Finding the Right Curve: Optimal Design of Constant Function Market Makers

Constant Function Market Makers (CFMMs) are a crucial tool for creating exchange markets, have been deployed effectively in the context of prediction markets, and are now especially prominent within the modern Decentralized Finance ecosystem. We show that for any set of beliefs about future asset prices, there exists an optimal CFMM trading function that maximizes the fraction of trades that a CFMM can settle. This trading function is the optimal solution of a convex program. This program therefore gives a tractable framework for market-makers to compile their belief-distribution on the future prices of the underlying assets into the trading function of a maximally capital-efficient CFMM. Our optimization framework further extends to capture the tradeoffs between fee revenue, arbitrage loss, and opportunity costs of liquidity providers. Analyzing the program shows how consideration of profit and loss qualitatively distort the optimal liquidity allocation. Our model additionally explains the diversity of CFMM designs that appear in practice. We show that careful analysis of our convex program enables inference of a market-maker's beliefs about future asset prices, and show that these beliefs mirror the folklore intuition for several widely used CFMMs. Developing the program requires a new notion of the liquidity of a CFMM at any price point, and the core technical challenge is in the analysis of the KKT conditions of an optimization over an infinite-dimensional Banach space.