When Does The Tail Wag The Dog? Curvature and Market Making

Abstract

In this paper, we give a simple but very general definition of 'price stability' for a class of markets. This class of markets includes the popular constant function market makers (CFMMs) such as Uniswap, Curve, and Balancer, used extensively in decentralized finance (DeFi), which now have daily trading volumes in the billions of dollars. We show that our definition of price stability is deeply connected to the curvature of the trading function used in the CFMM, making the folk intuition that "flatter CFMMs are more price stable" more concrete. We also show that this definition gives sufficient conditions for the profitability of liquidity providers, and, similar to the classical market microstructure literature, gives bounds on the edge of informed traders and bounds on the losses of liquidity providers. We also show how these bounds help explain some of the behaviors observed in decentralized finance in the second half of 2020, including the rise of 'yield farming ' and 'vampire attacks.'