A Comparison of Impermant Loss for Various CFMMs

Decentralized exchanges are one of the revolutionary inventions in cryptocurrency trading. Decentralized exchanges present a way to set prices mathematically without maintaining an order book. Order books have been a widely-known method of determining prices for over 400 years in the stock markets and even in decentralized cryptocurrency exchanges. Recently, a new type of decentralized exchange devised a constant function market maker to mathematically determine prices. Among them, constant product market makers are the most widely used. In addition, several constant function market makers have been proposed. However, there was no discussion about the desirable requirements of a constant function market maker in terms of impermanent loss. In this paper, we discuss the desirable requirements of a constant function market maker and analyze the impermanent loss of various market makers. Note that two market makers, LMSR and CCMM, show impermanent gain in the mathematical derivation, but their price functions violate the principle of supply and demand. Therefore, AMMs with impermanent gain while following the principle of supply and demand need to be studied further. A comparison study of impermanent loss for some CFMMs shows interesting results, too.