Fix a positive VIX tenor and a coercive continuous-path class. We ask when an all-strike SPX call surface and a continuously rolling VIX call surface are generated by one risk-neutral filtered law. Constructive joint calibration is formulated where consecutive windows abut, and there window-by-window solutions suffice; listed weekly windows instead share twenty-three of their thirty days. This paper proposes a coherence theory for the structure that gluing ignores. The state carries log spot, the quadratic variation of its stochastic logarithm, and one nonnegative additive conditional budget field
so every overlapping window is a projection of one tower, and the tower yields an explicit overlap inequality between consecutive VIX smiles that complete vanilla call surfaces alone can violate. Certification is then a dichotomy. On a finite grid the quote image is compact and convex, the alternative returns either a calibrating law or a strict pathwise separator, and every calibrating finite law has an exact positive continuous conditional-Bass realization preserving its skeleton and all window budgets. Inside one fixed compact class, a calibrated continuum law exists if and only if every determining finite quote restriction is feasible within the structural class, so failure is witnessed by finitely many quotes and structural rows together with the class tail bound; every such calibrated law is already a stochastic-clock model, and absolute continuity of the clock is exactly the additional condition for a calendar-time Brownian representation. The analysis is filtration-sensitive throughout: what a model reveals, and when, is part of its identity, which yields a payoff-typed early-revelation diagnostic and closedness of the feasible surface set in the enhanced topology. All statements are relative to one declared class, and examples show that this cannot be relaxed.Contents
The traded VIX complex is a rolling market. With weekly VIX expiries, listed contracts reference
thirty-day forward-variance windows whose start dates are only days apart, so the quoted windows overlap
heavily. In the continuous positive-martingale idealization, a VIX fixing at time with tenor is the
conditional future log-variance budget
where is the quadratic variation, or bracket, of the stochastic
logarithm . Overlapping windows therefore price increments of one bracket, and their quotes are linked
across fixing dates by the tower property of conditional expectation, not only by static convex
order.
For SPX alone the static side is classical: complete call prices identify the one-time
marginals, and convex order characterizes whether some martingale carries them (Breeden and
Litzenberger, 1978; Strassen, 1965; Kellerer, 1972). VIX admits no such static reduction: validity of
every SPX and VIX marginal separately does not settle joint generability, because the VIX readout is
conditional and filtration-sensitive.
Constructive joint calibration is formulated where the windows tile. Guyon (2024) closes the
one-window problem exactly, with entropic selection, and continuous-time, dual, and computational
counterparts follow (Guo et al., 2022; Guyon, 2025; Zitridis, 2025); the multi-maturity extensions place
consecutive maturities one tenor apart, so that the windows abut (Bourgey and Guyon, 2024; Dong
et al., 2025; Zhang, 2025). In that exact tiled regime, provided adjacent blocks use the same
prescribed intermediate SPX marginal, local and global feasibility coincide, and a global law exists
whenever every local block does (Acharya et al., 2026). The listed weekly complex is not in
that regime: consecutive thirty-day windows share twenty-three days, and an extension to
overlapping spans requires a model of the variance allocated to the overlap together with martingale
rows at every adjacent event time (Acharya et al., 2026). That object is the subject of this
paper.
The object every overlapping window projects from is one budget field
nonnegative, additive in , and
a martingale in its valuation time. The question of this paper is whether a complete SPX call surface and
the entire rolling fixed-tenor VIX call surface are generated by one filtered risk-neutral law carrying this
field, and, when they are not, how failure is witnessed. This paper proposes a coherence theory for the
structure that gluing ignores.
The hypotheses are sharp, and Section 4 shows it. Example 4.1 gives a fully specified four-date panel
of SPX smiles, VIX smiles at three rolling fixings, and three pairwise dispersion claims: deleting any one
pairwise claim leaves a market exactly calibratable by a positive continuous martingale, yet
no law matches the whole panel, and the failure carries an explicit finite witness with gap
one half. Neither the individual smiles nor any leave-one-cross-claim-out test detects it. Its
windows abut and its cross-window claims are not vanilla options, so the theory below is stated
both for declared panels that may carry such joint claims (Section 2) and for vanilla surfaces
(Section 3); Example 4.2 exhibits the same failure with complete vanilla smiles alone on genuinely
overlapping windows, through an explicit overlap inequality (Proposition 3.5) that the mean
rows and every one-window test miss. Ordinary marginal-hypergraph combinatorics does not
supply the failure: rolling windows are intervals, so the hypergraph they form is acyclic and the
classical marginal problem (Vorob’ev, 1962) would grant extension rather than obstruct it
(Remark 4.3).
