Why and How Are Prices in Smart Contract Determined Mathematically?

Abstract

It has been a decade since decentralized finance emerged. With the advent of smart contracts, numerous financial products are being built on blockchains. Despite limitations such as gas fee restrictions and the need for oracles, smart contracts are bringing about financial innovation. Smart contracts are a crucial tool for implementing financial automation, ideally suited for eliminating intermediaries and implementing atomic transactions. For a transaction to occur, a price must be determined. Over the years, the Black-Scholes equation, which determines options pricing, the market scoring rules (e.g., LMSR) that enable prediction markets, and the constant product formula (e.g., CPMM), which is at the heart of automatic market makers (AMMs), have been developed. Prices are highly subjective, and in reality, multiple prices exist for a single product. However, in decentralized finance, a single price is mathematically determined in a specific situation and accepted without resistance by the market, a remarkable phenomenon. This paper examines why prices must be mathematically determined and why they remain consistent with real-world prices. It also ex-amines how these prices are determined mathematically. Further-more, it examines the price determination mechanism from a cybernetic perspective. In particular, we analyze the phenomenon in which prediction market prices are also used as automatic market makers, and clearly distinguish the difference between the use of market scoring rules and constant product formulas. This paper demonstrates the existence of both path-independent and path-de-pendent prices. While path-independent prices have been extensively studied, research on path-independent pricing has been sparse.