NEST: Stale-Robust Tokenized-Stock Measurement for a Crypto-Treasury LETF Stack

NEST: Stale-Robust Tokenized-Stock Measurement for a Crypto-Treasury LETF Stack⋆ Doohwi Cha1[0009−0000−2513−6965] , Minsuk Sung2[0009−0007−2282−6873] , Minjae Lee3[0000−0002−8937−9605] , Hyeonjun Yeo4[0009−0005−9368−306X] , Hanhee Kim3[0009−0008−1280−7043] , Seunghan Son3[0009−0000−2902−9999] , Donghwa Seo5[0009−0003−5718−1535] , and Juyeong Lee6[0000−0002−4619−925X] 1

arXiv:2610.21491v1 [q-fin.TR] 15 Oct 2026

Mirae Asset Securities, Seoul, South Korea; doohwicha@gmail.com Korea University, Seoul, South Korea; minsuksung@korea.ac.kr 3 Independent Researcher, Seoul, South Korea; abraxasnz13@gmail.com, crypt0xb0@gmail.com, qzpm5n@g.skku.edu 4 Seoul National University, Seoul, South Korea; hj.yeo30@gmail.com 5 DS Investment & Securities, Seoul, South Korea; properitas95@kaist.ac.kr 6 EY Consulting, Seoul, South Korea; juyeong.lee0839@gmail.com 2

Abstract. Tokenized equities trade while listed markets are closed, so their marks could improve off-hours measurement along layered exposures that link Bitcoin, crypto-treasury stocks, and leveraged ETFs. A token return and the next opening gap, however, share the price move from the previous close to the token mark, so part of their apparent fit is mechanical. We develop NEST, a stale-robust protocol that separates this shared-history fit from conditional measurement value. A shared-history identity and a variance-clock bound show how much fit timing alone can generate. The partial determination index (PDI), the squared partial correlation between token and gap given public controls, measures what remains, and a finite-sample frontier expresses the partial-F decision in PDI units. Product, provenance, timestamp, and execution audits decide whether an informative mark is admissible. For MSTRx, a tracker certificate on MicroStrategy (MSTR), a token-only R2 of 0.916 shrinks to a modest increment once Bitcoin, mark age, and a pre-09:00 MSTR premarket proxy are controlled: the token explains 15.7% of the remaining gap variation (1% frontier 0.031), and 14.0% to 17.0% under stricter clock variants, including one in which every token mark precedes 09:00. A tradelevel reconstruction on Gate gives 25.2%; NVDAX and AAPLX clear their frontiers, whereas TSLAX does not. An admitted mark updates, rather than replaces, BTC-unit exposure diagnostics for the BTC–MSTR–LETF chain. Keywords: Tokenized stocks · Asynchronous markets · Stale prices · Partial R2 · Leveraged ETFs ⋆

Extended version of a paper accepted at MARBLE 2026 (Lecture Notes in Operations Research, Springer Nature); the changes are listed in the Declarations.

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1

Introduction

Tokenized equities extend observable prices beyond listed trading hours. A tracker certificate on a U.S. stock can trade on crypto venues overnight, on weekends, and on exchange holidays, when the listed share does not trade at all. Whether such off-hours marks carry information is a practical question for anyone who monitors exposure while the cash market is shut. We study it for a particularly levered case, the off-hours MSTRx mark in the chain that links Bitcoin, a crypto-treasury stock, and a leveraged ETF, Ct (BTC) −→ Et = mt (qt Ct − Dt ) (MSTR) −→ Lt (β-LETF),

(1)

where Ct is the Bitcoin price, qt the firm’s Bitcoin holdings, Dt its debt, mt the market premium over net Bitcoin value, and Lt a leveraged ETF that resets daily to β times MSTR’s return. The balance sheet and the premium transmit Bitcoin moves to MSTR, and the daily reset magnifies the MSTR response. The question is whether MSTRx adds information beyond Bitcoin and MSTR themselves. High raw fit does not answer that question. Both the token return and the next opening gap start at (or, for the token, near) the previous MSTR close, so they share the previous-close-to-mark price move (Fig. 1). That common prefix raises fit mechanically, even when the token says nothing about the move after its mark. Three estimands are therefore easy to conflate: whether the token reflects information by its mark (interval incorporation), whether it explains the gap beyond public information observable at the cutoff (conditional informativeness), and whether it predicts later shocks (post-mark forecasting). NEST targets the second, after characterizing the first. NEST (Nested Exposure Slope and Thresholds) is a stale-robust admission protocol. A shared-history identity and a variance-clock bound quantify the fit that timing alone can create. The partial determination index (PDI) and its finitesample frontier then test what the mark adds beyond public inputs observable by the cutoff—Bitcoin, the age of the mark, and MSTR’s own premarket price. Product rights, price provenance, timestamps, and execution costs form the admissibility layer that decides whether a statistical increment may be used. An admitted mark then updates BTC-unit exposure diagnostics, including the Bitcoin move that would extinguish a daily-reset LETF in one interval. The empirical answer is a modest, complementary increment. On 222 strictcutoff windows from July 2025 to June 2026, the token-only regression has R2 = 0.916. MSTR’s pre-09:00 premarket proxy absorbs most of that fit, yet MSTRx still explains 15.7% of the gap variation left by Bitcoin, mark age, and the proxy (PDI = 0.157; HAC t = 4.61; 1% frontier 0.031). The increment is 14.0% when every token mark is required to precede the proxy’s 09:00 bar end, 25.2% for a trade-level reconstruction on Gate, and statistically the same on weekdays and weekends. Across other xStocks it is product specific: NVDAX and AAPLX clear their frontiers and TSLAX does not. The paper makes three contributions. 1. Conceptual. It shows that shared timing, not information, can generate most of the raw fit between an off-hours mark and the next opening gap, and it

