Math 1-pager

A practical guide to Ekubo's internal math

Ekubo represents the current state of each pool with two values: sqrt_ratio and liquidity. The sqrt_ratio is the square root of the current price in terms of token1 / token0, and the liquidity is a measure of how much of the two tokens is available for trading at the current price.

Ekubo supports pair prices between 2**-128 and 2**128, which means the square root of the price can be anywhere between 2**-64 and 2**64. We use a fixed point number type for storing the price with 128 bits after the radix. For example, sqrt_ratio for the price 1 is represented as 1<<128.

$$sqrt\_ratio = \sqrt{y/x}$$

$$liquidity = \sqrt{x*y}$$

The value sqrt_ratio is defined as sqrt(y/x) and liquidity is defined as sqrt(x*y). This is a different method of tracking x and y in x * y = k that is much easier to think about when we introduce the concept of concentrated liquidity: the act of swapping only changes the price, and adding and removing x and y only updates the liquidity. With these two values defined as such, all other formulae can be derived from it (e.g.: the amount of x in the pool, the differences in x/y between prices, etc.)

Ekubo does not consider token decimals in any of its calculations. If token0 and token1 have different decimals, that just means the price of 1 is actually 10**token1_decimals / 10**token0_decimals.

Given sqrt_ratio and liquidity, we can easily compute the equivalent amount of x and y reserves.

$$x = liquidity \div sqrt\_ratio$$

$$y = liquidity \times sqrt\_ratio$$

A constant-liquidity AMM such as Uniswap V2 can be trivially implemented using these representations instead of x and y, and Ekubo with full range liquidity is a more efficient version of the usual AMM implementation that tracks x and y and does not support concentrated liquidity.

When the price moves due to swapping, positions are entered and exited, and Ekubo adjusts the liquidity mid-trade. Thus, trades must be executed iteratively: a user's swap is broken up into pieces trading through areas of the curve with constant liquidity. Each time we cross a position boundary, we update the current liquidity. How do we define position boundaries? That's where ticks come in.

Ticks divide the entire price range into discrete regions. They can be defined however an AMM designer wishes, but in Ekubo prices are divided up logarithmically so that each tick is equally spaced. The base of the log for Ekubo is set to 1.000001. You can compute the sqrt_ratio corresponding to ticks using decimal math libraries, for example in TypeScript:

tick_to_sqrt_ratio.ts

import {Decimal} from 'decimal.js-light';

const tick = 1234;

// A fixed point .128 number has at most 128 bits after the decimal, 
// which translates to about 10**38.5 in decimal.
// That means ~78 decimals of precision should be able to represent
// any price with full precision.
// Note there can be loss of precision for intermediate calculations,
// but this should be sufficient for just computing the price.
Decimal.set({ precision: 78 });

const sqrt_ratio_x128 = 
    new Decimal('1.000001')
        .sqrt()
        .pow(tick)
        .mul(new Decimal(2).pow(128));

The inverse can be computed (tick from sqrt_ratio), by doing taking the logarithm of sqrt_ratio in base 1.000001.

Once you've broken up the price range into pieces, positions are just a combination of an amount of liquidity and lower/upper tick boundaries.