1.1Main results
Three themes run through the results. Certification is a dichotomy: every alternative below returns either
a calibrating law or a strict separator, and both are explicit objects rather than existence statements;
“certificate” is used in both senses, but every realization statement below concerns only a feasible law
certificate. The analysis is filtration-sensitive throughout: the budget field is a prediction coordinate, and
what a model reveals, and when, is part of its identity (Section 2.3, Theorem 3.11, Remarks 3.12
and 4.3). Everything is relative to one declared coercive class, and the examples show that this cannot be
relaxed.
The first result is finite-dimensional but architecture-free. For a finite grid and a finite continuous
raw-payoff vector, conditional martingality, diagonal log-contract identities, and every budget
tower are affine rows (Lemma 2.1). Under a common -moment envelope their feasible quote
image is compact and convex. Membership is equivalent to all support inequalities, and failure
admits a strict pathwise structural separator (Theorem 2.2). A conditional Bass construction
then turns every feasible discrete law—including boundary and atomic ones—into a positive
continuous model without changing the grid skeleton or any declared VIX budget (Lemma 2.8 and
Theorem 2.9). For a finite payoff panel, exact recursive cubature can first replace the law
by a finite nonrecombining skeleton while preserving the quote vector and coercive moment
(Lemma A.3).
The second and headline result concerns the complete surface. We augment path space by the bracket
and the full jointly continuous budget field and fix one lower-semicontinuous coercive envelope. Within
that single compact class, the following are equivalent (Theorem 3.6):
(i)
one law calibrates all SPX and rolling fixed-tenor VIX calls;
(ii)
every finite subset of one countable determining quote panel is feasible in the same class;
(iii)
no finite strict support-functional witness exists; and
(iv)
one positive continuous stochastic-clock model generates both surfaces and its own VIX budgets.
The finiteness is on the quote side: each restriction in (ii) is still tested against the full structural class, so
(i)(ii) is a compact finite-intersection statement rather than a finite-dimensional market test; the
genuinely finite object is the witness of failure in (iii).
The third result locates the model boundary (Theorem 3.9). If the clock has density , the model can be
written on an extension, and absolute continuity of the clock is exactly what decides this: a singular clock
gives an exact continuous model but no calendar-time volatility density. This separates weak realization
from smooth, elliptic, or low-dimensional diffusion realization; Examples 4.5 and 4.6 inhabit the two
sides.
Further results support these. The tower has a market-facing consequence: an explicit overlap
inequality between consecutive VIX smiles and the SPX log strip over their non-overlapping part
(Proposition 3.5), void without overlap, not implied by the mean rows, evaluable from the two complete
VIX strips and the intervening complete SPX log-strip increment, and violated by a vanilla-only all-strike
panel on overlapping windows (Example 4.2). The finite alternative has a band form covering bid–ask and confidence panels, in which a witness is a separator that clears the quoted spread (Proposition 2.5). The
feasible surface set is closed in the enhanced topology carrying spot, quadratic variation, budget, and
filtration information (Theorem 3.11). The no-early-revelation step of the realization is isolated as a
payoff-typed diagnostic: the revelation defect vanishes row by row exactly under conditional
mean-independence (Proposition 2.11), spot martingality can survive a premature mark while the budget
row fails (Example 2.12), and panel-safety is the exact checkable requirement for preserving conditional
values (Definition 2.13).
The proofs combine classical tools—compact separation, conditional randomization, Bass
interpolation, continuous-martingale time change—with the one common rolling budget, full-history
testing, finite obstruction, and the exact feasible-law-to-model boundary.
