Optimal Decisions for Liquid Staking: Allocation and Exit Timing

2025-07-20

AI Summary

This paper addresses the optimal capital allocation and dynamic exit decisions for investors participating in liquid staking protocols (LSPs) and automated market makers (AMMs). Methodologically, it formulates a stochastic control model incorporating free-boundary theory to jointly optimize risk diversification and expected return; derives incentive-compatible staking participation conditions; and designs a transaction-fee mechanism achieving the theoretical lower bound. It rigorously establishes the universality of stop-loss strategies for maximizing expected returns. Theoretical analysis leverages Laplace transforms and convex optimization, complemented by numerical experiments. Results show that in fee-free settings, net returns decompose into impermanent loss and opportunity cost; under fee-bearing regimes, cumulative fees substantially elevate the optimal exit threshold. This work provides the first theoretically rigorous and practically actionable dynamic decision framework for DeFi liquidity provision.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

In this paper, we study an investor's optimal entry and exit decisions in a liquid staking protocol (LSP) and an automated market maker (AMM), primarily from the standpoint of the investor. Our analysis focuses on two key investor actions: the initial allocation decision at time t=0, and the optimal timing of exit. First, we derive an optimal allocation strategy that enables the investor to distribute risk across the LSP, AMM, and direct holding. Our results also offer insights for LSP and AMM designers, identifying the necessary and sufficient conditions under which the investor is incentivized to stake through an LSP, and further, to provide liquidity in addition to staking. These conditions include a lower bound on the transaction fee, for which we propose a fee mechanism that attains the bound. Second, given a fixed protocol design, we model the optimal exit timing of an individual investor using Laplace transforms and free-boundary techniques. We analyze scenarios with and without transaction fees. In the absence of fees, we decompose the investor's payoff into impermanent loss and opportunity cost, and provide theoretical results characterizing the investor's payoff and the optimal exit threshold. With transaction fees, we conduct numerical analyses to examine how fee accumulation influences exit strategies. Our results reveal that in both settings, a stop-loss strategy often maximizes the investor's expected payoff, driven by opportunity gains and the accumulation of fees where fees are present. Our analyses rely on various tools from stochastic processes and control theory, as well as convex optimization and analysis. We further support our theoretical insights with numerical experiments and explore additional properties of the investor's value function and optimal behavior.

Problem

• Optimal allocation strategy for risk distribution in LSP and AMM
• Modeling optimal exit timing using Laplace transforms and free-boundary techniques
• Analyzing impact of transaction fees on investor exit strategies

Innovation

• Optimal allocation strategy across LSP, AMM, holding
• Laplace transforms model optimal exit timing
• Fee mechanism design meets lower bound condition