# 3-CLPs

Cubic-Concentrated Liquidity Pools or 3-CLPs

## Description of 3-CLPs

**Cubic CLPs, or 3-CLPs, concentrate the liquidity of three assets to a pricing range. **The first 3-CLPs are designed for symmetric price ranges [α,1/α] on the three asset pairs in the pool. A given pool is parameterized by the pricing parameter α and the three assets that make up the pool** **[1].

Given quantities of real reserves (x,y,z) in the pool and the pool's pricing parameter α, the offset a can be calculated [2]. This offset describes the amount the pool adds to real reserves to form the virtual reserves that achieve the pricing range.

Symmetric 3-CLPs use the following invariant: (x+a)(y+a)(z+a) = L^3

Like in any Balancer or Curve pool with more than two assets, the prices between different pairs of assets in the 3-CLP interact. For example, if asset x is traded against asset z, the pool’s spot price of asset y vs asset z is going to change. An implication of this is that the combinations of prices that are simultaneously offered by the pool at any point in time are constrained by a mathematical relationship.

**Because of this relationship, understanding the multi-dimensional pricing bounds of the 3-CLP requires some thought. **For the 2-CLP, the pricing bounds of the pool are very easy to understand: they are simply an interval of prices on a line, as shown below.

**The graphic below visualizes the feasible pricing region of a 3-CLP, which is the region of spot prices that a 3-CLP may quote.** Here we chose α=0.5, so that the pool does not price below 0.5 or above 2.0.

The feasible pricing region is parameterized by the prices of asset x and y, respectively, denoted in units of asset z. The third price pair, of x denoted in units of y, is the quotient of the two prices: px/y=px/z/py/z. In the figure, each included asset is represented with a different color.

The lines indicate price combinations where the respective reserve asset x, y, or z is exhausted in the pool, and the shaded regions indicate where the respective reserve asset has a positive balance in the pool.

The region where all colors overlap is the feasible pricing region - these are combinations of prices that the pool can quote. At the corner points, the pool only holds a single asset and two of the three asset pairs realize either the minimum or the maximum price bound (i.e., they are equal to α=0.5 or 1/α=2.0) [3]. Further documentation is available here.

## Benefits of 3-CLPs

**3-CLPs amplify the benefits of 2-CLPs:**

**Capital efficiency***:*By concentrating liquidity between three assets, 3-CLPs achieve a 50% capital efficiency improvement from aggregating the third asset compared to a similar setup of trading pairs as 2-CLPs.: Trading among three assets is more gas efficient than connecting two trades through two different 2-CLPs.**Gas efficiency**: 3-CLPs remain comparatively simple in architecture and user experience.**User experience**

## Risks of 3-CLPs

**Using 3-CLPs also comes with certain risks, including the following. **These risks include those of the 2-CLPs; we only explain these risks briefly. See the section on 2-CLPs for further details.

: the risk that an exploit or bug puts deposited assets at risk.**Smart contract risk**: the risk that the portfolio strategy implied by the pool has worse payoffs than a different strategy (such as just holding the assets).**Strategy risk**: the risk that permanent price changes imply a loss to LPers because arbitrageurs take stale price quotes.**Adverse selection risk**

**Compared to a 2-CLP, LPers in a 3-CLP are exposed to additional risks: **

since the 3-CLP contains an additional asset, LPers are exposed to strategy and adverse selection risk due to two additional price pairs.**Asset interaction risk:**

For example, if the price of one of the assets drops to a sufficiently low level, the pool will end up with only that least valuable asset (see the figure above). This phenomenon occurs in many multi-asset AMMs, such as the Curve stablecoin 3-pool. Just like strategy risk in the 2-pool, it can be reduced by choosing fundamentally connected assets and appropriate price bounds.

Note that the pricing region of the 3-CLP is more complex than for other pools; this may mean that arbitrageurs need more time to get familiar with the functioning of the pool before it runs smoothly.

## Technical Specification

To read about the mathematical specification and implementation, see the below resources

Technical Documents## Notes

[1] For technical reasons, the parameter provided to the smart contract is $\sqrt[3]{\alpha}$ instead of α itself.

[2] The offsets are computed as $a = L \sqrt[3]{\alpha}$

[3] The figure is asymmetric, with the curve for z having a different shape than the curves for x and y, because we chose to express the relative pool prices in terms of asset z. This is only a property of how the figure was made and does not affect the functioning of the pool; within the pool, the three assets take on symmetric roles.

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