NEST: Stale-Robust Tokenized-Stock Measurement Yd : move through the token mark (shared by Gd and Td ) 16:00 close, d−1

overnight or weekend

3

Zd : post-mark move

08:59 09:00 token mark bar end

09:25 cutoff

09:30 open, d

mark age Ad (median 26 min) Gd = Yd + Zd : MSTR previous close to open (target) Td = Yd + εd : MSTRx return to its mark Ed = Yd + vd : MSTR premarket proxy (08:00-ET bar close) Bd : Bitcoin return to the cutoff

Fig. 1. Information clock for one closed-market window (schematic, not to scale). The opening gap Gd and the token return Td share the move Yd from the previous close to the token mark, which raises raw fit mechanically (Proposition 1). NEST asks whether Td still explains Gd after conditioning on information observable by the 09:25 cutoff: the Bitcoin return Bd , the mark age Ad , and MSTR’s premarket proxy Ed . In the 218-window premarket sample the composite mark falls between 08:56 and 09:15 ET, and 214 of the 218 marks precede 09:00.

bounds that fit by the share of closed-window variance realized by the mark (Propositions 1–2). 2. Measurement. It recasts the conditional question as a partial R2 with an exact finite-sample frontier and power function, gives the interpretation of a positive PDI against a noisy public benchmark, and implements a same-clock design whose timestamps cannot see the open (Lemmas 1–2, Theorem 1, Corollary 1, Sects. 5–6). 3. Operational. It audits product rights, price provenance, a named venue, and execution cost, and it translates an admitted mark into BTC-unit exposure and one-reset wipeout thresholds for the LETF layer (Sect. 4, Table 7). Section 2 relates NEST to the price-discovery and tokenized-asset literatures. Section 3 develops the measurement model and Section 4 the stack and admissibility diagnostics. Section 5 describes the data and the same-clock design, Section 6 reports the results, and Sections 7–9 discuss implications, limitations, and conclusions.

2

Related Literature

Price discovery across markets. Information shares and component shares allocate innovations in a common efficient price across cointegrated markets [16,14]. Putnin, š [23] shows that these metrics mix information leadership with differences in noise, so a market can rank highly by being less noisy rather than by moving first. NEST does not allocate a common price. It asks whether one intermittent mark adds explanatory power for a later target, conditional on measured public information, and its noisy-control lemma makes the noise channel explicit.

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Asynchronous and sequential trading. Nonsynchronous trading biases betas and return moments [24,18]. When trading hours do not overlap, state-space models recover around-the-clock price discovery from noncontemporaneous quotes [20]; sequential-market measures [25] and information shares for partially overlapping hours [10] adapt the price-discovery toolkit to markets that open one after another; and asynchronous updates can overstate the discovery attributed to one market [15]. The closest antecedent in spirit is the evidence that after-hours and pre-open trading in U.S. stocks is informative despite low volume [3]. Interval-R2 measures how much of a return is incorporated within a subperiod [4]. NEST differs in the object it evaluates: an aggregator-produced reference mark whose return shares a prefix with its target, which makes raw fit uninformative until the shared history is diagnosed. Tokenized stocks. Cong et al. [8] document that weekend token moves anticipate subsequent stock openings, and Dombrowski and Arshadi [11] report weak tracking and arbitrage limits when listed markets are closed. A contemporaneous daily-frequency study of matched xStock–underlying pairs attributes most long-run price discovery to the listed equity [17]. These results leave open the conditional contribution of an off-hours mark once its shared timing and the listed stock’s own premarket price are taken into account; NEST supplies a test of exactly that contribution together with the audits needed to use it. Leveraged ETFs. Daily-reset funds compound path-dependently, and their returns over longer horizons decay with realized variance [5,1]. In the stack (1) the LETF’s underlier is itself a balance-sheet-levered claim on Bitcoin, which is why an offhours measurement of MSTR’s Bitcoin exposure matters for the LETF layer.

3

Measurement Model

Table 1 aligns each variable with the time at which it is observed. Returns are logarithmic unless stated otherwise. 3.1

Mechanical Fit from Shared History

Write the opening gap as Gd = Yd + Zd and the token return as Td = Yd + εd , where Yd is the move through the token mark, Zd the subsequent move to the open, and εd mark-specific measurement noise from stale constituents, aggregation, or microstructure. Proposition 1 (Shared-history identity). Suppose Yd , Zd , εd are mean zero 2 and pairwise uncorrelated, with variances σY2 , σZ , σε2 , and that Var(Gd ) and Var(Td ) are positive. The population regression of Gd on Td has slope, fit, and post-mark covariance bT =

σY2 , 2 σY + σε2

RT2 =

σY4 , 2 2 (σY + σZ )(σY2 + σε2 )

Cov(Zd , Td ) = 0.

(2)

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Table 1. Core variables and their measurement times. Symbol

Meaning and timing

Gd

MSTR opening gap: previous official close to the 09:30 ET open of day d token return from the mark nearest the previous 16:00 ET close to the last mark at or before 09:25 ET Bitcoin return over the same window; age of the token end mark at the 09:25 cutoff MSTR premarket proxy: close of the 08:00-ET hourly bar relative to the previous close move through the token mark; later move to the open; mark-specific noise prespecified non-token controls: (1, Bd , Ad ), or (1, Bd , Ad , Ed ) in the premarket test partial determination index: squared partial correlation of G and T given S effective MSTR slope on Bitcoin; LETF daily target leverage

Td Bd , Ad Ed Yd , Zd , εd S PDIT |S κ, β

2 Proof. Pairwise orthogonality gives Cov(G, T ) = σY2 , Var(G) = σY2 + σZ , and Var(T ) = σY2 +σε2 . Substituting into the population slope bT = Cov(G, T )/ Var(T ) and the squared correlation gives the first two expressions, and Cov(Z, Y + ε) = 0 gives the third. ⊔ ⊓

Raw fit can therefore be high even though the token has zero covariance with everything that happens after its mark: fit is governed by how large the shared move is relative to the post-mark move and to mark noise. Because volatility does not accrue uniformly in calendar time—it is concentrated around news and around the U.S. premarket—the relevant timing screen uses a variance clock. Proposition 2 (Variance-clock bound). Take the closed window as [0, 1]. Let Mt be a square-integrable process with orthogonal increments and variance clock V (t) = Var(Mt − M0 ). For a mark time θ ∈ [0, 1], set G = M1 − M0 and T = Mθ − M0 + ε, where ε is mean zero, has variance σε2 , and is uncorrelated with the increments of M . If V (1) > 0 and V (θ) + σε2 > 0, then R2 (G | T ) =

V (θ)2 V (θ) ≤ , V (1){V (θ) + σε2 } V (1)

Cov(M1 − Mθ , T ) = 0.

