Optimal Control of Reserve Asset Portfolios for Stablecoins

1. Problem Statement

Stablecoin issuers face a fundamental treasury management dilemma: they must maintain par convertibility (the ability to redeem coins for $1 on demand) to preserve the peg, while simultaneously earning sufficient yield on reserves to operate sustainably.

• The Trade-off: Holding 100% cash ensures liquidity but incurs opportunity costs (zero yield). Holding short-term government bills (T-bills) generates yield but introduces duration risk and settlement lags.
• The Dynamics: Redemption and minting flows are not random; they exhibit clustering (self-excitation), where large transactions trigger further transactions (bank-run dynamics). Peg deviations occur when immediate cash reserves are insufficient to cover expected outflows within a settlement window, causing the market price to deviate from par.
• The Objective: Develop a control framework that dynamically allocates reserves between cash and T-bills and sets mint/burn fees to minimize peg deviations and transaction frictions while maximizing yield, all under regulatory constraints (e.g., the U.S. GENIUS Act).

2. Methodology

The paper proposes a Stochastic Model Predictive Control (MPC) framework integrated with Pontryagin's Maximum Principle (PMP).

A. Mathematical Modeling

• Flow Dynamics: Mint and redemption events are modeled as Hawkes processes (self-exciting point processes). The intensity of future events depends on past events and the current peg deviation ($\Delta P$), capturing the feedback loop where instability accelerates redemptions.
• State Variables: The system state includes liquid cash reserves ($R_{liq}$), bill reserves ($R_{bill}$), peg deviation ($\Delta P$), and the Hawkes intensities for mints ($\lambda_M$) and redemptions ($\lambda_R$).
• Peg Dynamics: The peg deviation evolves deterministically based on a "fair value" deviation ($\Delta P_{fair}$), which is a function of the liquidity shortfall relative to outstanding supply, and a corrective term driven by the mint/burn fee spread ($\delta$).
• Control Variables:
  1. Reallocation Rate ($\omega$): The flow of assets between cash and bills.
  2. Fee Spread ($\delta$): The spread charged on minting or burning to dampen order flow imbalances.

B. Optimization Framework

• Objective Function: Minimize an infinite-horizon discounted cost functional comprising:
  • Quadratic penalty on peg deviations (reputational/convertibility risk).
  • Quadratic penalty on fee spreads (user friction).
  • Linear and quadratic costs on reallocation (transaction costs/market impact).
  • Negative yield from T-bills (maximizing carry).
• Moment Closure: To make the stochastic problem tractable, the authors replace random flows with their conditional expectations (filtered estimates of Hawkes intensities). This transforms the stochastic control problem into a deterministic optimal control problem solvable over a rolling horizon.
• Solution via PMP: The authors apply Pontryagin's Maximum Principle to derive the optimal control laws. They introduce shadow prices (costates) representing the marginal value of cash, bills, and peg stability.

C. Key Theoretical Results

• Soft-Thresholding Structure: The optimal reallocation control ($\omega^*$) follows a soft-thresholding rule. No rebalancing occurs if the difference in shadow prices (marginal value of bills vs. cash) lies within a "deadzone" defined by transaction costs. Rebalancing only occurs when this differential exceeds the threshold, scaling linearly with the excess.
• Settlement Windows: Recognizing that real-world treasury operations occur at discrete intervals (e.g., Fedwire windows), the continuous-time policy is discretized. The optimal policy for a window becomes an average of the shadow prices over that period, maintaining the soft-thresholding structure.
• Monotone Stress-Response: The authors prove a Monotone Stress-Response Property: as expected redemption pressure increases (or settlement windows lengthen), the optimal policy shifts predictably and monotonically toward holding more cash. This ensures the system does not oscillate wildly but transitions smoothly from yield-seeking to liquidity-preserving modes.

3. Key Contributions

1. Unified Control Framework: First to jointly optimize reserve allocation (cash vs. bills) and fee setting under stochastic, self-exciting flow dynamics.
2. Operational Realism: The model explicitly incorporates settlement windows and transaction costs, moving beyond theoretical continuous-time models to a "sampled-data" implementation ready for treasury desks.
3. Analytical Policy Structure: Derivation of a soft-thresholding policy that provides a clear, auditable rule for when to rebalance, avoiding the "chatter" of continuous trading.
4. Regulatory Alignment: The framework is designed to comply with strict regulatory requirements (like the GENIUS Act) regarding eligible high-quality liquid assets (HQLA).
5. Stress-Testing Capability: The inclusion of Hawkes processes allows the model to naturally simulate "bank-run" scenarios where redemptions cluster and self-reinforce.

4. Numerical Results

The authors simulated the controller over 92 days across three scenarios: Single Shock, Prolonged Clustering (near-critical branching ratio), and False Alarm (transient noise).

• Performance vs. Benchmarks:
  • Optimal Control: Successfully maintained the peg (0% depeg frequency) in all scenarios. It captured significant yield during calm periods (by holding T-bills) and aggressively built cash reserves when stress signals appeared.
  • Maximum Yield Strategy: Failed catastrophically in stress scenarios. By holding too much in T-bills, it could not absorb clustered redemptions, triggering self-reinforcing depegs and massive losses.
  • Maximum Liquidity Strategy: Maintained the peg but generated zero revenue by holding 100% cash.
  • Equal Weights (50/50): Performed better than max yield but failed to adapt dynamically. It suffered depegs in single-shock scenarios because its fixed target could not scale cash buffers fast enough, and it missed yield opportunities during calm periods.
• Responsiveness: The optimal controller detected stress early (responding within ~0.67 days in prolonged clustering) and reversed course quickly once the threat subsided, avoiding unnecessary cash hoarding during false alarms.

5. Significance and Implications

• Practical Implementation: The paper bridges the gap between academic optimal control theory and practical treasury management. The derived "soft-thresholding" rule is simple enough to be implemented in daily operational workflows and is easily auditable by regulators.
• Resilience: The framework demonstrates that stablecoins can be both profitable and stable. By anticipating clustered flows via Hawkes processes, issuers can hold higher-yielding assets without sacrificing peg stability.
• Policy Design: It provides a quantitative basis for setting reserve ratios and fee structures dynamically, rather than relying on static, rule-of-thumb heuristics (e.g., "always hold 10% cash").
• Regulatory Insight: The model supports the argument that dynamic, stress-aware liquidity management is superior to static capital requirements, offering a path to comply with regulations like the GENIUS Act while optimizing economic efficiency.

In conclusion, Hammerl presents a robust, mathematically rigorous, and operationally feasible solution to the stablecoin treasury problem, proving that dynamic, predictive control significantly outperforms static heuristics in maintaining peg stability and maximizing issuer revenue.