserves’ proportions drastically.
Advantages:
- No Slippage: Ideal for stablecoins or assets with tightly correlated values.
- Arbitrage-Friendly: Maintains prices close to the market.
Limitations:
- Vulnerability to Arbitrage: Can be drained easily if asset prices deviate significantly.
- Limited Use Cases: Not suitable for volatile or uncorrelated assets.
See also:Wikipedia - Constant function market maker
Constant Mean Market Maker (CMMM / Balancer)
Balancer pools generalize the constant product formula by allowing multiple assets with custom weightings. The formula is: \(\prod_{i=1}^{n} r_i^{w_i} = k\) Here:
riis the reserve of the i-th asset.wiis the weight of the i-th asset.
Example:
For a liquidity pool with three assets, the equation would be the following \(\begin{aligned} (x*y*z)^{⅓} = k \end{aligned}\) . This allows for variable exposure to different assets in the pool and enables swaps between any of the pool’s assets.
See also Chainlink - What Are Automated Market Makers (AMMs)?
How It Works:
- Balancer pools can hold more than two tokens and allocate custom weightings for each asset.
- Prices adjust based on the weighted reserves.
Advantages:
- Diversified Liquidity: Supports multiple assets in one pool.
- Custom Weighting: Liquidity providers can tailor pools to specific strategies.
Limitations:
- Complexity: Requires more sophisticated calculations.
- Higher Gas Costs: Computationally intensive, leading to increased transaction fees.
StableSwap (Curve)
Curve Finance introduced the StableSwap formula, optimized for assets with similar prices, such as stablecoins or wrapped tokens. Its formula combines constant sum and constant product mechanisms, ensuring minimal slippage.
Curve AMMs combine both a CPMM and CSMM to offers offers low-price-impact swaps between tokens that have a relatively stable 1:1 exchange rate
How It Works
- At small price deviations, it behaves like a constant sum AMM, reducing slippage.
- For larger deviations, it transitions to a constant product AMM to maintain stability.
Pros/Cons
Advantages
- Low Slippage: Ideal for stablecoin trading.
- Efficient Liquidity Use: Concentrates liquidity near the equilibrium price.
Limitations
- Specialized Use Case: Not versatile for uncorrelated assets.
- Complex Implementation: Requires precise calibration.
Details
The constant D represents the total amount of coins in the pool when their prices are equal. To minimize price slippage, an “amplified” invariant should have low curvature, leading to smaller price changes. A “zero slippage” invariant corresponds to infinite leverage, but it’s essentially a constant-price or constant-sum invariant. On the other hand, a constant-product invariant corresponds to zero leverage.
To find a balance between these two extremes, the proposed solution introduces a dimensionless leverage parameter (χ). By adjusting χ, the invariant can smoothly transition between a constant-product invariant (when χ = 0) and a constant-sum invariant (when χ = ∞). To ensure consistency, the constant-sum invariant is multiplied by χDn−1, where D represents the total value of coins in the pool and n is the number of coins. This results in an invariant that incorporates both characteristics and allows for flexible adjustment of leverage.
Graph
Graphs are from the StableSwap whitepaper
Comparison of StableSwap invariant with Uniswap (constant-product) and constant price invariants. The portfolio consists of coins X and Y which have the “ideal” price of 1.0. There are x = 5 and y = 5 coins loaded up initially. As x decreases, y increases, and the price is the derivative dy/dx.
The price slippage (Fig. 2) is much smaller, if compared to constant-product invariant. The StableSwap invariant has an “amplification coefficient” parameter: the lower it is, the closer the invariant is to the constant product. When calculating slippage, we use a practical value of A = 100. This is somewhat comparable to using Uniswap with 100x leverage.
Price slippage: Uniswap invariant (dashed line) vs Stableswap (solid line)
Hybrid Models
Several AMMs, like Bancor and Kyber Network, use hybrid formulas that combine elements of the above models or introduce innovative mechanisms. These include:
- Dynamic Fees: Adjusting fees based on market conditions.
- Concentrated Liquidity: Allowing liquidity providers to specify price ranges for their funds (e.g., Uniswap v3).
Advantages:
- Flexibility: Adapts to various market conditions.
- Efficiency: Optimizes liquidity distribution.
Limitations:
- High Complexity: May be challenging to understand and implement.
- Technical Risks: Increased likelihood of bugs or vulnerabilities.
Hybrid Constant Function Market Maker (HCFMM) is mentioned in a KPMG report, but I haven’t really found any further explanation
Summary tab
Here is a summary of the different formula
| Formula Type | Apps | Formula | Graph | Best Use Case | Advantages | Limitations |
|---|---|---|---|---|---|---|
| Constant Product (CPMM) | Uniswap, bancor | x⋅y=k | Hyperbole | General-purpose, token swaps | Simple, continuous liquidity, price discovery | High slippage for large trades, impermanent loss |
| Constant Sum | x+y=k | Function affine | Stablecoin or tightly correlated assets | No slippage within the pool | Vulnerable to arbitrage, unsuitable for volatile assets | |
| **Constant Mean ** | Balancer | See article | Multi-token pools | Diversified liquidity, custom weighting | Complex, higher gas fees | |
| StableSwap | Combination of constant sum and product | hyperbola | Stablecoins, low-volatility pairs | Low slippage, efficient liquidity | Limited to specific asset types (e.g stablecoin) |
Conclusion
The choice of formula for an AMM depends on the specific use case and asset characteristics. While the constant product formula remains a foundational approach, innovations like StableSwap and custom weightings are paving the way for more specialized and efficient trading mechanisms. As DeFi evolves, we can expect further experimentation and hybridization of AMM models to cater to diverse needs.
By understanding the nuances of these formulas, developers and users alike can better navigate the dynamic world of decentralized trading.