(3)

Proof. Set Y = Mθ − M0 and Z = M1 − Mθ . Orthogonal increments give Cov(Y, Z) = 0, Var(Y ) = V (θ), and Var(G) = V (1), so Proposition 1 gives the equality. The bound follows from V (θ)/{V (θ) + σε2 } ≤ 1. ⊔ ⊓ The bound V (θ)/V (1) is the share of closed-window variance realized by the time of the mark. A mark that arrives late in the variance clock can fit the gap almost perfectly without any information about the remainder, so the bound is a timing screen, not evidence of measurement value. Square-integrable martingales, such as efficient log prices, satisfy the orthogonal-increment condition.

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Conditional Measurement: the Partial Determination Index

Let S be an n × p full-rank matrix containing an intercept and the prespecified non-token controls, let MS = I − S(S ′ S)−1 S ′ be the residual maker, and define R02 = R2 (G | S) and R12 = R2 (G | S, T ). Lemma 1 (PDI decomposition). If MS G ̸= 0 and MS T ̸= 0, then PDIT |S ≡ Corr2 (MS G, MS T ) =

R12 − R02 , 1 − R02

(4)

so that ∆R2 ≡ R12 − R02 = (1 − R02 ) PDIT |S . e = MS G and Te = MS T . The restricted residual sum of squares is Proof. Let G ′e e RSS0 = G G. By the Frisch–Waugh–Lovell theorem [12,19], adding T lowers it e ′ Te)2 /(Te′ Te), so (RSS0 − RSS1 )/RSS0 = Corr2 (G, e Te). With an intercept by (G in S, RSS0 = (1 − R02 ) T SS and RSS1 = (1 − R12 ) T SS, which gives (4). ⊔ ⊓ The PDI is the classical coefficient of partial determination. It measures the share of gap variation left unexplained by S that the token captures, and ∆R2 converts it back into an absolute fit gain. Both are conditional on the measured controls; neither identifies which market originated the information. The next result shows how a noisy public benchmark shapes the interpretation. Lemma 2 (Noisy-control interpretation). Let G = Y + Z, T = Y + u, and E = Y + v with centered, pairwise-orthogonal components, and write (a, b, c, d) = (Var Y, Var Z, Var u, Var v) with a, b, c > 0 and d ≥ 0. With h = ad/(a + d), PDIT |E =

h2 . (b + h)(c + h)

(5)

It is zero at d = 0, increases with the control noise d, decreases in b and c, and converges to a2 /{(a + b)(a + c)} as d → ∞. Proof. Projecting on E removes {a/(a + d)}E from both G and T . The residual covariance is a−a2 /(a+d) = h, and the residual variances are a+b−a2 /(a+d) = b+h and c+h, which gives (5). Since ∂ log PDI/∂h = 2/h−1/(b+h)−1/(c+h) > 0 and h increases in d from 0 to a, the comparative statics follow. ⊔ ⊓ A positive PDI against a noisy benchmark therefore means improvement over that benchmark, not exclusive information origin: if the proxy observed the shared move without noise (d = 0), a token with no post-mark information would have PDI = 0. Sampling uncertainty is expressed on the same scale. Theorem 1 (Finite-sample PDI frontier and power). Assume that n > p+1, that [S T ] has rank p+1, and that G = Sγ+T δ+η with η | S, T ∼ N (0, σ 2 In ). d T |S denote the sample version of Lemma 1, let ν = n − p − 1, and define Let PDI the signal share θ = δ 2 T ′ MS T /(δ 2 T ′ MS T + nσ 2 ) ∈ [0, 1). Then FPDI =

 nθ  d T |S ν PDI ′ ∼ F1,ν , d T |S 1−θ 1 − PDI

cα (n, p) =

F1,ν;1−α , ν + F1,ν;1−α

(6)

NEST: Stale-Robust Tokenized-Stock Measurement

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′ where F1,ν (λ) is the noncentral F distribution with noncentrality λ. Under H0 : δ = 0, equivalently θ = 0, it is the central F1,ν , and the level-α partial-F test d T |S > cα (n, p). rejects exactly when PDI