1.2Relation to adjacent results
The problem descends from model-free SPX/VIX bounds (De Marco and Henry-Labordère, 2015; Guyon
et al., 2017; Papanicolaou, 2018) and from Guyon (2024), who proves the exact one-window
feasibility-and-arbitrage alternative with entropic selection. Its continuous-time, dual, and computational
successors (Guo et al., 2022; Guyon, 2025; Zitridis, 2025) are formulated for one window, with further
maturities noted as possible at the cost of one state per maturity; multi-maturity constructions iterate or
chain consecutive maturities, the gluing argument for which places those maturities exactly one tenor
apart, so that the windows abut (Bourgey and Guyon, 2024; Dong et al., 2025; Zhang, 2025); in that
exact tiled setting with prescribed common seam marginals, local and global nonemptiness coincide,
while the exact overlapping-window reference construction is left open (Acharya et al., 2026);
term-structure no-arbitrage between VIX futures and equity derivatives is expressed by Badran and
Goldys (2015) as drift restrictions on model coefficients; and term-structure consistency inside a
fixed architecture is studied by Papanicolaou (2022) and Forde and Smith (2023). Closest in
spirit, Zaugg and Grzelak (2026) build a parametric VIX-first model on the rolling-window
definition, with single-window consistency enforced by a backward Feynman–Kac equation;
positivity of the recovered variance and general existence are left open, and no feasibility
characterization is given. None of the works reviewed here proves a nonparametric, complete-surface,
common-class alternative for the rolling pair with overlapping windows, which is what is established
below.
The key methods are those of martingale-transport duality (Beiglböck et al., 2013), the Bass
embedding (Bass, 1983), adapted topologies for filtration-sensitive quantities (Backhoff-Veraguas
et al., 2020), and Tchakaloff cubature (Bayer and Teichmann, 2006). The comparison is feature-specific
and is not a global priority claim.
1.3Scope
All prices are discounted and deterministic carry is absorbed in the forward numeraire. VIX means the
idealized conditional log contract in (1); exchange discretization, jumps, rates, dividends, and settlement
conventions are outside the theorem. We fix one and do not take . Existence is always relative to one
stated compact class. A separator is a model-class certificate; it is an executable arbitrage only when every
structural and coercive term is traded at the assumed cost. No uniqueness, physical law, statistical identification, or hedging conclusion is claimed. The mathematical theorems are not empirical claims: they
neither assert that historical SPX/VIX data satisfy the maintained conditions nor select a forecasting
law.
Throughout, denotes a candidate risk-neutral law and denotes the bracket clock. This convention is
maintained in both the finite and continuous formulations below.
The paper follows the three results. Section 2 treats the finite grid: the structural rows (Lemma 2.1),
the quote-space alternative and its band form (Theorem 2.2, Proposition 2.5), the conditional-Bass
realization of every structural finite law (Theorem 2.9), and the early-revelation diagnostic that the
realization isolates (Section 2.3). Section 3 passes to complete surfaces: the certificate class and its
identification (Definition 3.2, Lemma 3.3), the overlap inequality between consecutive VIX smiles
(Proposition 3.5), the finite-intersection alternative (Theorem 3.6), the clock boundary (Theorem 3.9),
and closedness in the enhanced topology (Theorem 3.11). Section 4 sharpens each hypothesis by example,
Section 5 records what the theorems change, what is checkable from market input, and the limitations,
and Section 6 concludes. Appendices A and B supply the finite cubature reduction and the details of the
Bass realization.
2The finite common-budget market
On a finite grid every structural row is affine and the alternative is finite-dimensional; this section proves
it, realizes each structural law as a continuous model, and isolates the information step that the
realization depends on. Section 3 lifts the same architecture to complete surfaces.
Fix a grid and write . At date store
where is the date- conditional budget rate for
cell . The root is deterministic. Let and on the corresponding Polish path space . Define
Let be the linear span of these residuals multiplied by bounded continuous functions of at their
conditioning date.
Put
For and , define
Only nonnegativity of the marks is imposed; no strictly positive lower bound on budgets is needed
anywhere below. Nonnegativity is not a feasibility test: Remark 2.3 shows that the structural class can be
empty with nonnegative root budgets.
Lemma 2.1 (Structural rows are the common budget) .For , membership in is equivalent to
Consequently is a positive discrete martingale and, for every ,
Thus is the model’s own conditional
log-contract volatility for every grid-aligned window, simultaneously.
Proof.The conditional equations imply annihilation after conditioning. Conversely, fix one residual
at date and define the finite signed measure on the Polish history space. Its finiteness follows
from and . Annihilation against every bounded continuous history function says that integrates
every such function to zero. These functions determine finite signed Borel measures, hence and .
Apply this to (3)–(5).
