Designing funding rates for perpetual futures in cryptocurrency markets

Hyungbin Park; Jaehyun Kim

REVIEW 3 major objections 4 minor 23 references

Designing funding rates for perpetual futures in cryptocurrency markets

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Funding rates can be designed so a perpetual future's price exactly tracks a chosen target, with a unique replicating portfolio.

desk verdict A genuinely useful funding-rate design that pins perpetual futures to arbitrary targets, but the path-dependent half of the main theorem is stated under an impossible inequality and needs correction before it can be used. read the letter →

arxiv 2506.08573 v1 pith:2M2XG6WP submitted 2025-06-10 q-fin.MF q-fin.PR

Jaehyun Kim , Hyungbin Park This is my paper

classification q-fin.MFq-fin.PR MSC 91G2060H1060H3091G80

keywords perpetualfuturesfundingratedesigninfinite-horizonbackwardstochasticdifferentialequationspath-dependentratesreplicatingportfolioscryptocurrencyderivativesarbitragepricing

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 futures dominate cryptocurrency derivatives, but their funding-rate mechanism has had little formal theory. This paper shows that an issuer who wants a perpetual future to trade at a chosen target value $\varphi(s,X_s)$ — for example an index, a power of an asset, or an FX rate — can design a funding rate that makes that target the unique no-arbitrage price. The design adds a strong penalty term proportional to the gap between target and futures price, together with a generator term that cancels the target's drift, and the paper proves via infinite-horizon backward stochastic differential equations that the resulting replicating portfolio exists and is unique. It also treats the practical version in which the funding rate is averaged over the past eight hours, showing it has a unique price and converges to the instantaneous design with an explicit $O(\sqrt{\delta})$ error. The upshot is a concrete recipe for exchanges to pin perpetual prices to desired benchmarks and hedge them.

What carries the argument

The load-bearing object is the infinite-horizon backward stochastic differential equation $$Y(s)=Y(T)-\int_s^T\big(r(u,X_u)Y(u)-\Phi(u,X_u,Y(u))\big)\,du-\int_s^T Z(u)\,dB(u),$$ whose unknown pair $(Y,Z)$ is the price and hedge of a funding portfolio. The funding-rate design (4.1) is built so that the functional Itô formula makes $Y(s)=\varphi(s,X_s)$ an exact solution; the term $H(\varphi,y)$ is a penalty that vanishes at the target and, for $\ell$ large enough, supplies the strong monotonicity that forces uniqueness. In the path-dependent case the delay $\frac{1}{\delta}\int_{u-\delta}^{u}Y(v)\,dv$ turns the BSDE into an infinite-horizon delayed BSDE, solved by a Picard contraction using Burkholder–Davis–Gundy estimates and a generalized Gronwall lemma.

What would settle it

Take the paper's own Black–Scholes example ($r=0.02$, $\sigma=0.3$, target $\varphi(x)=x$, $\ell$ between 1.26227 and 15.75125, $\delta=1/1095$) and simulate the delayed funding-rate BSDE. If the realized pathwise deviation exceeds the claimed bound $\|Y^\delta-X\|_T \le (3.68432\|X\|_T+0.84216)\sqrt{\delta}$, or if the $Z^\delta$ bound in (4.7) fails, then the explicit convergence estimate is wrong; absent a numerical value for the BDG constant $M_{\rho\vee2}$, the paper cannot certify that a chosen $\ell$ actually lies in the uniqueness regime.

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Extended reading notes

Core claim

The paper's central claim is that a funding rate of the form $$\Phi(s,\gamma,y)=H(\varphi(s,\gamma),y)-\partial_s\varphi(s,\gamma)-\tfrac12\operatorname{tr}\big(\$\sigma$\$\sigma$^\top\partial_{xx}\varphi\big)(s,\gamma)-r(s,\gamma)\partial_x\varphi(s,\gamma)\gamma(s)+r(s,\gamma)y$$ anchors the perpetual future price to the target $\varphi(s,X_s)$. Here $H$ is any gap-penalty function satisfying $H(y_1,y_2)=0$ when $y_1=y_2$, Lipschitz in $y_2$, and strongly decreasing in $y_2$ with strength $\ell$; the linear case $H(y_1,y_2)=\ell(y_1-y_2)$ gives the constant-proportion funding rate used by exchanges. Theorem 4.1 proves that when $\ell$ satisfies condition (4.2), the risk-neutral pricing BSDE has exactly one solution, and that solution is $Y(s)=\varphi(s,X_s)$, $Z(s)=(\partial_x\varphi\,\sigma)(s,X_s)$; the second component is the hedge ratio. Theorem 4.3 proves that replacing $\Phi$ by its 8-hour average $\Phi^\delta$ leaves the price unique and sends the price and hedge to the instantaneous ones with an explicit $O(\sqrt{\delta})$ error.