d = (RSS0 − RSS1 )/RSS0 , hence ν PDI/(1 d d = Proof. By Lemma 1, PDI − PDI) e = MS G = Teδ + MS η (RSS0 − RSS1 )/(RSS1 /ν). Conditional on (S, T ), G ′e 2 ′e e e e with T = MS T , so RSS0 − RSS1 = (T G) /(T T ) is σ 2 times a noncentral χ21 variable with noncentrality δ 2 T ′ MS T /σ 2 = nθ/(1 − θ), while RSS1 /σ 2 ∼ χ2ν is independent of it because the two quadratic forms project η onto orthogonal ′ subspaces. Their scaled ratio is therefore F1,ν (nθ/(1 − θ)). Since x 7→ νx/(1 − x) d > cα (n, p). is increasing on [0, 1), FPDI > F1,ν;1−α is equivalent to PDI ⊓ ⊔ Corollary 1 (Minimum detectable PDI). Under the assumptions of Theo′ rem 1, the power of the level-α test, Pr{F1,ν (nθ/(1−θ)) > F1,ν;1−α }, is continuous and strictly increasing in θ, rising from α toward one. For every target power π ∈ (α, 1) there is therefore a unique minimum detectable signal share θπ (n, p, α), and for large ν, θπ ≈ K 2 /(n + K 2 ) with K = z1−α/2 + zπ . Proof. The noncentral F distribution is stochastically increasing in its noncentrality, which is continuous and increasing in θ and diverges as θ → 1; this gives existence signed square root of the numerator √ and uniqueness. As ν → ∞, the √ is N ( λ, 1), so power is approximately Φ( λ − z1−α/2 ); equating it to π gives λ = K 2 and θπ = K 2 /(n + K 2 ). ⊓ ⊔ Theorem 1 restates the classical nested-model test in PDI units and adds its power function; the signal share θ is the population counterpart of the PDI. The frontier lets a reader compare an estimated share directly with the share required for significance, and Corollary 1 turns it into a design tool. At the 1% level and 80% power, 218 windows detect any PDI above 0.052, whereas 48 windows detect only PDIs above 0.21, so a small subsample that fails its frontier is uninformative unless its PDI is compared with θ80 . Because daily return errors are heteroskedastic and possibly autocorrelated, every frontier decision is also paired with a Newey–West HAC t statistic with five lags on the token coefficient [22]. 3.3

The Admission Protocol

NEST admits an off-hours mark as a measurement input in four steps. (i) Clock: every input must be observable by the cutoff, and the shared-history and varianceclock diagnostics are reported next to raw fit. (ii) Conditioning: the prespecified controls S include the strongest cutoff-observable public benchmark, here MSTR’s own premarket price. (iii) Decision: the mark clears when the estimated PDI exceeds the frontier c.01 (n, p) and the HAC check agrees. (iv) Admissibility: product rights, price provenance, a named-venue replication, and execution costs determine how far a cleared mark can be used (Sect. 4). The protocol does not ask the token to be the dominant venue; it asks whether the mark is a reliable, conditional input.

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Stack Exposure and Admissibility

An admitted mark is useful only if it can be mapped into the exposures that the stack (1) creates. Let gk be MSTR’s simple return in reset interval k, β the LETF’s target leverage, and f its fee rate per interval. Lemma 3 (Daily-reset identity). On paths with 1 + βgk > 0 for every k, N

Y LN = (1 + βgk ) e−f N , L0 k=1

X  EN β(β − 1) X 2 LN = β log − gk − f N + O |gk |3 . log L0 E0 2 k

(7)

k

Proof. The product is the definition of a daily reset with a continuously accrued fee. Expanding log(1 + βg) = βg − β 2 g 2 /2 + O(|g|3 ) and β log(1 + g) = βg − βg 2 /2 + O(|g|3 ), and summing over k, gives the second expression [5,1]. ⊔ ⊓ For leveraged or inverse funds (β > 1 or β < 0) the quadratic term is a variance drag. The equity layer adds balance-sheet leverage. With E = m(qC − D) and a locally linear premium response d log m = ρm d log C + e, the effective slope of MSTR on Bitcoin is κ = λE + ρm ,

λE =

qC , qC − D

∂λE qD =− < 0, ∂C (qC − D)2

(8)

for q, D > 0 and qC > D. A Bitcoin drawdown therefore raises book leverage mechanically, and the premium can reinforce or offset that channel. Corollary 2 (One-reset wipeout threshold). If the premium is constant over a reset interval and q, D are fixed, the Bitcoin move that sets a β-LETF to zero in that interval is (∆C/C)wipe = −1/(βλE ). Replacing λE by κ gives the first-order comparison −1/(βκ) when the premium moves with Bitcoin. Proof. With constant m, ∆E/E = q∆C/(qC − D) = λE ∆C/C exactly, and the LETF value reaches zero when 1 + βg = 0. With a moving premium, g ≈ κ ∆C/C to first order. ⊔ ⊓ The threshold is a link-specific exposure diagnostic: it depends on which estimate of κ—book leverage, the listed-session beta, or the slope implied by an admitted off-hours mark—is used to update it (Sect. 6.7). Executable band. Statistical informativeness does not make a token price enforceable. Let Pt be the per-unit token price, Vtref the per-unit reference value of the underlying, τ the settlement delay, r the riskless rate, and k − (τ, x), k + (τ, x) the present-value per-unit all-in costs of the buy–redeem and borrow–sell–create strategies at size x, with k + including borrowing and funding.

NEST: Stale-Robust Tokenized-Stock Measurement

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Proposition 3 (Executable band conditional on access). Suppose an eligible trader can redeem size x, and can borrow and sell x tokens at t while committing to create the same size for delivery at t + τ , with both requests locking Vtref at t. Absence of arbitrage then implies e−rτ Vtref − k − (τ, x) ≤ Pt ≤ e−rτ Vtref + k + (τ, x).

(9)

Proof. Below the lower bound, buying and redeeming earns a positive present value after redemption costs; above the upper bound, borrowing and selling now and covering with the locked creation at t + τ does so after creation, borrowing, and funding costs. Either trade contradicts absence of arbitrage. ⊔ ⊓ Without the stated locking, eligibility, and inventory, (9) is only a friction benchmark. Creation and redemption of MSTRx are restricted by KYC, whitelisting, fees, and minimum sizes (Sect. 5), so the band is not enforceable for most holders. NEST therefore reports these conditions and a named-venue execution cost instead of assuming arbitrage.

