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With the emergence of decentralized finance, new trading mechanisms calledAutomated Market Makers have appeared. The most popular Automated Market Makersare Constant Function Market Makers. They have been studied both theoreticallyand empirically. In particular, the concept of impermanent loss has emerged andexplains part of the profit and loss of liquidity providers in ConstantFunction Market Makers. In this paper, we propose another mechanism in whichprice discovery does not solely rely on liquidity takers but also on anexternal exchange rate or price oracle. We also propose to compare thedifferent mechanisms from the point of view of liquidity providers by using amean / variance analysis of their profit and loss compared to that of agentsholding assets outside of Automated Market Makers. In particular, inspired byMarkowitz' modern portfolio theory, we manage to obtain an efficient frontierfor the performance of liquidity providers in the idealized case of a perfectoracle. Beyond that idealized case, we show that even when the oracle is laggedand in the presence of adverse selection by liquidity takers and systematicarbitrageurs, optimized oracle-based mechanisms perform better than popularConstant Function Market Makers.

Automated Market Makers: Mean-Variance Analysis of LPs Payoffs and Design of Pricing Functions