Optimal Decisions for Liquid Staking: Allocation and Exit Timing

Paper: arXiv 2507.14810

Authors: Ruofei Ma, Zhebiao Cai, Wenpin Tang, David Yao

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.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 3.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework using Laplace transforms, stochastic control, and free-boundary problems to solve optimal allocation and exit strategies in Liquid Staking Protocols and AMMs. While it features strong mathematical derivations and useful protocol design insights, its empirical validation relies primarily on illustrative numerical experiments rather than extensive historical market backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal: Optimize Investor Allocation & Exit Timing in LSPs and AMMs"] --> B["Data Inputs: Transaction Fees, Yields, Impermanent Loss & Asset Volatility"]
    B --> C["Methodology: Stochastic Control, Free-Boundary Problems & Convex Optimization"]
    C --> D["Computational Analysis: Analyze Payoffs With & Without Transaction Fees"]
    D --> E1["Outcome 1: Optimal Allocation & Fee Bounds Driving LSP/AMM Participation"]
    D --> E2["Outcome 2: Stop-Loss Exit Strategy Maximizes Payoffs via Fee & Yield Accumulation"]

Cited References (32)

1. SoK: Liquid Staking Tokens (LSTs) and Emerging Trends in Restaking (2024) — link
2. The economics of liquid staking derivatives: Basis determinants and price discovery (2025)
3. Liquid staking solutions now have more TVL than DEXs: Finance Redefined (2023) — link
4. Exploring the Market Dynamics of Liquid Staking Derivatives (LSDs) (2024) — link
5. Automated Market Making and Arbitrage Profits in the Presence of Fees (2023) — link
6. Automated Market Making and Loss-Versus-Rebalancing (2024) — link
7. The Adoption of Blockchain-based Decentralized Exchanges (2021) — link
8. A Myersonian Framework for Optimal Liquidity Provision in Automated Market Makers — → 2303.00208

2 of 32 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.