5

Data and Same-Clock Design

What the mark represents. MSTRx is a variable-denomination tracker certificate issued by Backed Assets (JE) Ltd.; it is not MSTR common stock and carries no voting rights [2,26]. The issuer reports segregated MSTR collateral. On EVM chains the token rebases and on Solana it uses the Token-2022 Scaled UI extension, so corporate actions are passed through by scaling rather than by price formation [27]. Primary creation and redemption are subject to KYC, whitelisting, fees, and minimum sizes. An informative secondary mark is therefore a measurement of a collateralized tracker outside listed hours, not of unrestricted ownership or of an enforceable primary-market arbitrage. Price sources. The main signal is DefiLlama’s archive of CoinGecko’s multi-venue MSTRx reference price [9]. CoinGecko normalizes venue prices, removes outliers, and applies volume weights, but does not publish the historical constituents and weights behind each mark [7]. Every observed MSTRx confidence value equals 0.99, so the confidence field screens availability, not liquidity. Bitcoin returns use DefiLlama’s archived CoinGecko Bitcoin price at the same targets. The target joins Financial Modeling Prep’s previous official MSTR close to the next 09:30 ET official open. To obtain venue evidence that does not depend on the composite’s undisclosed composition, we reconstruct 694,246 MSTRX–USDT trades from Gate’s twelve official monthly trade files for July 2025–June 2026 and take the last trade at or before each target, at most 15 minutes old; 24 order-book snapshots at the window endpoints provide a cost check [13]. A clock that cannot see the open. Candidate windows join adjacent U.S. trading sessions from July 2, 2025 through June 5, 2026. Token and Bitcoin marks are requested at the previous 16:00 ET close and at a 09:25 ET cutoff, five minutes

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Table 2. Sample construction for MSTR/MSTRx. Availability screens precede the backward cutoff repair; the 222-window row is the primary sample. Screen

Retained

Removed

Role

Adjacent U.S. session windows All four price endpoints observed Confidence at least 0.8 Endpoint lags at most 60 min

234 233 233 232

0 1 0 1

Universe Missing token end mark Availability screen Start mark 421 min old

Token end mark by 09:25 ET Both token marks by their targets

222 219

10 3

Primary sample Stricter clock

before the open. An earlier nearest-mark rule could select prices observed after the cutoff: of the 232 windows that pass the availability screens, 10 used a token end mark after 09:25, and 8 of these a mark at or after the 09:30 open. The primary design instead requires the token end mark to be available by the cutoff, and a stricter sensitivity requires both token marks to precede their targets. Table 2 keeps the economically different exclusions visible. The composite updates roughly hourly, so the end mark is typically the 08:59 print: its age at the cutoff has median 26.0 minutes and standard deviation 1.5 minutes. The two windows removed by the availability screens are calmer than the retained ones (mean absolute gap 1.50% versus 1.87%; mean absolute Bitcoin move 1.08% versus 1.53%), and the Spearman correlations of mark lag with the absolute gap and the absolute Bitcoin move are insignificant (p = 0.655 and 0.642). With two exclusions this diagnostic has little power, so we report it rather than rely on it. The three windows that start after a 13:00 ET early close [21] and the 41 strictly pre-target Friday-to-Monday windows enter as sensitivities. The public benchmark the token must beat. MSTR’s own premarket price is the economically relevant alternative. Historical Financial Modeling Prep data begin at the regular open, so we archive Yahoo Finance’s hourly extended-hours chart [28] and define Ed = log(Pd08 /Pd−1 ), where Pd08 is the close of the bar labelled 08:00 ET. Wherever the next bar is observed (217 of the 218 matched windows) it starts at 09:00, so Pd08 is the last value before 09:00. The proxy is coarse: the provider does not document whether bars are built from trades or quotes, every matched bar reports zero volume, and four of the 222 strict windows lack an 08:00 bar (their mean absolute gap is 2.63% against 1.83%; Mann–Whitney p = 0.99). The 09:00 bar close is excluded because it is mechanically the open (its correlation with the official opening price exceeds 0.9999). This leaves 218 common windows, on which the token and the proxy are added in both orders after Bitcoin and mark age. By Lemma 2, a surviving token increment is improvement over this observed proxy and leaves the origin of information open. Clock audit. The composite end marks in the 218-window sample fall between 08:56 and 09:15 ET. In 214 windows they precede 09:00 by 0.85 to 3.15 minutes (median 0.97), so the token mark and the nominal end of the proxy bar essentially

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coincide; in the remaining four the mark is 0.2 to 15.8 minutes later than 09:00. Section 6.2 reports the test without those four windows. The Bitcoin control is measured to the 09:25 cutoff, about 26 minutes after the token mark; this asymmetry works against the token, because Bitcoin information arriving after the mark is credited to the controls. Bitcoin end marks are at most 41 seconds after 09:25 (112 windows) and always precede the open. Venue and portability designs. The Gate premarket comparison fixes the Gate end price at the last trade by 08:59 ET, aligned with the end of the proxy bar. A same-clock comparison evaluates Gate at the composite’s own timestamps and conditions each source on the other. The cross-name exercise repeats the premarket-conditioned test for NVDAX, TSLAX, and AAPLX with the same strictly backward rule and each underlying’s own 08:00-ET bar, giving 204, 197, and 224 windows. Expanding-window and frozen July 2026 evaluations test whether the broader Bitcoin-and-age increment persists out of sample.

6

Results

6.1

From Raw Fit to Conditional Value

The raw relation shows why the timing diagnostics are needed. On the 222-window strict sample, the token-only regression has R2 = 0.916. Relative to Bitcoin and mark age, MSTRx adds ∆R2 = 0.133 and has PDI = 0.620 (HAC t = 10.92), far above its 1% frontier of 0.030 (Table 3). This comparison, however, omits the listed stock’s own premarket price. The premarket benchmark is the harder test. On the 218 common windows, Bitcoin and age explain 77.9% of the gap variation and adding the premarket proxy raises this to 90.8%. MSTRx then adds ∆R2 = 0.014, so the full model explains 92.3%. The increment is small in absolute terms but clears the frontier decisively: the token explains 15.7% of the variation left unexplained (PDI = 0.157, F = 39.6, HAC t = 4.61, c.01 = 0.031). Figure 2 shows the contrast between the raw relation and the residualized one. Token and proxy are highly correlated (0.970), and the reverse increment of the proxy given the token is PDI = 0.082: the two prices overlap substantially without either spanning the other. MSTRx is a complementary off-hours measurement, not a dominant price-discovery venue. 6.2

