Stable Math

First introduced by Curve, the Stableswap is a hybrid algorithm, to bring solution to the problems of slippage and fixed liquidity. The Stableswap hybrid combines both Constant Product and Constant Sum models, and the following chart shows the Stableswap algorithm in relation to constant product and constant sum invariants.

The amplification parameter, $A$, defines the degree to which the Stable Math curve approximates the Constant Product curve (when $A=0$), or the Constant Sum curve (when $A\rightarrow \infty$).

• Constant Sum: When the liquidity pool portfolio is balanced, the algorithm functions as a Constant Sum formula; x + y = k. You can observe the StableSwap blue line staying close to the Constant Sum red line, and the price is stable.

• Constant Product: As the liquidity pool portfolio becomes imbalanced, the StableSwap algorithm functions as a Constant Product formula; x * y = k. You can observe the StableSwap blue line now resembling the Constant Product purple line, and the price becoming expensive.

Invariant

Since the Stable Math equation is quite complex, determining the invariant, $D$, is typically done iteratively.

$$A \cdot n^n \cdot \sum{x_i} +D = A \cdot D \cdot n^n + { \frac{D^{n+1}}{{n}^{n}\cdot \prod{x_i} } }$$

Where:

• $n$ is the number of tokens

• $x_i$ is is balance of token $i$

• $A$ is the amplification parameter