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REVIEW 4 major objections 6 minor 1 cited by

Perpetual Demand Lending Pools

T0 review · 4 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read This paper formalizes perpetual demand lending pools and proves that a delta-hedged LP position matches or beats the unhedged Sharpe ratio whenever two simple conditions on fee revenue and hedge variance hold.

desk verdict Useful first formalization of PDLPs with coherent fee and arbitrage analysis, but the central delta-hedging theorem rests on an invalid variance step; worth reviewing but not as-is. read the letter →

arxiv 2502.06028 v2 pith:C5UXYC65 submitted 2025-02-09 cs.GT q-fin.PMq-fin.RM

Tarun Chitra , Theo Diamandis , Nathan Sheng , Luke Sterle , Kamil Yusubov This is my paper · ORCID
classification cs.GTq-fin.PMq-fin.RM
keywords perpetualfuturesdecentralizedfinanceliquiditypoolsdeltahedgingSharperatiotargetweightmechanismfundingratearbitrageloss-versus-rebalancing
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Perpetual demand lending pools let traders borrow pool assets to open perpetual futures positions, and they now hold over $2.5 billion. The paper gives the first formal model of these pools, covering funding-rate arbitrage, share creation and redemption, and the target-weight mechanisms used by GMX, Jupiter, and Hyperliquid. Its main result is a pair of sufficient conditions under which a delta-hedged LP portfolio has a Sharpe ratio at least as high as the unhedged portfolio: fee revenue must be at least the risk-aversion-scaled maximum eigenvalue of the asset covariance times expected portfolio delta, and the hedge's variance must be at most four times the portfolio's variance. If these conditions hold in practice, the model explains why hedged PDLP strategies have proliferated and suggests how dynamic fees and pool splitting can improve capital efficiency.

What carries the argument

The load-bearing objects are the PDLP itself (a reserve portfolio $R$, target portfolio $\pi$, lending fee $f$, and outstanding loans), the target weight mechanism that discounts share creation or redemption to push reserves toward target weights, and the mean-variance delta-hedging update $\pi_{\mathrm{new}} = (f/\gamma)\Sigma^{-1}\ell - \Delta$. The Sharpe-ratio argument runs through two inequalities: condition (14) ensures the hedged expectation is non-decreasing, and the variance condition ensures the hedged variance is non-increasing. The target weight mechanism does the work of bounding delta exposure, while the covariance eigenvalue condition connects loan demand to hedge profitability.

What would settle it

On a live PDLP, compute the sample variance of a delta-hedged position and the unhedged portfolio over rolling windows. If $\mathrm{Var}[p^T\pi] > 4\,\mathrm{Var}[p^T R]$ in any sustained window, or if measured fee revenue falls below $\gamma\lambda_{\max}\mathbb{E}[p^T\Delta]$, the sufficient conditions of Claim 4.1 fail and the Sharpe improvement is not guaranteed by the paper's argument.

Watch

Extended reading notes

Core claim

The paper's central claim is that PDLPs are easier to delta hedge than constant-function market makers, and that this is a structural property of the mechanism rather than an accident. Concretely, Claim 4.1 shows the Sharpe ratio of a hedged LP position is at least that of an unhedged position whenever expected fee revenue $f\mathbb{E}[p^T\ell]$ is at least $\gamma\lambda_{\max}\mathbb{E}[p^T\Delta]$ and the variance of the offsetting portfolio is at most four times the unhedged variance. In the same framework, the target weight mechanism bounds an LP's delta exposure: under the assumptions of Claim 3.4, the new portfolio's delta is at most $(1/2 - f)$ times the original delta. The paper also characterizes the fee interval that keeps both funding-rate arbitrageurs and LPs profitable, with fees inversely proportional to open interest.

Load-bearing premise

The Sharpe-ratio result depends on an asserted negative correlation between the hedge and the unhedged portfolio, together with a variance bound $\mathrm{Var}[p^T\pi] \leq 4\,\mathrm{Var}[p^T R]$ that is assumed rather than derived from PDLP mechanics, and the proof is done without transaction costs.