Clock Audit of the Premarket Comparison

Table 4 varies the clock of the premarket comparison. Dropping the four windows whose token mark is later than 09:00 lowers the PDI from 0.157 to 0.140 (HAC t = 4.36), still well above the frontier of 0.031; in this sample every token mark precedes the nominal end of the proxy bar by at most about three minutes. Restricting to windows whose Bitcoin end mark is at or before 09:25 halves the sample but leaves the PDI at 0.168. Requiring both token marks to precede their targets, tightening the matching tolerance, winsorizing, or replacing the proxy by

12

D. Cha et al.

Table 3. Conditional incremental fit. R02 is the fit of the conditioning set S, ∆R2 the absolute gain from adding the token, and PDI the share of the remaining variation that the token explains; “Clears” means PDI > c.01 (n, p). The Gate premarket row ends the Gate return at 08:59 ET. HAC t uses five Newey–West lags. Signal; conditioning set S

N

R02

∆R2

PDI

c.01

HAC t

Clears

MSTRx composite BTC, age BTC, age, premarket

222 218

0.785 0.908

0.133 0.014

0.620 0.157

0.030 0.031

10.92 4.61

yes yes

Gate trade, MSTRX–USDT BTC, age BTC, age, premarket (08:59)

225 216

0.778 0.908

0.176 0.023

0.792 0.252

0.030 0.031

26.78 6.85

yes yes

Other xStocks; S: BTC, age, own premarket NVDAX 204 0.891 AAPLX 224 0.749 TSLAX 197 0.896

0.018 0.023 0.001

0.169 0.092 0.010

0.033 0.030 0.034

4.85 5.16 0.80

yes yes no

the open of the 09:00 bar leaves the PDI between 0.150 and 0.170. The reverse increment of the proxy remains positive throughout, consistent with two noisy measurements of the same closed-market move. The increment does not depend on the type of closure. The 170 weekday windows give PDI = 0.122 and the 48 weekend-or-holiday windows 0.112, and in the full premarket model a token-by-weekend interaction is far from significant (HAC t = −0.14, p = 0.89; token loadings 0.52 on weekdays and 0.49 on weekends). The weekend estimate stays below its own frontier of 0.145 only because 48 windows are too few: they detect a PDI with 80% power at the 1% level only above θ80 = 0.21 (Corollary 1), and at the weekday value the test has 46% power. Weekend windows therefore carry the same conditional increment as weekdays, measured with less precision. 6.3

Sensitivity and Closure Length

Table 5 changes the clock and the sample within the broader Bitcoin-and-age specification. Lag limits of 30 to 60 minutes, the exclusion of early-close starts, and winsorization leave the PDI between 0.613 and 0.629. Requiring all four endpoints to precede their targets cuts the sample to 67 windows, because Bitcoin marks often sit seconds after their targets, and still gives 0.445 against a frontier of 0.101. Splitting at the January 26, 2026 vintage of CoinGecko’s aggregation methodology gives 0.634 before and 0.559 after. Without the premarket proxy, the increment is largest across closures: 0.784 for multi-day closures and 0.824 for the 41 Friday-to-Monday windows, when Bitcoin and mark age say least about the listed stock’s weekend move. Section 6.2 shows that this weekend advantage comes from the controls rather than from the token: once Monday’s premarket price is included, the token’s increment is the same on weekends and weekdays.

NEST: Stale-Robust Tokenized-Stock Measurement (a) Raw relation

(b) After BTC, age, premarket proxy

R 2 = 0:913

Opening gap, residualized (pp)

MSTR opening gap (%)

7.5 5.0 2.5 0.0 ¡2.5 ¡5.0 ¡7.5

¡5

13

0

PDI = 0:157 (c: 01 = 0:031) 2 1 0 ¡1 ¡2

5

MSTRx closed-window return (%)

¡2

0

2

MSTRx return, residualized (pp)

Fig. 2. Raw and conditional fit on the 218-window premarket sample. (a) The MSTR opening gap against the MSTRx closed-window return; most of this fit is the shared previous-close-to-mark move. (b) The same two series after residualizing both on an intercept, the Bitcoin return, mark age, and the premarket proxy (Frisch–Waugh–Lovell); the squared correlation of the residuals is the PDI and the line has the full-model token slope 0.57.

Resampling checks address concentration rather than generality. In the primary sample, leave-one-out PDIs range from 0.590 to 0.635, and none of the 221 nontrivial circular shifts of the token series produces a PDI above 0.043. For the 41 Friday-to-Monday windows, circular-block bootstrap 95% intervals run from 0.664 to 0.919 with three-window blocks and from 0.683 to 0.918 with six-window blocks, and leave-one-out PDIs range from 0.793 to 0.859. The circular-shift p-value of 0.024 equals 1/41, the smallest value this test can attain, so a 1% claim is not available from the subsample alone. 6.4

Venue Construction and Execution

Gate’s trade-level mark reproduces the result on a named venue. With Bitcoin and age only it has PDI = 0.792; ending the Gate return at 08:59 ET to align it with the proxy, it clears the premarket-conditioned frontier with PDI = 0.252 (HAC t = 6.85; Table 3). The Gate and composite returns have correlation 0.976 on identical windows, with a median absolute difference of 0.30%. At common timestamps, each source adds only a small increment to the other (PDI = 0.100 for Gate given the composite and 0.057 for the reverse; Table 6), so the two sources largely carry the same information. A 09:25 Gate trade gives PDI = 0.518 against the proxy, but that comparison also credits about 24 minutes of extra freshness and is not a same-clock test. Statistical informativeness is not executable value. Across the 24 endpoint snapshots, the median quoted spread is 19.1 basis points and simulated market orders cost 23.7/20.8 basis points to buy/sell $1,000 and 72.0/49.3 basis points

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D. Cha et al.