Load-bearing premise

The entire construction hinges on the issuer being able to choose the gap-penalty strength $\ell$ large enough, but the paper's threshold for 'large enough' is an infimum involving a Burkholder–Davis–Gundy constant whose numerical value is never given, and the path-dependent theorem's printed small-$\delta$ condition has to be replaced by the appendix's corrected linear-in-$\delta$ version to make sense.

Editorial extensions

If this is right

Issuers can choose any sufficiently smooth target functional $\varphi$ — an index, a power of an asset, an FX rate, or a CFMM value — and the funding rate in (4.1) makes that target the unique no-arbitrage perpetual price, with a replicating portfolio obtained by holding $\partial_x\varphi(s,X_s)$ units of the underlying assets.
The same funding rate ensures a unique price among all portfolios whose value grows at most polynomially in the underlying assets, and increasing the penalty strength $\ell$ widens the class of admissible portfolios for which uniqueness holds.
An 8-hour averaged funding rate, the practical form used by exchanges, produces its own unique price and hedge; as the averaging window $\delta$ shrinks, both converge to the instantaneous design with an explicit $O(\sqrt{\delta})$ error bound.
Because the funding-rate formula can be expressed through quadratic variations and the money-market account only, it can be implemented without estimating drift or volatility functionals.
The framework covers targets that are not tradable, such as a geometric mean CFMM value, so issuers can hedge exposures to baskets rather than single assets.

Reading between the lines

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

A practical next step would be to compute the Burkholder–Davis–Gundy constant $M_2$ numerically; the threshold (4.2) depends on it, but no value is supplied, so a calibrated value would turn the existence condition into a checkable number for exchanges.
The same design logic extends naturally to capped funding rates: choosing $H$ as a piecewise-linear function with different slopes above and below the target keeps the uniqueness argument intact while bounding the funding payment, since Assumption 4.1 already allows such $H$.
The viscosity-solution result for path-dependent PDEs (Theorem 3.3) suggests that the funding-rate design might work for non-smooth targets such as payoffs with kinks, if the uniqueness argument can be pushed through the viscosity framework rather than requiring $C^{1,2}_p$ smoothness.
The explicit constants in the $O(\sqrt{\delta})$ bound grow with moments of the underlying process, so the practical tightness of an 8-hour window depends on volatility and horizon; the paper does not quantify that dependence in real-market parameter ranges.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The thing to know: this paper has a real, usable idea. By adding a strongly monotone penalty term H(φ(s,X_s), y) to the funding rate, the authors make the target φ the unique solution of the pricing BSDE, and they back it with an honest existence-uniqueness proof. That is new for non-tradable targets, and the design is genuinely constructive — the funding rate is explicit, no parameters are fitted to data, and the replicating portfolio comes out of the same construction. The path-dependent extension, an 8-hour averaged funding rate modeled by a delayed BSDE, is also new territory, and the O(√δ) convergence estimate is a reasonable thing to want. The paper gives credit where due, and the citation pattern looks fair; Angeris et al. (2023) is properly positioned as the model-free baseline that this uniqueness mechanism improves on.