Editorial extensions

If this is right

  • A PDLP whose fees are set inversely proportional to open interest can sustain an equilibrium where both funding-rate arbitrageurs and liquidity providers earn nonnegative profit.
  • When Claim 4.1's conditions hold, liquidity providers can earn lending fees without taking on extra price risk, which directly explains the growth of hedged PDLP vaults.
  • Because target weight mechanisms bound LP delta, PDLP share tokens such as JLP can serve as higher-quality collateral in lending protocols.
  • Splitting a PDLP into multiple pools improves the delta-hedged Sharpe ratio when each sub-portfolio's conditional covariance has a smaller minimum eigenvalue than the other's full covariance (Claim B.1).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension is to estimate $\Sigma$, $\ell$, and $\Delta$ on live JLP or HLP data and check whether the two conditions of Claim 4.1 hold; if they fail in high-volatility regimes, the 'easy to hedge' conclusion becomes regime-dependent.
  • Because the main Sharpe proof omits transaction costs, a natural extension adds positive rebalancing costs $c$ to the mean-variance update and asks whether a threshold fee exists below which hedging stops improving Sharpe.
  • The target-weight mechanism resembles no-regret online learning on the simplex, so dynamic target weights could plausibly be designed with regret guarantees, a direction the paper leaves open.
  • By analogy with loss-versus-rebalancing, a multi-period model could quantify a 'loss-versus-target' that dynamic fees would need to compensate.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. This paper formalizes Perpetual Demand Lending Pools (PDLPs), a class of DeFi mechanisms used by GMX, Jupiter, Hyperliquid, and dYdX. It models the perpetuals exchange funding rate, pool arbitrage, and target weight mechanisms (TWMs), and analyzes share creation and redemption arbitrage. The main theoretical contribution is a set of sufficient conditions under which a delta-hedged PDLP LP position has no worse Sharpe ratio than the unhedged position, offered as an explanation for the observed proliferation of hedged PDLP strategies. The paper also discusses when splitting a single pool into multiple pools improves delta-hedged returns.

Significance. If the results held, the paper would provide a useful first formalization of an important and rapidly growing DeFi primitive, and the target-weight/delta-hedging analysis could inform protocol design. The explicit optimization formulations for funding-rate and share arbitrage are valuable, and the empirical context (fee/TVL data) makes the questions concrete. The paper does not fit parameters to data and many bounds are stated in terms of protocol parameters, which is a strength. However, the main delta-hedging theorem is not proven, and several supporting claims contain proof errors, so the paper's headline conclusion currently rests on unverified assumptions.

