Constant Function Market Makers: Multi-Asset Trades via Convex Optimization

Abstract

The rise of Ethereum and other blockchains that support smart contracts has led to the creation of decentralized exchanges (DEXs), such as Uniswap, Balancer, Curve, mStable, and SushiSwap, which enable agents to trade cryptocurrencies without trusting a centralized authority. While traditional exchanges use order books to match and execute trades, DEXs are typically organized as constant function market makers (CFMMs). CFMMs accept and reject proposed trades based on the evaluation of a function that depends on the proposed trade and the current reserves of the DEX. For trades that involve only two assets, CFMMs are easy to understand, via two functions that give the quantity of one asset that must be tendered to receive a given quantity of the other, and vice versa. When more than two assets are being exchanged, it is harder to understand the landscape of possible trades. We observe that various problems of choosing a multi-asset trade can be formulated as convex optimization problems, and can therefore be reliably and efficiently solved.

Differential Liquidity Provision in Uniswap v3 and Implications for Contract Design

“…In addition, their work also provides expressions for potential LP earnings under simple price changes in v2 contracts. This work is extended in [3], where the authors provide similar guarantees for a more general class of CFMMs, as well as in [2], where similar results and techniques are extended to CFMMs supporting multi-asset trades. In [4] the authors also study the implications of the curvature in reserve curves for traders and LPs, providing concrete tradeoffs for when high and low curvature regimes favor each of these two classes of agents.…”

“…Definition 1 (Uniswap v2 Reserve Curve). For L > 0 units of liquidity held in the contract, we let R (2) (L) denote the v2 reserve curve at liquidity L between token A and token B, with…”

“…Based on the following correspondence, each point on reserve curve R (2) (L) can be identified by (x, y) assets of token A and token B or equivalently (L, P ) and the amount of liquidity and the contract price.…”

Explainable AI via Learning to Optimize

“…Step 2: Choose Algorithm. The proximal-gradient scheme generates a sequence {z k } converging to a limit which solves (5). By simplifying and combining terms, the proximal-gradient method can be written via the iteration 4…”

“…where λ > 0 is a step-size and W is a matrix defined in terms 5 of W 1 , W 2 , and A . From the update on the right hand side of (6), we see the step size λ can be "absorbed" into the tunable matrix W and the shrink function parameter can be set to θ > 0.…”