Elliptic Concentrated Liquidity Pools (E-CLP)

Gyroscope designed the elliptic concentrated liquidity pool (E-CLP) which is an automated market maker (AMM) where trading takes place along part of an ellipse curve. Mathematically, the ellipse is represented as a shifted deformation of a circle. The motivation behind using ellipses is twofold: configurability and ease of implementation. Ellipses offer adjustable parameters, making them adaptable for various needs. Additionally, they can be implemented efficiently on the blockchain despite this configurability.

While Uniswap V3's concentrated liquidity allows for better capital utilization, lower slippage, and higher trading volumes, thus generating more fees for liquidity providers (LPs). StableSwap offers a straightforward user experience where LPs can simply deposit funds without actively managing their positions.

$(x + \frac{L}{\sqrt\beta})\cdot(y+ \frac{L}{\sqrt\alpha})$

$4A(x+y) + D = 4AD+ \frac{D^3}{4xy}$

By integrating these advantages, the goal is to create a system where LPs benefit from efficient capital use and automated liquidity adjustments, especially as prices change with yield-bearing assets, thereby eliminating the need for manual intervention. This hybrid model leverages intelligent defaults and preconfigured settings to ensure that even less experienced users can participate effectively, allowing for broader market participation and enhanced liquidity.

Elliptical Trading curve

The approach involves taking an ellipse, rotating it, and then translating it in a manner akin to the operations in Uniswap V3 or the 2 CLP variant. The ellipse is adjusted to intersect the axis at points where the tangents correspond to the specified price bounds. This methodology is elucidated through an equation resembling that of a circle, expressed as:

\begin{equation} \left( e_x \cdot A \begin{pmatrix} x \\ y \end{pmatrix} - r \chi \right)^2 + \left( e_y \cdot A \begin{pmatrix} x \\ y \end{pmatrix} - r \chi \right)^2 = r^2 \end{equation}

In this equation, the terms inside the parentheses represent the transformed coordinates of the ellipse. The expression $\left( e_x \cdot A \begin{pmatrix} x \\ y \end{pmatrix} - r \chi \right)^2$ and $\left( e_y \cdot A \begin{pmatrix} x \\ y \end{pmatrix} - r \chi \right)^2 = r^2$ denote the adjusted $(x)$and $(y)$-coordinates, respectively, after the linear transformation by matrix $[A]$. This transformation morphs the circle into an ellipse. The value of $(\chi)$ is computed to achieve the precise offset required for the desired configuration, ensuring that the ellipse aligns correctly with the specified price bounds.

E-CLP Parameters

1. Price Range: The price range is defined by the points $[\alpha , \beta]$at which the ellipse intersects the $x$ and $y$ -axes. These intersections, determined by the slopes of the ellipse at these points, establish the boundaries within which liquidity is concentrated and trading occurs.

2. Peg/ Peak Liquidity Price: The "peg price", or "peak liquidity price", refers to the point on the ellipse curve where it is flattest. This point represents the highest concentration of liquidity, indicating that the largest amount of liquidity is provided at or near this price. It is the central point around which liquidity is most densely allocated, ensuring optimal trading conditions and minimal slippage at this price level.

3. Amplification: It refers to the degree to which the ellipse is stretched from a circle. It is quantified by a parameter, often denoted as lambda $(\lambda)$, which determines the shape of the ellipse. A larger amplification factor $(\lambda)$ results in an ellipse that is longer and flatter, meaning that prices remain close to the "peak price" for a longer range. Conversely, a smaller amplification factor leads to a more gradual transition between price bounds, spreading the liquidity more evenly across the range. This parameter allows for precise control over the concentration and distribution of liquidity within the specified trading range.

Step by Step construction of E-CLP

E-CLPs are designed to focus liquidity primarily around the $1 peg for GYD. However, they can also accommodate some price fluctuations by incorporating a small amount of liquidity at prices slightly above and below $1.

Rate Scaling

The concept is orthogonal to the trading curve and ellipses discussions and is particularly important for yield-bearing assets. It is designed to manage yield-bearing or rebasing tokens in liquidity pools. It ensures that the trading ranges for yield-bearing or rebasing tokens remain accurate and functional over time, accounting for the changing value of these assets and preventing them from drifting out of their designated trading ranges.

