Optimal Allocation Strategy in DeFi

• Optimal Allocation Strategy is a framework that balances risk between staking and liquidity provision by integrating market dynamics and fee thresholds.
• It defines precise staking criteria where returns outweigh holding, and liquidity benchmarks are met when fee accumulation surpasses set thresholds.
• The strategy employs optimal stopping techniques and numerical experiments to guide investors and protocol designers in timing exits and managing risk.

constant product market maker shows an investor’s final “liquidity‐share” portfolio is given by an optimized expression: $2\sqrt{xy\,D_tP_t}$ This reflects both the external price dynamics and the discount factor.

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1. Conditions for Staking and Liquidity Provision The paper outlines conditions under which an investor is incentivized to stake through an LSP and provide liquidity in an AMM.

Staking Incentives: The investor prefers staking over holding ETH if: $r + g - \rho - m > \ln P_0 > 0$ where $r$ is the staking reward rate, $g$ is the token price growth rate, $m$ is the delay in staking rewards, and $\rho$ is the discount rate. This inequality ensures the discounted expected return from staking exceeds that of holding ETH.

Liquidity Provision: For liquidity provision to be beneficial, the accumulated transaction fees should exceed a threshold. Specifically, the investor is incentivized to supply liquidity if: $$\int_{0}^{t} \phi_s e^{-\rho s} \, ds > P_0\Bigl(e^{(g - m - \rho)t}(e^{rt}+1) - 2\,e^{\bigl(\frac{1}{2}g - \frac{1}{8}\sigma^2 -\frac{1}{2}m - \rho\bigr)t}\Bigr)$$ The paper provides a specific fee mechanism $\phi_t$ that achieves this bound, thereby supporting both LSP staking and AMM liquidity provision.

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1. Optimal Exit Timing For optimal exit timing, the investor must consider when to withdraw from staking or AMM to maximize payoffs.

Optimal Stopping Problem: This is framed using techniques such as Laplace transforms and free-boundary analysis. The investor's goal is formulated to optimize: $$W(t,P_t)= S(t,P_t) - e^{-\rho t}\,\frac{e^{rt}\,D_tP_t}{P_0}$$ where $S(t,P_t)$ accounts for current portfolio value including fees and rewards.

Stop-Loss Strategy: In the presence of transaction fees, results indicate a stop-loss strategy often maximizes payoff. This strategy involves withdrawing when the price falls to a critical lower threshold due to the opportunity cost being lower than the trading fee accumulation benefits, which drive the decision under transaction costs.

Numerical experiments show the optimal threshold, influenced by parameters such as fee rates and market conditions.

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1. Theoretical and Numerical Insights Key theoretical findings are supported by numerical analysis:

• Balanced Allocation: The allocation strategy $a^\star = \frac{1}{2}$ is optimal, dividing risk evenly across staking and direct holding, ensuring balanced exposure to staking, fees, and market dynamics.
• Incentive Alignment: Conditions $r+g-\rho-m > \ln P_0$ ensure economic incentive for staking over holding, while liquidity is further justified if trading fees surpass the defined threshold.
• Decomposition of Payoff: The payoff is effectively captured through impermanent loss (negative) and opportunity cost (positive under specific conditions), providing clarity on net benefits.
• Timing Sensitivity: Empirical data illustrates how the optimal exit time varies with market conditions, illustrating robustness and adaptability of the strategy.

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1. Implications for Investors and Protocol Designers The analysis presents tangible insights for both investors and designers of LSPs and AMMs:

• For Investors: Adopting a balanced allocation mitigates risk while maximizing potential returns from both staking and holding. The stop-loss strategy optimizes exit timing, taking advantage of fee accumulation under specified conditions.
• For Protocol Designers: The paper suggests structuring fee mechanisms to meet the derived lower bounds ensures adequate economic incentives for participants, aligning with optimal investment strategy outcomes.
• Strategic Flexibility: By offering clear boundaries and mechanisms for risk distribution and timing, the model supports dynamic and data-driven decision-making in volatile markets.

In summary, the paper provides a rigorous framework for optimal allocation and exit timing in liquid staking and AMM settings, grounded in stochastic calculus and optimization techniques that cater to investors' need for maximizing expected returns while managing inherent risks.