What Must the Price in Decentralized Exchanges Be?

Decentralized exchanges are one of the most remarkable revolutionary inventions in cryptocurrency trading. Decentralized exchanges present a way to set prices mathematically without maintaining the order book-based trading system. The order book-based trading system has been a widely-known method to determine prices in the stock markets for over 400 years. Recently, a new type of decentralized exchange has devised a constant function market maker to determine prices mathematically. Among them, the method of constant product market makers is the most widely used one. In addition, several constant function market makers have been proposed. However, there was little discussion about the desirable properties of a constant function market maker regarding price and impermanent loss. In this paper, we discuss the desirable price and its effects on impermanent loss. This study considers two types of prices: the reference price and the actual price. We show that impermanent gain can be achievable under certain conditions and prove it mathematically. Two examples are provided to show that impermanent gain is achievable. One question is whether there is agreement on applying actual price when calculating value gains.