Now the soft spots, in proportion. The stress-test note is right and it lands on the load-bearing part of the paper. Assumption 4.2(ii) is printed as e^{ρ((ℓ−r)^2 + ½|ℓ−r|ℓ + 2|ℓ−r|)δ} < 1, which cannot hold for any positive δ because the exponent is strictly positive. Appendix D.1 silently works with L6((ℓ−r)^2 + ... )δ < 1, so the theorem statement and the proof are not even using the same hypothesis. That makes Theorem 4.3(ii) vacuous as stated, and the corrected condition is not accessible because L6 is never quantified. Separately, condition (4.2) in Theorem 4.1 contains the BDG constant M_{ρ∨2} with no numerical upper bound in the paper, so a practitioner cannot actually verify the required lower bound on ℓ. These are not cosmetics: the uniqueness and convergence arguments depend on those exact inequalities. The instantaneous-spot construction with a chosen ℓ is mathematically sound — I checked the main line of the proof and it holds. The path-dependent claims are plausible but rest on a misprint plus an unquantified constant.

The paper is for two audiences: crypto-derivative researchers who want a rigorous pricing framework for funding rates, and exchange engineers who want a principled design recipe. Both can get value once the assumptions are fixed. This deserves referee time — the core idea is solid enough that a serious referee should engage with it, but the path-dependent section needs a careful correction before the paper can be trusted as a design guide.

My recommendation: send it out, but the report should require the authors to fix Assumption 4.2(ii), make the corrected L6-version part of the theorem statement, and either quantify the BDG constants or state the ℓ-condition in a checkable form. Conditional accept is the right posture.

Referee Report

3 major / 4 minor

Summary. The paper studies the design of funding rates for cryptocurrency perpetual futures. It introduces a funding portfolio whose wealth process is governed by an infinite-horizon BSDE, proves existence and uniqueness of solutions under a monotonicity condition, and proposes an explicit funding rate functional (4.1) that makes a prescribed target process φ(s,X_s) the unique perpetual-future price. It then studies a path-dependent variant Φ^δ obtained by averaging Φ over a window δ (the common 8-hour window), proves existence and uniqueness of the associated delayed BSDE, and establishes an O(√δ) closeness result between the prices generated by Φ and Φ^δ. Applications to power-index futures, foreign-exchange futures, and geometric-mean CFMMs are developed.

Significance. If the results are stated correctly, the paper makes a useful contribution: it gives a rigorous, arbitrage-based construction of funding rates that anchor perpetual-future prices to possibly non-tradable targets, with replicating portfolios and a quantitative comparison between instantaneous and averaged funding. The core verification that (4.1) makes Y=φ, Z=(∂_xφ σ) solve the risk-neutral pricing BSDE is clean and the uniqueness argument via strong monotonicity is standard. The path-dependent extension is potentially the most practical contribution, since the 8-hour averaging window is the exchange convention. However, as printed, the path-dependent theorems rest on a hypothesis that cannot be satisfied, and the checkable/numerical content of the design conditions is incomplete.

major comments (3)

[Assumption 4.2(ii) and Theorem 4.3] Assumption 4.2(ii) states the two inequalities 1/3 (e^{|6(ℓ−r)^2−2ℓ+2|δ}−1)/(|6(ℓ−r)^2−2ℓ+2|δ)<1 and e^{ρ((ℓ−r)^2 + ½|ℓ−r|ℓ + 2|ℓ−r|)δ}<1. For any ℓ,r>0 and δ>0, the exponent in the second inequality is strictly positive, so e^{positive·δ}<1 is false. Thus the hypotheses of Theorem 4.3 are empty: no admissible (ℓ,δ) exists. Appendix D.1 silently replaces this with L6((ℓ−r)^2+½|ℓ−r|ℓ+2|ℓ−r|)δ<1, where L6 is the constant from Proposition A.1, but L6 is not quantified and the corrected condition is not part of the theorem statement. This is a load-bearing error, not a typo: the uniqueness, existence, and O(√δ) bounds in Theorem 4.3 all rely on this small-window condition.
[Theorem 4.1 and Assumption 3.4(v), condition (4.2)] The sufficient condition on the penalty strength ℓ is ℓ > inf_{K>0}(K + ½(C_r/√(2K) + M_{ρ∨2}C_3)^2)ρ, where M_{ρ∨2} is the Burkholder–Davis–Gundy constant from (3.7). No numerical value or explicit upper bound for M_q is given anywhere in the paper, so the condition cannot be checked by an issuer who wants to select ℓ. The same issue affects Corollary 4.2, which uses M_{p+2}. Since the paper advertises ℓ as a design parameter and gives a numerical threshold in Section 5.1, the authors should either provide a computable bound for M_q (for example via Doob's maximal inequality for q>1 and a standard estimate for q=1) or replace the condition by a more explicit, checkable sufficient condition.
[Section 5.1 numerical example] The Black–Scholes example states that condition (4.2) is satisfied for ℓ>0.26227 and that Assumption 4.2 holds for 1.26227<ℓ<15.75125, but no derivation of these numbers is provided. Given that Assumption 4.2(ii) as printed is unsatisfiable, the claimed admissible interval for (ℓ,δ) is unsupported. Moreover, the value 0.26227 depends on the unquantified constant M_{ρ∨2} from (4.2), so the numerical claim is not reproducible from the paper. This example is the main practical illustration of the path-dependent result and should be corrected together with Assumption 4.2.

