📐Principles

Motivation behind the design

Principle 1

On a small enough time interval, impermanent loss (IL) in a constant function pool is exactly equal to pool-enabled arbitrage profits.

However, over longer time intervals IL and arbitrage behave differently due to differences in price-path dependence.

What is price-path dependence?

Price-path dependence means simply that the result (i.e. pool position value) is dependent on the path that the price takes. Constant function automated market makers are price-path independent, meaning that if the value of the pool is $X at price $A, the value of the pool will always be equal to $X if the price returns to $A, regardless of the path that the price takes.

Principle 2

Impermanent loss is not price-path dependent, and is proportional to the relative price change between two tokens in a pool. Regardless of the price-path, if prices return to the same relative value as when the liquidity was provided, then IL will be zero. This impermanence is a property of swaps taking place along a fixed price curve.

Principle 3

Arbitrage opportunities are price-path dependent. Opportunities enabled by a pool are proportional to the cumulative price change distance. Put simply, if the value of a token first increases by $1 and then decreases by $1 to the original price, the total price change distance would be $2.

Principle 4

From 1, 2, and 3, it follows that: Arbitrage profits place an upper bound on impermanent loss. They are equivalent in the case where relative pricing moves monotonically in one direction, indefinitely. In all other cases, the path dependence of arbitrage opportunities means that they will be larger.

Conceptually, this is similar to drawing lines between two points on a map: two straight lines will always be equal in length, though a line with any curvature or deviation will make it longer than its straight line counterpart.

We can now set out to design a dex which reduces arbitrage opportunities, in order to effectively reduce impermanent loss. Equivalently, we can reduce IL by capturing arbitrage opportunities for liquidity providers.