Iterating the tower row gives . Condition the diagonal identity at date back to date , multiply
by , sum, and telescope the log returns. This proves (8).∎
2.1Finite-panel feasibility and strict separation
Let be continuous on and satisfy . It may contain raw SPX calls, VIX futures and calls, or continuous
joint grid payoffs. Define
with the convention ; when the supremum is a maximum, by the compactness
proved next.
Theorem 2.2 (Finite quote-space alternative) .Exactly one of the following two cases holds.
(a)
. Then and are compact and convex, and
If , there are , a finite structural gain , , and such that
Conversely, (10) excludes every calibrating law in .
(b)
. Then , and for every the pathwise form (10) holds with arbitrarily negative, already at , where the certificate reads with : emptiness is certified by finitely many structural rows together with the moment-envelope constraint, without using quote rows.
Proof.Case (a) first. The sublevel sets of are compact. Markov’s inequality gives tightness of ,
and lower semicontinuity of makes it closed, hence compact. Moreover,
Therefore expectations
of continuous functions dominated by are continuous on . The structural rows define a closed
affine subset, and the quote map is continuous and affine. This proves compactness and convexity.
Formula (9) is finite-dimensional separation.
For completeness, let . Sion’s minimax theorem (Sion, 1958) on the compact convex set and the linear
space gives ∎
where the right-hand maximum exists because is compact and nonempty. Indeed the inner
infimum in the reverse order equals on and off it, by scaling a violated row. For each fixed , the
concave value function
has a nonnegative supergradient at the interior point . Applying the
supergradient inequality to Dirac masses yields
Writing the pointwise supremum as the least cash
gives the pathwise dual. Strict separation of from supplies a positive gap; choosing a dual
tuple within that gap proves (10). Integration under a calibrating law proves the converse. For
case (b), if , the inner infimum of the reverse order is at every , so (11) holds in and the
minimax value is : for every some has , and the one-moment dual for that supplies with and
.
Remark 2.3 (A checkable nonemptiness criterion) . The standing condition does not by itself
make nonempty. With , , , , , and one has , yet the spot and diagonal rows force and for every
structural law, hence and . A sufficient condition is explicit: the reference law with independent
lognormal cell returns and frozen marks satisfies every structural row, so whenever , a finite
lognormal-moment computation from the root budgets. In the example, ; an explicit emptiness
certificate is , , , and none with exists, since satisfies the rows.
Remark 2.4 (Structural witness versus traded arbitrage) .The portion of is an ordinary
discounted SPX trading gain. The diagonal and tower portions are Lagrange multipliers for the
model’s log-budget identities. They are traded gains only if the corresponding forward log contracts
and budget claims are available. Likewise is a tail certificate unless a dominating payoff can be
bought for at most . The theorem always supplies a strict mathematical witness relative to the
class; literal execution needs these extra instruments.
Proposition 2.5 (Band and noisy panels) .Let componentwise and . No nonemptiness is assumed.
Exactly one of the following holds:
(i)
some has ; or
(ii)
there is with
In case (ii) the witness upgrades to the pathwise form (10) with .
Proof.If then (i) fails and (ii) holds with , with the pathwise upgrade of the empty case of
Theorem 2.2. Otherwise and are compact and convex, so they are disjoint if and only if some linear
functional strictly separates them, which is (ii). The pathwise upgrade is Theorem 2.2 applied to
inside the gap .∎
Remark 2.6 (Transaction costs and statistical bands) . is the revenue of assembling at the
quoted spread: positions sold at the bid and bought at the ask. A band witness is therefore a
separator that clears the spread. Statistical confidence bands give the same statement conditionally
on coverage: whenever the bands jointly cover the true prices, band infeasibility certifies that no
law in the class matches them; constructing such bands from data is outside this paper. The
reproducibility package implements exactly this band form, in elastic-programming shape, on the
four-date panel of Example 4.1 and on listed SPX/VIX quotes.
Remark 2.7 (Root budgets as declared inputs) . The root state , including every root mark , is a
declared input, matching the continuum situation where complete SPX smiles force the root strip
through (31); a finite panel need not determine it. A panel without declared root budgets is handled
in the same framework by dropping the deterministic-root restriction on the marks: the structural
rows remain affine, remains compact and convex, and Theorem 2.2 and Proposition 2.5 apply
verbatim, with the same extended-real convention if the relaxed class is empty. Nonemptiness of
the structural class with a random root implies nonemptiness with some deterministic root: testing
the rows against a countable separating class and conditioning on yields, outside one null set,
deterministic-root conditional laws satisfying every structural row, while enforces the moment cap
on a set of root values of positive measure. Quoted prices, by contrast, are matched only on average
across roots, so calibrated feasibility is genuinely weaker with a random root than with a declared
one and is decided directly in the relaxed class, which is the correct primitive when root budgets
are unquoted.
