The Geometry of Constant Function Market Makers

TL;DR

This paper introduces a broad geometric framework for constant function market makers (CFMMs), revealing their fundamental properties, composition rules, and a simple geometric description that applies even without path-independence.

Contribution

It provides a general geometric axiomatic framework for CFMMs, extending known results and establishing new properties without requiring differentiability or homogeneity.

Findings

Every CFMM has a unique nondecreasing, concave, homogeneous trading function.

CFMMs satisfy intuitive geometric composition rules.

Path-independent CFMMs have a simple geometric description independent of trading history.

Abstract

Constant function market makers (CFMMs) are the most popular type of decentralized trading venue for cryptocurrency tokens. In this paper, we give a very general geometric framework (or 'axioms') which encompass and generalize many of the known results for CFMMs in the literature, without requiring strong conditions such as differentiability or homogeneity. One particular consequence of this framework is that every CFMM has a (unique) canonical trading function that is nondecreasing, concave, and homogeneous, showing that many results known only for homogeneous trading functions are actually fully general. We also show that CFMMs satisfy a number of intuitive and geometric composition rules, and give a new proof, via conic duality, of the equivalence of the portfolio value function and the trading function. Many results are extended to the general setting where the CFMM is not assumed to be path-independent, but only one trade is allowed. Finally, we show that all 'path-independent' CFMMs have a simple geometric description that does not depend on any notion of a 'trading history'.