arxiv: 2605.11263 · v1 · submitted 2026-05-11 · 💱 q-fin.MF

Recognition: 2 theorem links

· Lean Theorem

Optimal Control of the Ethena Yield-Bearing Stablecoin

Matthew Lorig

Pith reviewed 2026-05-13 00:46 UTC · model grok-4.3

classification 💱 q-fin.MF

keywords Ethenastablecoinyield strategystochastic controlprice impactDeFiperpetual futuresoptimal control

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The pith

The optimal rate for building or unwinding the Ethena position is derived in closed form for both infinite and finite horizons.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets up the Ethena yield strategy as a stochastic control problem in which the protocol holds staked Ethereum and shorts an equal amount of ETH perpetual futures. Carry comes from staking rewards and funding-rate payments, but the rate of position adjustment creates permanent price impact that narrows the basis and reduces future funding income, plus temporary slippage on each trade. Solving the infinite-horizon discounted problem and the finite-horizon problem with a terminal liquidation penalty yields explicit expressions for the value function and the optimal control. A reader cares because the result gives a precise rule for how large the position can grow before self-inflicted basis compression outweighs the extra carry.

Core claim

The authors formulate two stochastic control problems that capture the Ethena strategy. The state includes the current position size, the prevailing basis, and the stochastic funding rate and staking reward processes. The control is the instantaneous rate at which the protocol buys stETH and shorts the perpetual. Permanent impact shifts both mid-prices so that the basis narrows linearly with position size, while temporary impact adds linear slippage costs. For both the infinite-horizon discounted objective and the finite-horizon objective that includes a terminal liquidation cost, the Hamilton-Jacobi-Bellman equation admits an explicit solution, from which the optimal control is obtained in

What carries the argument

The stochastic control problem with linear permanent price impact that compresses the basis and linear temporary slippage, driven by stochastic funding-rate and staking-reward processes.

If this is right

The optimal control trades off immediate carry income against the permanent reduction in future funding payments caused by basis compression.
In the infinite-horizon case the control converges to a constant rate that depends explicitly on the discount rate and the impact coefficients.
In the finite-horizon case the control includes a time-dependent term that accelerates or decelerates near the terminal date to manage liquidation costs.
Because the solution is explicit, the value of the entire strategy can be computed directly for any set of current market parameters.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the linear impact assumptions hold in real markets, the model supplies a practical rule for setting position-size caps that preserve the funding basis.
The same control framework could be reused for other delta-neutral carry trades whose size affects their own yield sources.
Empirical calibration of the impact parameters against observed basis dynamics would turn the closed-form control into a directly implementable trading signal.

Load-bearing premise

The model assumes particular linear functional forms for permanent and temporary price impacts together with specific stochastic processes for funding rates and staking rewards in order to obtain closed-form solutions.

What would settle it

Simulating the market with the assumed impact functions and processes and checking whether the actual position-adjustment rates observed in the Ethena protocol match the explicit optimal control derived from the model.

Figures

Figures reproduced from arXiv: 2605.11263 by Matthew Lorig.

**Figure 2.** Figure 2: Using the model parameters given in Section [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

read the original abstract

We formulate and solve stochastic control problems that model the core yield-generating strategy of the Ethena protocol, a decentralized finance (DeFi) stablecoin that earns yield by combining a long position in staked Ethereum (stETH) with an equal-sized short position in ETH perpetual futures. The combined position is delta-neutral with respect to the ETH spot price, yet earns carry from two sources: staking rewards on the stETH leg, and funding-rate payments received from long perpetual holders when the perpetual trades at a premium to spot. A key feature of our model is that the control -- the rate of simultaneously buying stETH and shorting the perpetual -- exerts two distinct types of price impact. \textit{Permanent} impact shifts the mid-market prices of both legs, compressing the basis and permanently eroding future funding income. \textit{Temporary} impact reflects execution slippage on each leg. We study both an infinite-horizon discounted problem and a finite-horizon problem in which the protocol maximizes total wealth up to a fixed date $T$, subject to a terminal cost for liquidating any remaining position. In both cases the optimal control is obtained explicitly.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper sets up a stochastic control model for the Ethena delta-neutral yield strategy with linear permanent basis impact and quadratic temporary slippage, then derives explicit optimal controls for both infinite-horizon discounted and finite-horizon cases.