Table 4. Clock audit of the premarket-conditioned test. S contains an intercept, the Bitcoin return, and mark age; E is the premarket proxy. PDIE|S,T is the reverse increment of the proxy given the token, and θ80 the smallest PDI detectable at the 1% level with 80% power (Corollary 1). The last two rows split the sample by calendar; a token-by-weekend interaction has HAC t = −0.14. Clock or sample variant

N

PDIT |S,E

c.01

θ80

HAC t

Clears

Baseline (mark by 09:25)

218

0.157

0.031

0.052

4.61

yes

0.082

Token mark by 09:00 BTC end mark by 09:25 Both marks by targets Marks within 30 min Winsorized at 1%/99% Proxy: 09:00-bar open

214 106 215 217 218 217

0.140 0.168 0.170 0.168 0.165 0.150

0.031 0.064 0.031 0.031 0.031 0.031

0.053 0.102 0.052 0.052 0.052 0.052

4.36 4.46 5.25 5.18 5.13 4.45

yes yes yes yes yes yes

0.092 0.025 0.072 0.074 0.085 0.082

Weekdays only Weekend or holiday

170 48

0.122 0.112

0.040 0.145

0.066 0.208

4.49 2.24

yes no

0.081 0.203

PDIE|S,T

for $10,000; in the thinnest snapshot one side of the book held only about $436 within 100 basis points of the midpoint. The mark is therefore an informative measurement input for monitoring, but not evidence that size could be traded at the mark off hours. 6.5

Temporal Validation

Expanding-window forecasts that start with 60, 80, or 100 windows reduce out-of-sample mean squared error by 59.8%, 57.6%, and 61.3% relative to the Bitcoin-and-age model (Clark–West t = 5.46, 4.79, and 4.43 [6]). The token’s information also persists after the estimation window. In July 2026, coefficients frozen on June 5 match 21 of 22 realized gap signs and reduce mean squared error by 77.8% relative to the frozen Bitcoin-and-age model, and the raw MSTRx return alone cuts that model’s root mean squared error by a similar margin (0.0081 against 0.0178; the frozen model attains 0.0084). The holdout thus confirms the token’s out-of-sample value without separating the fitted model from the raw token, as expected when the token loading is close to one (0.91). A frozen Gate model agrees with the composite direction in all 21 windows covered by Gate’s official July archive and matches 20 realized signs. These evaluations concern the Bitcoin-and-age specification; a premarket-conditioned holdout requires a later premarket sample. 6.6

Portability Across Products

Figure 3 places the frontier decisions on one scale. NVDAX (PDI = 0.169) and AAPLX (0.092) clear their frontiers, whereas TSLAX does not (PDI = 0.010 against 0.034; HAC t = 0.80). For these non-crypto names, Bitcoin and age

NEST: Stale-Robust Tokenized-Stock Measurement

15

Table 5. Clock, sample, and closure sensitivity of the Bitcoin-and-age specification. The 480-minute lag row admits one window with a stale start mark and is reported for completeness. Check

N

R02

PDI

c.01

HAC t

Clock and filter Both token marks by targets All four endpoints by targets Endpoint lags at most 30 min Endpoint lags at most 480 min Excluding early-close starts Winsorized at 1%/99%

219 67 221 223 219 222

0.786 0.781 0.785 0.785 0.785 0.785

0.629 0.445 0.629 0.528 0.621 0.613

0.030 0.101 0.030 0.030 0.030 0.030

12.12 5.93 11.71 5.80 10.81 10.72

Reference-price vintage Before Jan. 26, 2026 Jan. 26, 2026 onward

132 90

0.715 0.856

0.634 0.559

0.051 0.075

7.46 9.80

Closure length One-day closures Multi-day closures Spanning a weekend Friday to Monday

173 49 47 41

0.801 0.778 0.777 0.769

0.538 0.784 0.787 0.824

0.039 0.138 0.145 0.166

8.64 10.00 10.03 9.61

explain little of the gap, so the token’s Bitcoin-and-age PDIs are large (0.836, 0.759, and 0.717), but the own-stock premarket proxy absorbs most of that fit. The protocol thus travels across products while allowing its answer to differ: admission is product specific. 6.7

From an Admitted Mark to Stack Exposure

Table 7 turns the admitted mark into the exposure diagnostics of Section 4. Book leverage is 1.19 on reported first-quarter 2026 debt and 1.15 after the announced May 25, 2026 note retirement; the premium beta of 0.103 is imprecise (standard error 0.084). The closed-window slope of MSTRx on Bitcoin in the strict sample is 1.077, close to book leverage and well below the 60-day listed-session beta of 1.474. For a two-times LETF these slopes imply one-reset Bitcoin wipeout moves between −46.4% and −33.9%. The dispersion is the point: the monitored risk depends on which link of the BTC–MSTR–LETF chain the admitted mark is used to update, and the threshold is an exposure diagnostic, not a return forecast.

7

Discussion

The central finding is not the high raw association between MSTRx and the next MSTR open. Once shared timing and the listed stock’s own premarket price are taken into account, most of that association is absorbed by public information, and what remains is smaller but stable: 14.0% to 17.0% of the residual gap variation

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D. Cha et al.