major comments (4)
  1. [§4.1, Claim 4.1] Claim 4.1's proof that variance is non-increasing is invalid. The text asserts that negative correlation between p^Tπ and p^T R plus Var[p^Tπ] ≤ 4 Var[p^T R] implies Var[p^Tπ + p^T R] ≤ Var[p^T R]. This is false: the required inequality is Var[p^Tπ] + 2 Cov(p^Tπ, p^T R) ≤ 0, i.e., Cov ≤ -Var[p^Tπ]/2. Negative correlation only gives Cov < 0, and |Cov| ≤ σπ σR provides an upper bound on -2Cov, not the lower bound. For example, with σ_R = 1, σ_π = 2, and Cov = -0.1, both stated hypotheses hold yet Var(p^Tπ + p^T R) = 4.6 > 1. Moreover, the negative-correlation 'fact' is not a consequence of the model: with π = (f/γ)Σ^{-1}ℓ - Δ and Δ = R, Cov(p^Tπ, p^T R) = (f/γ) Cov(p^TΣ^{-1}ℓ, p^T R) - Var(p^T R), which can be positive when the fee-driven term dominates. In fact, p^T(π + R) = (f/γ)p^TΣ^{-1}ℓ when Δ = R, so the hedged portfolio's variance does not depend on the variance of R in the way the proof assumes. The central abstract claim that PDLPs are 'easy to delta hedge' is therefore unsupported.
  2. [§2.1 and §2.3.1] The funding rate sign convention is inconsistent. Equation (2) defines γL = κ(L/S - p/p0), and the text says positive γL means shorts pay longs. But the example in §2.1 says γL = 3κ > 0 means 'longs pay shorts,' and in §2.3.1, after a price increase γL < 0, the text says 'short positions must pay long positions' and has the arbitrageur open a long position to capture the funding rate. Under the stated convention a negative funding rate is paid by longs to shorts, so the arbitrageur should open a short position and would pay, not receive, the fee fℓ used in the LP profit expression. This reversal invalidates the fee upper bound (4) as stated and the associated claim that fees inversely proportional to open interest ensure arbitrageur profitability.
  3. [§3.3, Claim 3.3] The proof of Claim 3.3 does not establish the claimed bound. The derivation yields terms of order G²/µ (e.g., '8G²/µ' before the final inequality), which are not bounded by the claimed (40 + 8/C)∥p∥₂ G³/µ² without additional assumptions. The final inequality also replaces the hypothesis min_i p_i ≥ C with ∥p∥₂ without explanation, and C does not appear in the proof. In addition, Claim 3.1's first bound is stated as 2G/µ under the hypothesis max ∥∇f∥₂² ≤ G, but the proof yields 2√G/µ. These errors leave the approximate-arbitrage results in §3.3–§3.4 without a valid quantitative guarantee.
  4. [§3.5, Claim 3.4] Claim 3.4's proof has an inequality-direction error. Under F ≤ 1, the factor 1/(1+F) is at least 1/2, not at most 1/2, so the first inequality ∇pV_new ≤ (1/2 + f)R - ... is not justified; an upper bound on ∇pV_new would require 1/(1+F) ≤ 1/2, which is false. Also, the assumption ∇pF ≥ 8fR/(p^T R) is an ad hoc sufficient condition that is not derived from the TWM mechanism or from protocol parameters, and the claim that 'TWMs reduce volatility' is therefore not established.
minor comments (6)
  1. [Abstract vs. §5] The abstract says the pools generated 'over $890 million in fees in 2024,' while §5 says 'over $700 million'; these figures should be reconciled.
  2. [§2.1 example] The liquidation price is stated as '$1,5000' instead of '$1,500.'
  3. [§3.3, Claim 3.1] The hypothesis 'Suppose f(δ) = 0' should presumably be f(0) = 0, since the proof uses f(0) = 0.
  4. [§3.3] The text defines δ*_H = min_δ h(δ) for a concave h; this should be a maximization, since the optimum of a concave function is a maximum.
  5. [Appendix B, Claim B.1] Claim B.1 uses σmin(Σ_Y) without defining Y, and the proof's sufficient condition σmax(A) < σmin(Σ/A) does not match the stated conditions; the statement and proof should be aligned.
  6. [§4.1] There are several typos in this section, including 'the delta hedge in negatively correlated' for 'is negatively correlated,' and the proof would benefit from stating explicitly that the delta of the unhedged portfolio is Δ = R.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the paper's sufficient conditions are not fitted inputs or self-referential claims; Claim 4.1's variance step is a correctness gap, not circularity, and Gauntlet self-citations are motivational only.

full rationale

The paper's derivation chain is sufficiency-based rather than circular. Section 2.3 derives fee bounds from the funding-rate and swap models; Section 3 derives TWM arbitrage and delta bounds from stated assumptions on F; Section 4 solves the mean-variance hedge (Eq. 13) and states sufficient conditions (Eq. 14 and Claim 4.1). None of these conditions is defined in terms of the conclusion: Condition 1 is a lower bound on fee revenue, Condition 2 is a bound on the hedge's own variance, and each is a genuine hypothesis rather than a restatement of the target Sharpe-ratio inequality. The variance subproof in Section 4.1 is, however, not valid: negative correlation and Var[p^T pi] <= 4 Var[p^T R] do not imply Var[p^T pi + p^T R] <= Var[p^T R]; the required inequality is Var[p^T pi] <= -2 Cov(p^T pi, p^T R). This is a missing proof / correctness risk, not a circular reduction, because the hypotheses are not the conclusion under another name. The paper's self-citations, notably [Gau24c] (Gauntlet's hedged JLP whitepaper) used to motivate the proliferation claim and as a practical example, are not load-bearing: the formal inequalities in Claim 3.4 and Claim 4.1 do not assume or invoke that whitepaper. The formalization of GMX/Jupiter target-weight mechanisms is descriptive and accompanied by new optimization bounds, so it is not a renaming-only contribution. No equation was found that reduces to its own input by construction; hence the circularity score is minimal, with the small score reflecting only the presence of minor, non-load-bearing self-citations.