2.2Every finite structural law has a continuous realization
The finite alternative does not yet say that a structural law is a continuous model. We now supply that
implication. The construction uses a conditional version of the Bass embedding (Bass, 1983); unlike an
endpoint interpolation performed after calibration, it also preserves the successor budget marks and does
not reveal their randomizer at the left edge.
Let be standard Borel and let be a probability kernel on satisfying
Let be its generalized quantile,
the standard-normal CDF, and
Lemma 2.8 (Conditional Bass edge and log budget) .For a Brownian motion , conditional on the
process
is a continuous strictly positive true martingale from to terminal law . It is jointly measurable in and, on ,
Moreover,
Proof.Measurability of the parameterized quantile follows from
The generalized-inverse identity
gives the terminal law. Conditional on , the terminal Brownian value is , so . The moment closes
the martingale and gives a continuous version up to . A nonnegative martingale with a strictly positive terminal value cannot hit zero, proving strict positivity. Gaussian convolution and Hölder’s
inequality justify differentiation under the integral on compact subsets of , and Itô’s formula gives
the displayed SDE.
Write , localize , and apply Itô’s formula: ∎
The stopped logarithms are uniformly integrable.
Indeed , while conditional Jensen gives . Letting and using monotone convergence proves (15).
This is a complete proof of the Bass fact needed below; no optimality property of Bass martingales
is used.
Theorem 2.9 (Exact finite common-budget realization) .Let . There exists a filtered model with a
continuous positive true martingale and a continuous finite-dimensional Gaussian memory factor such
that:
(i)
its full grid skeleton has law exactly ;
(ii)
conditional on the continuous filtration at , the successor state has the same kernel as under conditional on ;
(iii)
on each open grid cell,
for a Borel time-inhomogeneous coefficient; and
(iv)
every grid-aligned window is exact:
Consequently every finite feasible raw quote panel has an exact continuous history-state realization. If the
panel contains finitely many raw payoffs, the finite-cubature reduction in Lemma A.3may first be applied,
yielding a realization with a finite nonrecombining skeleton. No relative-interior, density, or uniform
ellipticity condition is required.
Proof.Disintegrate into Borel successor kernels . Parameterized disintegration separates each
kernel into its successor-spot marginal and a conditional kernel on the fiber having that spot. The
standard randomization lemma for probability kernels on standard Borel spaces realizes the fiber
kernel by a Borel map of one independent uniform variable.
For each grid cell take two fresh Brownian coordinates. Normalize their increments over the
cell to processes running from to a standard normal endpoint. Interpolate the successor-spot
marginal with Lemma 2.8 using . At use to draw the remaining successor mark conditionally
on the realized spot. Freeze both coordinates after the cell. Recursing over produces a continuous
Gaussian memory factor of dimension at most and the exact skeleton law .
The active endpoint randomizers are independent of the past at ; therefore
This is the
no-early-revelation step. The Bass conditional-expectation identity inside each cell and the conditional
mean row across boundaries make a global true martingale. On cell , with , Itô’s formula gives
which is
(16) after reading from the frozen memory.
By (15), the conditional integrated variance on cell is ∎
Condition back to , use the common tower for ,
and sum over . Lemma 2.1 then yields (17). All grid payoffs retain their prices because the complete
skeleton law is unchanged.
Remark 2.10 (Exact level of realization) .The coefficient may switch at grid boundaries and
degenerate near an atomic endpoint. Time plus the Gaussian memory is a finite-dimensional Markov
state, but no small factor count or smooth uniformly elliptic coefficient follows. The conclusion is
exact continuous weak realization, not a classical regular diffusion theorem.
2.3Early revelation: a payoff-typed diagnostic
The construction above consumes its randomizers at the left edge of each cell, and (18) records the
consequence. Because a premature mark can preserve some declared rows while destroying
others, we state the failure mode in the form in which an implementation can be tested against
it.