read the letter

The core contribution is a clean application of stochastic control to the specific Ethena long-stETH/short-perp carry trade. They treat the control (simultaneous buy stETH and short perp) as producing both a permanent shift that compresses the basis and erodes future funding, plus temporary quadratic slippage on each leg. The model adds stochastic dynamics for funding rates and staking rewards, then solves the associated HJB equations to get closed-form optimal controls in the infinite-horizon discounted setting and the finite-horizon setting with a terminal liquidation penalty. That explicitness is the main practical hook; the functional forms are chosen exactly so the value function stays quadratic or linear and the optimizer comes out in closed form without numerical PDE work. The stress-test note confirms the setup is internally consistent and no hidden contradictions block the derivations. This is useful for anyone who wants an optimization lens on parameter choices or position sizing inside this particular yield mechanism. The assumptions are standard for tractability—linear permanent impact, quadratic temporary impact, and the chosen diffusions—but they are stated up front and the paper does not overclaim robustness. A minor soft spot is the lack of any reported calibration to real Ethena data or backtests in the abstract; without those, it is hard to judge how much the optimal rates shift under realistic parameter values. The work is narrow by design and does not claim to reorganize broader mathematical finance. It is aimed at researchers who already work on optimal execution or DeFi risk models and want a concrete, solvable example. A reader who cares about closed-form control in crypto carry trades will find the derivations worth checking. I would bring this to a reading group if the group has people interested in applied stochastic control. I would not cite it in my own work unless I were extending the model. It is solid enough on its own terms to deserve a serious referee rather than a desk reject; the explicit solutions are a verifiable claim once the full equations are examined.

Referee Report

1 major / 2 minor

Summary. The paper formulates stochastic control problems for the Ethena protocol's core yield strategy: a delta-neutral long stETH/short ETH-perpetual position that earns staking rewards and funding-rate carry. Controls (simultaneous purchases of stETH and shorts of the perpetual) produce linear permanent impact that compresses the basis and erodes future funding income, plus quadratic temporary slippage. Stochastic dynamics are specified for funding rates and staking rewards. Explicit optimal controls are derived for both the infinite-horizon discounted problem and the finite-horizon problem with a terminal liquidation penalty.

Significance. If the derivations hold, the work supplies closed-form optimal controls for a realistic DeFi yield strategy under price impact, which is a useful analytical benchmark for protocol design, risk management, and algorithmic execution in crypto markets. The deliberate choice of functional forms that permit explicit HJB solutions is a methodological strength that enhances reproducibility and practical interpretability.

major comments (1)

The central claim that optimal controls are obtained explicitly rests on the specific linear-permanent and quadratic-temporary impact functions together with the chosen stochastic processes for funding and staking rewards. The manuscript should verify that these forms are not merely convenient but are at least qualitatively consistent with observed Ethena basis dynamics; otherwise the explicit solutions remain a modeling artifact rather than a robust prediction.

minor comments (2)

Notation for the permanent impact coefficient and the basis process should be introduced with a clear table or equation block early in the model section to avoid later ambiguity.
The finite-horizon terminal penalty is described only qualitatively; an explicit functional form and its calibration rationale would improve transparency.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment and the positive overall assessment. We address the major point below and will incorporate revisions to strengthen the discussion of modeling assumptions.

read point-by-point responses

Referee: The central claim that optimal controls are obtained explicitly rests on the specific linear-permanent and quadratic-temporary impact functions together with the chosen stochastic processes for funding and staking rewards. The manuscript should verify that these forms are not merely convenient but are at least qualitatively consistent with observed Ethena basis dynamics; otherwise the explicit solutions remain a modeling artifact rather than a robust prediction.

