Constant Power Root Market Makers

TL;DR

This paper introduces the constant power root market maker, a flexible trading function that interpolates between various known market makers, balancing price slippage and impermanent loss for liquidity providers.

Contribution

It generalizes existing market makers as special cases of the constant power root function, providing new insights and formulas for liquidity provider metrics.

Findings

The power q interpolates between harmonic, geometric, and arithmetic means.

Adjusting q trades off between price slippage and impermanent loss.

Derived formulas for value, price, impact, and greeks of the market maker.

Abstract

The paper introduces a new type of constant function market maker, the constant power root market marker. We show that the constant sum (used by mStable), constant product (used by Uniswap and Balancer), constant reserve (HOLD-ing), and constant harmonic mean trading functions are special cases of the constant power root trading function. We derive the value function for liquidity providers, marginal price function, price impact function, impermanent loss function, and greeks for constant power root market markers. In particular, we find that as the power q varies from the range of -infinity to 1, the power root function interpolates between the harmonic (q=-1), geometric (q=0), and arithmetic (q=1) means. This provides a toggle that trades off between price slippage for traders and impermanent loss for liquidity providers. As the power q approaches 1, slippage is low and impermanent loss is high. As q approaches to -1, price slippage increases and impermanent loss decreases.