Fix a grid date and a mark on the same space, and enlarge the history to . For an integrable payoff
define the revelation defect
Proposition 2.11 (Revelation defect) .Let be integrable. Then
(i)
;
(ii)
if in addition , then
(iii)
almost surely if and only if , that is, conditional mean-independence of and given .
Proof.Since , the tower property gives , which is (i). Given (i), is conditionally centred, so its
conditional second moment is the stated conditional variance, and the unconditional identity follows
by taking expectations; the hypothesis is used only here, and is not implied by the standing
envelope of Section 2. Part (iii) is the definition of the conditional expectations in (19) being
equal.∎
Part (iii) is stated for one payoff, and that is exactly the point: mean independence is a property of a
row, not of the mark.
Example 2.12 (Spot survives, the budget does not) . At a fixing date let and let a fair coin
select between the two mean-one terminal laws
Both have mean one, so : revealing leaves the
martingale row intact. Their log budgets are and . With the coin hidden the squared VIX readout
is the average ; with the coin revealed it is or according to the branch, a ratio of . Conditional
mean-independence for spot therefore does not protect the diagonal row, and a model that still
calibrates every SPX quote can carry the wrong VIX.
Definition 2.13 (Panel-safe revelation) . Let be the declared quote vector and let be the
structural residuals conditioned at date . The mark is panel-safe at if for every .
Panel-safety is sufficient for the enlarged model to satisfy every declared structural row at
date and to calibrate every unconditional quote, and it is necessary and sufficient for it to
preserve every date-conditional value in . It is not necessary for unconditional calibration alone:
two branch laws with the same spot mean and the same log mean preserve the spot row, the
diagonal row, and every unconditional price while changing a conditional call value, and an
enlargement never alters an unconditional expectation. The conditional target is the relevant one
here, because a VIX fixing is itself a conditional value. Panel-safety is strictly weaker than
requiring for all bounded , which is conditional independence of from the future given — the
immersion or -hypothesis of the enlargement-of-filtrations literature — and strictly stronger than
mean-independence for the spot row alone, by Example 2.12. As a diagnostic it is finite and checkable: for
each declared row separately, compare the two conditional expectations in (19), using (20) to
report a defect magnitude when the row is square-integrable. Testing the spot row alone is not
sufficient.
Remark 2.14 (What the construction supplies) . Theorem 2.9 does not merely
achieve panel-safety. Its cell randomizers are independent of , so (18) preserves the entire successor
kernel, hence every row simultaneously and every undeclared payoff as well. The diagnostic above
is therefore addressed to constructions that introduce auxiliary marks by other means, not to the
one built here. The same sensitivity to what is revealed when reappears at the continuum level:
Theorem 3.11 is stated in the enhanced topology for exactly this reason, and Remark 3.12 gives
the continuous-time counterpart of Example 2.12.
3The complete rolling market
We now pass to complete surfaces. The state is enlarged so that the quadratic variation and the entire
budget field are path coordinates, making filtration information part of the path itself; the finite
alternative then lifts through compactness of one declared class, and every structural continuum law is
already a stochastic-clock model.
Fix a horizon and one tenor . SPX calls are declared on and VIX calls on . The quoted surfaces are
assumed continuous in maturity and strike, and every strike section is a valid call transform of an
integrable nonnegative law. Equality on a dense set of strikes determines the corresponding
marginal law because call transforms are -Lipschitz and their right derivatives recover survival
functions.
Put
The canonical space is the closed subset of
with coordinates satisfying a deterministic root ,
and, pathwise,
The root field is also continuous, nonnegative, and additive. Let and . The raw filtration records and
only budget valuations observed no later than the current time.
Assumption 3.1 (One common coercive class) . Choose . There is a lower-semicontinuous with compact
sublevel sets such that
Moreover is the increasing supremum of bounded continuous cylinder functions
on one countable dense coordinate set. Fix and set
Every finite restriction in the continuum theorem
uses this same .
The cylinder presentation enters only through monotone convergence: along the defining supremum,
so the moment cap is checkable on finite-dimensional marginals. Concrete sufficient choices append
coercive moduli of continuity for to the left side of (23). The theorem does not infer such moduli from
option prices.
For , let be the bounded continuous functions of finitely many canonical coordinates observed by
.
Definition 3.2 (Common-budget continuum certificate) . A law is structural if, for every admissible
time tuple and conditioning multiplier , it satisfies the following identities, with all expectations taken
under :
Write for the set of structural laws. It is calibrated if, in addition,
for every declared .