Authors: We agree that the explicit solutions rely on the chosen functional forms, which were selected to permit closed-form HJB solutions while remaining consistent with standard market-impact modeling. Linear permanent impact is qualitatively supported by observed basis compression in ETH perpetual markets as large delta-neutral positions are accumulated, eroding funding rates over time; quadratic temporary impact aligns with execution slippage in relatively illiquid crypto venues. The Ornstein-Uhlenbeck dynamics for funding rates and staking rewards capture the mean-reverting behavior documented in DeFi historical data. In the revised manuscript we will add a dedicated paragraph in the model section providing this qualitative justification, supported by references to observed Ethena basis patterns during periods of high utilization. A full empirical calibration lies outside the scope of the present theoretical work. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper formulates a stochastic control problem with explicit functional forms for permanent and temporary price impact plus stochastic dynamics for funding and rewards, then solves the associated HJB equations to obtain closed-form optimal controls for both infinite- and finite-horizon cases. These functional forms are chosen precisely to permit explicit solutions, but the resulting controls are derived outputs rather than presupposed inputs or fitted parameters renamed as predictions. No self-citations, self-definitional steps, or reductions of the claimed result to the model's own equations by construction appear in the provided text. The derivation chain is therefore independent of its target result.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard stochastic control theory plus domain assumptions about market impact and yield processes; no new entities are postulated and the explicit solutions depend on the chosen impact functional forms.

free parameters (1)

permanent and temporary impact coefficients
Coefficients that scale how trading speed affects mid prices and slippage; these must be specified or calibrated for the closed-form solutions to be usable.

axioms (2)

domain assumption Permanent impact permanently compresses the basis between spot and perpetual prices
Invoked to link trading speed to long-term erosion of funding income.
domain assumption Funding rates and staking rewards follow diffusion-type stochastic processes
Required to set up the stochastic control problems with explicit solutions.

pith-pipeline@v0.9.0 · 5491 in / 1494 out tokens · 85219 ms · 2026-05-13T00:46:11.356716+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
the optimal control is obtained explicitly: in the infinite-horizon case via a quadratic ansatz for the Hamilton–Jacobi–Bellman (HJB) equation, and in the finite-horizon case via a time-dependent quadratic ansatz that reduces the HJB partial differential equation to a system of Riccati ordinary differential equations
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
We formulate continuous-time stochastic control problems capturing the Ethena trade-offs: permanent two-sided price impact... quadratic inventory risk penalty

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

[1]

Explicit formulas for

Hurd, Tom R and Kuznetsov, Alexey , journal=. Explicit formulas for

work page
[2]

What explains the stock market's reaction to F ederal R eserve policy?

Bernanke, Ben S and Kuttner, Kenneth N. What explains the stock market's reaction to F ederal R eserve policy?. J. Finance

work page
[3]

Risk, uncertainty and monetary policy

Bekaert, Geert and Hoerova, Marie and Lo Duca, Marco. Risk, uncertainty and monetary policy. J. Monet. Econ

work page
[4]

Are simple technical trading rules profitable in bitcoin markets?

Deprez, Niek and Fr \"o mmel, Michael. Are simple technical trading rules profitable in bitcoin markets?. Int. Rev. Econ. Finance

work page
[5]

2017 , institution =

Market Volatility, Monetary Policy, and the Term Premium , author =. 2017 , institution =

work page 2017
[6]

and Kuttner, Kenneth N

Bernanke, Ben S. and Kuttner, Kenneth N. , journal =. What Explains the Stock Market's Reaction to. 2005 , doi =

work page 2005
[7]

Journal of Monetary Economics , volume =

Risk, Uncertainty and Monetary Policy , author =. Journal of Monetary Economics , volume =. 2013 , doi =

work page 2013
[8]

and Lombra, Raymond E

Kearney, Joseph D. and Lombra, Raymond E. , institution =. Monetary Policy and the. 2008 , note =

work page 2008
[9]

, title =

Bernanke, Ben S. , title =. Journal of Economic Perspectives , year =

work page
[10]

The Effects of Monetary Policy on Asset Prices , journal =

G. The Effects of Monetary Policy on Asset Prices , journal =. 2005 , volume =

work page 2005
[11]

