Application of drifted Brownian motion in dynamic pricing of constant function market makers

Abstract

This paper builds upon the price dynamics models proposed by Joseph Najnudel et al. Within this paper, we propose a reference price following geometric Brownian motion with drift μ. The Constant Function Market Makers (CFMM) price is bounded by γ·(reference price) and (1/γ)·(reference price) for some constant fee factor γ. Through the use of random walks and First-Step analyses, we derived the first hitting times of a reflecting Brownian motion with drift, as well as several numerical results including the expected value, variance, and limiting distribution of the discretized CFMM price process.