Replicating Monotonic Payoffs Without Oracles

TL;DR

This paper demonstrates that any monotonic payoff can be replicated using liquidity provider shares in CFMMs without oracles, providing explicit methods for constructing such payoffs and analyzing arbitrage profits.

Contribution

It introduces a method to replicate any monotonic payoff with CFMMs using only liquidity shares, eliminating the need for oracles or extra collateral.

Findings

Any monotonic payoff can be replicated with CFMMs.

Explicit trading functions for various payoffs are provided.

A formula for arbitrage profits against these CFMMs is derived.

Abstract

In this paper, we show that any monotonic payoff can be replicated using only liquidity provider shares in constant function market makers (CFMMs), without the need for additional collateral or oracles. Such payoffs include cash-or-nothing calls and capped calls, among many others, and we give an explicit method for finding a trading function matching these payoffs. For example, this method provides an easy way to show that the trading function for maintaining a portfolio where 50% of the portfolio is allocated in one asset and 50% in the other is exactly the constant product market maker (e.g., Uniswap) from first principles. We additionally provide a simple formula for the total earnings of an arbitrageur who is arbitraging against these CFMMs.