Stable Math

Overview

Stable Math is designed to allow for swaps between any assets that have the same price, or are "pegged" to the same asset. The most common examples are stablecoins that track US Dollars (DAI, USDT, USDC), and assets that track the price of Bitcoin (WBTC, renBTC, sBTC). Prices are determined by the pool balances, the amplification parameter, and amounts of the tokens that are being swapped.

In an ideal scenario, it would make sense to simply allow 1-to-1 trades for these assets; this would be a Constant Sum curve. In a worst case scenario where one or more of these assets loses their peg and their value diverges, it would make sense to enforce trade rules for uncorrelated assets; this would be a Constant Product curve, such as the one in Weighted Math.

Since most cases are neither ideal nor disasters, the Stable Math curve combines the Constant Sum and Constant Product curves and is designed to facilitate approximately 1-to-1 trades that incur large price changes only when token balances differ greatly. 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 $A\rightarrow \infty$).

StableSwap approaches Constant Product as A->0 and Constant Sum as A->∞

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

If you are interested in understanding how the trades affect the pool’s price and slippages, please refer to the study here and here.