Table 6. Venue construction, execution cost, and temporal validation. Order-book costs are medians across 24 endpoint snapshots for simulated market orders. Expandingwindow and July evaluations use the Bitcoin-and-age specification; MSE changes are relative to the same specification without the token. Check

Finding

Venue construction and execution Same clock: Gate vs. composite PDI 0.100 / 0.057 (Gate / at composite times composite); corr. 0.984 Gate at 08:59; S: BTC, age PDI 0.681 Gate at 09:25; S: BTC, age, PDI 0.518 premarket Gate order book ($1k) buy/sell cost 23.7/20.8 bp Gate order book ($10k)

buy/sell cost 72.0/49.3 bp

Temporal validation of the BTC-and-age specification Expanding window, start MSE −59.8/−57.6/−61.3% N = 60/80/100 July 2026, model frozen on signs 21/22; MSE −77.8% June 5 July 2026, RMSE (log points) July 2026, frozen Gate model

Design and comparison N = 218; S adds the competing mark N = 221 N = 220; trade 23.7 min after the proxy bar 24 snapshots; median spread 19.1 bp min. two-sided depth within 100 bp: $436 Clark–West t = 5.46/4.79/4.43 raw token and BTC-age model: 20/22, 20/22 BTC-age model 0.0178

frozen model 0.0084; raw token 0.0081 agrees with composite 21/21; MSE −88.1%; HAC signs 20/21 t = 1.54

across the clock variants in the upper panel of Table 4. This is consistent with daily-frequency evidence that listed equities lead tokenized trackers [17]. It also sharpens the weekend evidence that token moves anticipate openings [8]. The token’s large weekend advantage in the Bitcoin-and-age specification reflects how little Bitcoin and mark age reveal about a weekend move, and Monday’s premarket price absorbs that advantage. What remains is a conditional increment of the same size on weekends and weekdays (0.112 and 0.122; interaction p = 0.89): the token is a steady complement to the listed stock’s own premarket price, not a weekend-only signal. For risk monitoring, the practical implication is a sequence rather than a verdict. A desk that wants to use a 24/7 token price before the open should first check that the clock cannot see the target, then measure the token against the strongest public benchmark available at the cutoff, and only then translate an admitted mark into exposure. The same protocol applies to other asynchronous tokenized measurements—equity perpetual marks, tokenized funds, or oracle-fed collateral prices—whenever a mark and its target share a return prefix.

NEST: Stale-Robust Tokenized-Stock Measurement clears the 1% frontier

does not clear

17

frontier c: 01 (n; p)

S: BTC return, mark age MSTRx composite (N = 222) Gate trade (N = 225) S: BTC return, mark age, own-stock premarket proxy MSTRx composite (N = 218) MSTRx, mark by 09:00 (N = 214) Gate trade at 08:59 (N = 216) NVDAX (N = 204) AAPLX (N = 224) TSLAX (N = 197) 0.01

0.03

0.1

0.3

1

Partial determination index (log scale)

Fig. 3. Frontier decisions on a common scale. Each dot is an estimated PDI and each bar the 1% frontier c.01 (n, p) of Theorem 1 for that sample; filled dots clear it. The upper group conditions on Bitcoin and mark age, the lower group also on the own-stock premarket proxy.

8

Limitations

The evidence is bounded in five ways. First, the premarket proxy is coarse: an undocumented hourly chart with zero reported volume. A timestamp-matched consolidated quote series is the natural confirmatory benchmark, and by Lemma 2 a sharper proxy would lower the PDI if the current proxy is noisy. Second, the composite’s historical constituents are not disclosed; Gate supplies one named venue, not the whole token market. Third, the main asset is a single cryptotreasury stock over about eleven months, and calendar subsamples are small: the 48 weekend-or-holiday windows detect only PDIs above 0.21 at the 1% level with 80% power. Fourth, the out-of-sample evaluations cover the Bitcoin-and-age specification only; a premarket-conditioned holdout awaits a later premarket sample. Fifth, the executable band relies on primary-market access that most holders lack, so admission describes a monitoring input, not an arbitrage. NEST does not identify where information originates and does not forecast post-mark shocks.

9

Conclusion

Off-hours tokenized-stock marks can look highly informative for the next opening gap simply because both returns share the move up to the mark. NEST separates that shared history from conditional measurement value with a shared-history

18

D. Cha et al.

Table 7. Stack calibration. Each slope estimate is mapped into the one-reset Bitcoin move that sets a two-times LETF to zero, −1/(2κ) (Corollary 2). Balance-sheet inputs use reported Bitcoin holdings and total debt; listed-session estimates use daily data through June 5, 2026. Slope estimate Book leverage, Q1 2026 debt Book leverage, post-retirement debt Book leverage plus premium beta MSTRx closed-window slope (N = 222) 60-day listed-session beta

κ

s.e.

Wipeout

Link

1.191 1.150 1.294 1.077 1.474

– – 0.084 0.043 –

−42.0% −43.5% −38.7% −46.4% −33.9%

balance sheet balance sheet plus premium admitted mark listed session

identity, a variance-clock bound, the partial determination index, and a finitesample frontier, and it attaches product, provenance, timestamp, and execution audits to the statistical decision. Applied to MSTRx, it finds that public information absorbs most of the raw fit while a modest premarket-conditioned increment remains on the composite, on Gate, and for two of three other xStocks. MSTRx is therefore a complementary off-hours measurement input for the BTC–MSTR– LETF chain, and NEST is a reusable protocol for deciding when an asynchronous tokenized mark may enter a risk system. Relation to the conference version. A shorter version of this paper has been accepted for publication in the MARBLE 2026 proceedings (Lecture Notes in Operations Research, Springer Nature). This extended version adds the informationclock, conditional-fit, and frontier figures; a clock audit of the premarket comparison, including the tests without token marks later than 09:00 and with Bitcoin marks at or before the cutoff, and a calendar split with an interaction test (Table 4); the power function of the PDI test and the minimum detectable PDI (Theorem 1, Corollary 1); the full sample-construction, sensitivity, and validation tables; the stack calibration and wipeout thresholds based on the strict-sample token slope (Table 7); proofs of all results; and an expanded literature review. It also corrects the description of the Bitcoin data source and one table entry of the conference version. Data and code availability. Derived results and the analysis code are available from the authors on reasonable request. Vendor data must be obtained from the providers and cannot be redistributed. Disclaimer. The views expressed are those of the authors and not of their affiliated institutions. This paper is not investment advice. Competing interests. The authors have no competing interests to declare. Use of AI-assisted tools. An AI agent assisted with drafting and editing the extended text, generating tables and figures from archived results, and consistency checks. The authors reviewed all output and take full responsibility for the manuscript.

NEST: Stale-Robust Tokenized-Stock Measurement

19

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