Assumptions & free parameters 7 free parameters · 7 assumptions · 0 invented entities

The central claims rest on reasonable idealized descriptions of live protocols, including a linear funding rate, an unmanipulable oracle, and target-weight discount properties. They also rest on several ad hoc sufficient conditions, such as strong concavity, a gradient lower bound, and a variance bound, that are not verified and in some cases are nearly as strong as the conclusions they support.

free parameters (7)
  • κ (funding rate constant)
    Appears in the linear funding rate model (Eq. 2); chosen as a model parameter, not fitted to data.
  • B (price move bound)
    Assumed upper bound on relative price change in Section 2.3.1; used to derive the fee upper bound.
  • f (lending fee)
    Protocol parameter; the fee interval conditions in Section 2.3.3 are stated in terms of f.
  • L0 (initial open interest)
    State variable used in the funding-rate arbitrage bounds; not estimated.
  • γ (risk aversion)
    Parameter of the mean-variance hedging objective in Eq. (12).
  • Σ (covariance matrix)
    Input to the delta hedge; in practice estimated, but the paper does not provide an estimation procedure or data.
  • μ and G (strong concavity and gradient bounds)
    Assumptions in Claims 3.1 to 3.3; not measured for any live discount function.
assumptions (7)
  • domain assumption Linear funding rate model γ = κ(L/S − p/p0)
    Section 2.1, Eq. (2). Central to the fee arbitrage bounds; not all exchanges use a linear funding rate.
  • domain assumption Unmanipulable price oracle updates at fixed times
    Section 2.2 dynamics. Needed for the timing of funding-rate and PDLP arbitrage.
  • domain assumption Discount rate F is concave, F(p,w*,R,0)=0, and maximized at target weight
    Section 3.1 assumptions on the target weight mechanism; used in the creation and redemption arbitrage problems.
  • ad hoc to paper F is μ-strongly concave with gradient norm bounded by G
    Claims 3.1 to 3.3. The paper notes that the actual GMX GLP discount function is not strongly concave, so the claims rely on an approximation.
  • ad hoc to paper ∇pF ≥ 8fR/(pT R) and F ≤ 1
    Claim 3.4. Unverified for live TWMs; it essentially encodes the delta-reduction conclusion rather than deriving it from mechanism primitives.
  • ad hoc to paper Delta hedge is negatively correlated with the unhedged portfolio and satisfies Var[p^Tπ] ≤ 4 Var[p^T R]
    Section 4.1 variance proof and Condition 2 of Claim 4.1. These are asserted rather than derived from the PDLP mechanics.
  • domain assumption Zero transaction costs for the main delta-hedging result
    Section 4.1 states 'we will prove our main result without transaction costs'.

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Cite this review

Pith. "Pith review of Perpetual Demand Lending Pools." pith.science (2026). https://pith.science/paper/C5UXYC65

@misc{pith2026250206028,
  author       = {Pith},
  title        = {Pith review of: Perpetual Demand Lending Pools},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C5UXYC65}},
  note         = {Machine review of arXiv:2502.06028}
}
read the original abstract

Decentralized perpetuals protocols have collectively reached billions of dollars of daily trading volume, yet are still not serious competitors on the basis of trading volume with centralized venues such as Binance. One of the main reasons for this is the high cost of capital for market makers and sophisticated traders in decentralized settings. Recently, numerous decentralized finance protocols have been used to improve borrowing costs for perpetual futures traders. We formalize this class of mechanisms utilized by protocols such as Jupiter, Hyperliquid, and GMX, which we term~\emph{Perpetual Demand Lending Pools} (PDLPs). We then formalize a general target weight mechanism that generalizes what GMX and Jupiter are using in practice. We explicitly describe pool arbitrage and expected payoffs for arbitrageurs and liquidity providers within these mechanisms. Using this framework, we show that under general conditions, PDLPs are easy to delta hedge, partially explaining the proliferation of live hedged PDLP strategies. Our results suggest directions to improve capital efficiency in PDLPs via dynamic parametrization.

Figures

Figures reproduced from arXiv: 2502.06028 by the authors.

Figure 1
Figure 1. Total Value Locked for PDLPs (i.e. assets locked into PDLP pools) in 2024 (Link To Data). Background on existing PDLPs. PDLPs have amassed over $2.5 billion in assets and generated roughly $897.73 million dollars in aggregate fees1 since GMX launched in Septem￾ber 2021. Jupiter’s JLP pool and Hyperliquid’s HLP pool followed, each making changes to the original GMX PDLP mechanism. In the last year, the total value of… view at source ↗
Figure 2
Figure 2. Heatmap of the GMX GLP discount function with two assets for a target weight [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

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Pith tools

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