A Theoretical Approach to Stablecoin Design via Price Windows

TL;DR

This work studies price-window backed stablecoins, where mint and redeem prices are fixed at $1+\varepsilon_\alpha$ and $1-\varepsilon_\beta$, which confines the market price to $[1-\varepsilon_\beta,\,1+\varepsilon_\alpha]$ when reserves are adequate. It develops a theoretical framework around interchangeability of agents, introduces a sensitive speculator, and proves that without secondary stabilization, such designs cannot achieve both short- and long-term stability unless the backing-coin tail volatility is effectively tamed; stability requires the price-window width to scale with tail volatility, leading to inheritance of backing-coin volatility in the stablecoin price. The paper strengthens insights through a Case Study with IID price draws and simulations on real-world BTC/ETH data, showing that reserve depletion can occur in practical timeframes (e.g., hundreds of timesteps) and is accelerated by higher backing-coin volatility and patient, risk-tolerant behavior. The findings imply that price-window designs alone are insufficient for durable stability in volatile environments and highlight the necessity of secondary stabilization mechanisms (such as governance or dual-token models) to achieve robust stability, with quantified tradeoffs for reserve levels and transaction fees.

Abstract

In this paper, we explore the short- and long-term stability of backed stablecoins offering constant mint and redeem prices to all agents. We refer to such designs as price window-based, since the mint and redeem prices constrain the stablecoin's market equilibrium. We show that, without secondary stabilization mechanisms, price window designs cannot achieve both short- and long-term stability unless they are backed by already-stable reserves. In particular, the mechanism faces a tradeoff: either risk eventual reserve depletion through persistent arbitrage by a speculator, or widen the distance between mint and redeem prices enough to disincentivize arbitrage. In the latter case, however, the market price of the stablecoin inherits the volatility of its backing asset, with fluctuations that can be proportional to the backing asset's own volatility.

Figures (3)

• Figure 1: The log of the expected time to depletion for prices drawn independently at random from a normal distribution with $\mu = 100$ and various values of $\sigma^2$, given a speculator with $\delta \in [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7]$, $\Lambda = 0.5$, $n_0 = 10$, and initial reserves $R_0 = 1000$.
• Figure 2: The log of the expected time to depletion for prices drawn independently at random from a normal distribution with $\mu = 100$ and various values of $\sigma^2$, given a speculator with $\delta=0.2$, $\Lambda \in [0, 0.2, 0.4, 0.6, 0.8]$, $n_0 = 10$, and initial reserves $R_0 = 1000$.
• Figure 3: The results of our Gaussian random walk simulations for both Ethereum and Bitcoin. The mean step-size $\mu_{step}$ and random walk starting point were fixed for each currency throughout, and 100 random walk simulations were run for each value of $\sigma_\text{step}$ for up to 100,000 iterations. The initial reserves contained $R_0 = 1000$ coins, and the speculator was endowed with $n_0 = 100$ coins at the start of each run.

Theorems & Definitions (25)

• Definition 2.1: $\varepsilon$-Stability over a sequence
• Definition 2.2: Weak $\varepsilon$-stability
• Definition 3.1: Approximate optimality
• Definition 3.2: Speculator sensitivity
• Theorem 3.1
• Example 1
• proof : Proof of Theorem \ref{['thm:price-window-impossibility-result_constant-epsilon']}
• proof : Proof of (1)
• proof : Proof of (2)
• Lemma B.1