minor comments (4)

[Appendix D.1, proof of Theorem D.1] The contraction estimate contains the expression e^{Mδ}−1/(3Mδ), but the absolute value signs on M are missing in several displayed formulas; the surrounding text uses |M|, so the notation should be made consistent.
[Section 4.2, Eq. (4.4)] The notation Yδ_u and Yδ(u) is used interchangeably in the delayed BSDE (4.5); the authors should fix one convention, since the distinction between the stopped path and the point value is otherwise important in this section.
[Theorem 4.3] The sentence 'The upper bounds L1, L2, L3, L4, L5 specified in the theorem can be explicitly calculated' is not followed by the explicit expressions in the main text; the reader is referred to Appendix D.2, but the appendix defines them through auxiliary constants a(ρ), b(ρ) that are themselves lengthy. A concise explicit statement in the main text would improve usability.
[Throughout] There are several grammatical slips, e.g., 'we have the followings' in Theorem 4.3 and 'holds for all T ∈ (0,∞), δ∈(0,1) and s∈[0,T]' where the variable s is not present in the displayed inequality. These should be corrected in a polish pass.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: funding-rate design is an explicit construction verified by BSDE uniqueness; the only self-citation is non-load-bearing.

full rationale

The paper's central construction, Theorem 4.1, defines Φ in (4.1) by inserting H(φ,y) − ∂sφ − ½tr(σσᵀ∂xxφ) − r∂xφ·γ + r y. The proof then verifies that (φ, (∂xφ)σ) satisfies the BSDE because H(φ,φ)=0 by Assumption 4.1(i), and invokes the independent uniqueness result of Theorem 3.2 under (4.2) to conclude these are the unique solution. This is a design-and-verify theorem, not a fitted parameter called a prediction: φ is an input chosen by the issuer, not estimated from data, and ℓ and δ are design constants. Theorem 4.3 similarly derives an explicit O(√δ) bound for the averaged-rate price; its constants are explicit functions of model parameters and C5, not calibrated outputs. No step in the argument reduces to its own conclusion. The only self-citation is Alexander et al. (2020), which includes co-author Park, cited in the literature review as empirical background; it plays no role in Theorems 3.2, 4.1, or 4.3. Two limitations are worth flagging but they are not circularity: Assumption 4.2(ii) is printed as e^{ρ((ℓ−r)²+½|ℓ−r|ℓ+2|ℓ−r|)δ}<1 with a positive exponent, so as stated Theorem 4.3(ii) has no admissible (ℓ,δ); and condition (4.2) involves the unquantified BDG constant M_{ρ∨2}, so the asserted uniqueness regime is not numerically checkable. These are consistency and computability defects in the hypotheses, not instances of the derivation being equivalent to its inputs.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The paper's contribution is a design rule for funding rates, not an empirical law. The only free parameters are the design choices ℓ and δ; all other inputs are standard market primitives or regularity assumptions. No new physical or financial entities are introduced.

free parameters (2)

ℓ (penalty strength) = ℓ > 0.26227 in the Black-Scholes example; otherwise unspecified
Introduced in Assumption 4.1(iii) and required to satisfy (4.2) to guarantee uniqueness of the perpetual future price. It is a design parameter chosen by the issuer, not fitted to market data.
δ (averaging window) = 1/1095 (8 hours) in applications
The path-dependent funding rate averages the instantaneous rate over the last δ years. The value 1/1095 corresponds to the 8-hour window used by exchanges; δ is a practical parameter, not fitted to data.