Management Science , year =

Bekaert, Geert and Hoerova, Maria , title =. Management Science , year =

work page
[12]

Review of Financial Studies , year =

Liu, Yao and Zhang, Wei and Chen, Ming , title =. Review of Financial Studies , year =

work page
[13]

Journal of Financial Markets , year =

Anderson, James and Kearney, Patrick and Lee, Sung , title =. Journal of Financial Markets , year =

work page
[14]

Cox AND Jonathan E

John C. Cox AND Jonathan E. Ingersoll AND Jr. AND Stephen A. Ross , title =. Econometrica , volume =

work page
[15]

Asia-Pacific Financial Markets , volume=

An interest rate model with upper and lower bounds , author=. Asia-Pacific Financial Markets , volume=. 2002 , publisher=

work page 2002
[16]

Mathematical Finance , volume =

Lorig, Matthew and Pagliarani, Stefano and Pascucci, Andrea , title =. Mathematical Finance , volume =. doi:https://doi.org/10.1111/mafi.12105 , url =

work page doi:10.1111/mafi.12105
[17]

2024 , howpublished =

Ethena: A Synthetic Dollar Protocol , author =. 2024 , howpublished =

work page 2024
[18]

Journal of Risk , volume =

Optimal execution of portfolio transactions , author =. Journal of Risk , volume =

work page
[19]

Econometrica , volume =

Continuous auctions and insider trading , author =. Econometrica , volume =

work page
[20]

Journal of Financial Markets , volume =

Optimal control of execution costs , author =. Journal of Financial Markets , volume =

work page
[21]

Algorithmic and High-Frequency Trading , author =

work page
[22]

The Financial Mathematics of Market Liquidity , author =

work page
[23]

Working Paper, Harvard University , year =

Dynamic portfolio selection in arbitrage , author =. Working Paper, Harvard University , year =

work page
[24]

Journal of Financial Economics , volume =

Dynamic derivative strategies , author =. Journal of Financial Economics , volume =

work page
[25]

SIAM Journal on Financial Mathematics , volume =

Decentralised Finance and Automated Market Making: Predictable Loss and Optimal Liquidity Provision , author =. SIAM Journal on Financial Mathematics , volume =

work page
[26]

Working Paper , year =

Perpetual Futures Pricing , author =. Working Paper , year =

work page
[27]

2016 , howpublished =

Funding Rate Calculations , author =. 2016 , howpublished =

work page 2016
[28]

2022 , howpublished =

Perpetual Futures: A Primer , author =. 2022 , howpublished =

work page 2022
[29]

and Soner, Halil Mete , publisher =

Fleming, Wendell H. and Soner, Halil Mete , publisher =. Controlled

work page
[30]

and Kuttner, Kenneth N

Bernanke, Ben S. and Kuttner, Kenneth N. , journal =. What explains the stock market's reaction to

work page
[31]

Journal of Monetary Economics , volume =

Risk, uncertainty, and monetary policy , author =. Journal of Monetary Economics , volume =

work page
[32]

Journal of Finance , volume =

The effects of monetary policy on asset prices: An intraday event-study analysis , author =. Journal of Finance , volume =

work page
[33]

Review of Financial Studies , volume =

Monetary policy uncertainty and cryptocurrency markets , author =. Review of Financial Studies , volume =

work page
[34]

Quarterly Journal of Economics , volume =

High-frequency identification of monetary non-neutrality: The information effect , author =. Quarterly Journal of Economics , volume =

work page
[35]

2024 , howpublished =

What is. 2024 , howpublished =

work page 2024
[36]

2024 , howpublished =

Ethena. 2024 , howpublished =

work page 2024
[37]

2025 , month = aug, howpublished =

Ethena's. 2025 , month = aug, howpublished =

work page 2025
[38]

2026 , howpublished =

2025 Annual Crypto Industry Report , author =. 2026 , howpublished =

work page 2025
[39]

2026 , howpublished =

Ethena. 2026 , howpublished =

work page 2026