How does it work?

1. Yield-Bearing Tokens: Tokens like sDai, which accrue yield over time, increase in value because they represent investments in interest-bearing assets.

2. Rate Provider: A smart contract, known as the rate provider, implements a function called getRate. This function retrieves the current exchange rate between the yield-bearing token and its underlying asset (e.g., the amount of Dai one unit of sDai can be redeemed for).

3. Scaling Balances: Before any transaction within the liquidity pool, the balances of the tokens are adjusted according to the rate provided by the rate provider. This adjustment ensures that the price ranges within the pool reflect the current value of the yield-bearing tokens relative to their underlying assets.

4. Price Range Adjustment: By scaling the token balances, the effective price ranges in the pool remain accurate, corresponding to the underlying asset's value rather than the yield-bearing token’s nominal value.

5. Handling Limitations: This method is particularly useful when the underlying infrastructure (e.g., the Balancer Vault) does not support direct use of rebasing tokens due to internal accounting constraints.

Liquidity Desnsity

The liquidity density tool allows for a more nuanced understanding of liquidity dynamics, helping traders make informed decisions based on liquidity conditions. When dealing with stablecoin pairs or liquid staking tokens versus underlying pairs, traditional visual analysis might not reveal significant differences because the trading range is typically close to one.

Defintion: Liquidity density quantifies the amount of asset $y$ required to trade to change the price of asset $𝑥$ by one unit. Alternatively, in relative terms, it measures how much asset $𝑦$ can be traded to change the price of $𝑥$ by 1%.

High liquidity density indicates that a significant amount of asset $𝑦$ can be traded without causing significant price movement in asset $𝑥$, signifying high liquidity. Conversely, low liquidity density suggests increased price slippage and reduced liquidity, which are unfavorable for traders

$$\text{Liquidity Desnity} := LD := \frac{dy}{dp} \\\text{REaltive Liquidity Density} := LD_{rel}:=\frac{dy}{d\space log(p)}=LD.p$$

where

• $x,\space y$=reserve amounts of pool assets

• $p$ = $\frac{dy}{dx}$$dy/dx$=relative price

• All derivatives along the current trading curve

E-CLP use case:

Maximum Capital Efficiency in Secondary Markets:

E-CLP facilitates maximum capital efficiency in secondary markets by continuously providing liquidity to trading pairs, ensuring that assets can be bought or sold without significant price slippage. By dynamically adjusting liquidity provision based on market demand, E-CLP optimizes capital allocation, allowing traders to execute trades efficiently while maintaining stable asset prices.

Liquidity Provision within Mint and Redemption Quotes with Uneven Liquidity Distribution Concentrated Around the Peg:

In systems where liquidity is unevenly distributed around a pegged asset, such as stablecoins, E-CLP ensures consistent liquidity provision within mint and redemption quotes. This means that users can easily mint or redeem assets without experiencing significant price impact, regardless of the distribution of liquidity across the trading range. E-CLP's adaptive liquidity provision mechanism ensures smooth minting and redemption processes, enhancing user experience and confidence in the system.

Maximum Fee Accrual in Reserve Vaults:

E-CLP maximizes fee accrual in reserve vaults by concentrating liquidity within stablecoin pairs. By providing concentrated liquidity in high-demand pairs, such as stablecoin-to-stablecoin pairs, E-CLP facilitates frequent trading activity, resulting in increased fees generated for the reserve vaults. These accumulated fees bolster the health and sustainability of the reserve over time, providing a reliable source of revenue to support the stability and growth of the ecosystem.

Concentrated Liquidity in Reserve Vaults Populated by Stablecoins Means Increased Fees Which Bolster the Health of the Reserve Over Time:

E-CLP's focus on concentrating liquidity within reserve vaults populated by stablecoins leads to increased fees generated from trading activity. This concentrated liquidity ensures deep liquidity pools for stablecoin pairs, reducing slippage and enhancing trading efficiency. The accumulated fees contribute to the resilience and robustness of the reserve over time, providing a solid foundation for the ecosystem's stability and growth.