assumptions (6)

domain assumption Lipschitz continuity of drift and volatility (Assumption 3.1)
Ensures existence and uniqueness of the underlying wealth process X via standard SDE theory.
domain assumption Bounded short rate r (Assumption 3.2)
Used to control moments of X and to derive the growth estimates (A.1).
domain assumption Invertible volatility and Girsanov martingale condition (Assumption 3.3)
Needed to pass to the risk-neutral measure and write the pricing BSDE (3.6).
domain assumption Strong monotonicity and polynomial growth of the BSDE driver (Assumption 3.4)
Core conditions for the existence-uniqueness theorem; the monotonicity constant must exceed a BDG-dependent bound (4.2), which is not numerically specified.
ad hoc to paper Design conditions on H, including H(y,y)=0 and Lipschitz monotonicity (Assumption 4.1)
The penalty term introduced by the authors to enforce uniqueness of the futures price; a modeling choice, not an observed fact.
ad hoc to paper Small-window conditions on ℓ and δ (Assumption 4.2)
Technical assumptions for the delayed BSDE contraction and the √δ convergence. As printed, (4.2)(ii) is impossible because e^{positive}δ<1 cannot hold; the appendix uses a corrected form.

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Cite this review

Pith. "Pith review of Designing funding rates for perpetual futures in cryptocurrency markets." pith.science (2026). https://pith.science/paper/2M2XG6WP

@misc{pith2026250608573,
  author       = {Pith},
  title        = {Pith review of: Designing funding rates for perpetual futures in cryptocurrency markets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2M2XG6WP}},
  note         = {Machine review of arXiv:2506.08573}
}

read the original abstract

In cryptocurrency markets, a key challenge for perpetual future issuers is maintaining alignment between the perpetual future price and target value. This study addresses this challenge by exploring the relationship between funding rates and perpetual future prices. Our results demonstrate that by appropriately designing funding rates, the perpetual future price can remain aligned with the target value. We develop replicating portfolios for perpetual futures, offering issuers an effective method to hedge their positions. Additionally, we provide path-dependent funding rates as a practical alternative and investigate the difference between the original and path-dependent funding rates. To achieve these results, our study employs path-dependent infinite-horizon BSDEs in conjunction with arbitrage pricing theory. Our main results are obtained by establishing the existence and uniqueness of solutions to these BSDEs and analyzing the large-time behavior of these solutions.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Ackerer, D., Hugonnier, J., and Jermann, U. (2024). Perpetual futures pricing. Technical report, National Bureau of Economic Research

work page 2024

[2] [2]

Alexander, C., Choi, J., Park, H., and Sohn, S. (2020). BitMEX bitcoin derivatives: P rice discovery, informational efficiency, and hedging effectiveness. Journal of Futures Markets , 40(1):23--43

work page 2020

[3] [3]

and Chitra, T

Angeris, G. and Chitra, T. (2020). Improved price oracles: C onstant function market makers. In Proceedings of the 2nd ACM Conference on Advances in Financial Technologies , pages 80--91

work page 2020

[4] [4]

Angeris, G., Chitra, T., Evans, A., and Lorig, M. (2023). A primer on perpetuals. SIAM Journal on Financial Mathematics , 14(1):SC17--SC30

work page 2023

[5] [5]

Bally, V., Caramellino, L., Cont, R., Utzet, F., and Vives, J. (2016). Stochastic integration by parts and functional I t \^o calculus . Springer

work page 2016

[6] [6]

and Rogers, T

Butler, G. and Rogers, T. (1971). A generalization of a lemma of B ihari and applications to pointwise estimates for integral equations. J. Math. Anal. Appl , 33(1):77--81

work page 1971

[7] [7]

Christin, N., Routledge, B., Soska, K., and Zetlin-Jones, A. (2022). The crypto carry trade. Preprint at http://gerbil. life/papers/CarryTrade. v1 , 2

work page 2022

[8] [8]

Confortola, F., Cosso, A., and Fuhrman, M. (2019). Backward SDE s and infinite horizon stochastic optimal control. ESAIM: Control, Optimisation and Calculus of Variations , 25:31

work page 2019