Lemma 3.3 (Identification of bracket and budget) .For every , in the usual augmented canonical
filtration:
(i)
is a continuous square-integrable martingale and ;
(ii)
is a positive continuous true martingale; and
(iii)
for ,
In particular (29) is the model’s own rolling VIX.
Proof.It is enough to test (24)–(27) at a countable dense set of times against a countable separating
algebra of history cylinders. A signed-measure and monotone-class argument, as in Lemma 2.1,
upgrades those rows to conditional identities. Path continuity and the envelope then extend them
to all real times. Martingality transfers to the usual augmentation by the downward martingale
theorem along , path continuity, and the same domination; for the diagonal row, whose window
start moves with the conditioning date, the transfer also uses for .
Rows (24) and (25) say that and are true martingales. Since is continuous increasing from
zero, uniqueness in the quadratic-variation characterization gives . From the definition of , , so Itô’s
formula gives . Localize this positive local martingale. The envelope controls in , so dominated
convergence removes the localization and is true.
Corollary 3.4 (Fast necessary market identities) .Complete SPX smiles force
and
The right side of
(31) must be finite, continuous, nonnegative, and additive. These conditions are necessary, not sufficient:
they do not impose a common conditional tower.
Proof.Take expectations in (30) at time zero, then use the SPX and VIX marginal laws.∎
Proposition 3.5 (Overlap inequality) .Let and let , with , be two fixings whose windows overlap. Then,
almost surely,
and consequently, for every , with ,
If, in addition, is calibrated, every term in (34) is
determined by complete all-strike call strips: , and is the SPX log-strip increment over the
non-overlapping part . This is an exact all-strike identity; evaluation from a finite grid of exchange-listed
strikes additionally requires declared interpolation and tail bounds. At the inequality follows
from the two mean identities of Corollary 3.4, with slack , the expected budget of the trailing
cell dropped in the proof; for it is not implied by the mean rows (Example 4.2); and for the windows share no budget coordinate and the argument yields nothing beyond the mean
rows.
Proof.By additivity, . By Lemma 3.3, additivity, and (21), . Subtracting gives (33). For , ∎
and the
right side is nonnegative, so it dominates . Take expectations and use from Lemma 3.3. The strip
formula is Carr–Madan replication of the convex payoff , with , whose second-derivative measure
is on plus the atom carried by the jump of .
3.1Finite determination and pathwise obstruction
Choose a countable dense set containing and closed under admissible -shifts, and rational nonnegative
strikes. Continuity of and uniform integrability from Assumption 3.1 imply that structural rows and
quote rows on this countable panel determine all real times and strikes. Denote its quote indices by and
the corresponding payoffs and prices by .
Enumerate a countable determining collection of homogeneous structural residuals from (24)–(27) as ,
and let be their finite linear span. The envelope gives, for every and every finite quote combination ,
with ; the quadratic row is where binds.
Here a stochastic-clock model means a positive continuous true martingale whose stochastic logarithm
has continuous integrable bracket , with the forward budget given by conditional bracket increments. The
sharper calendar-time distinction is made in Section 3.2.
Theorem 3.6 (Complete-market common-budget alternative) .If , the following are equivalent:
(a)
there is a calibrated continuum certificate;
(b)
for every finite , there is satisfying for all ;
(c)
there is no finite set and such that
(d)
there is a positive continuous stochastic-clock model in the class whose conditional future bracket has the jointly continuous version and whose SPX/VIX call surfaces are .
In this nonempty-class case, if these equivalent conditions fail, the finite witness can be strengthened to a
pathwise infeasibility certificate: for some such finite , some , , and ,
If instead , there are , , and such that
Thus structural infeasibility is already witnessed by finitely
many structural rows together with the class bound, without using quote rows.
Proof.We first record compactness. Markov’s inequality and compact sublevels of make tight;
Portmanteau makes it closed. Hence it is compact. Every structural or quote payoff is continuous
and is dominated by for some . The bound
gives uniform integrability, so its expectation is weakly
continuous. Therefore is compact and convex.
Assume first . For each , let
These sets are closed in the compact space . Condition (b) is
their finite-intersection property, so it is equivalent to a nonempty total intersection. A law in
the intersection satisfies every dense quote row. Path continuity, surface continuity, the -Lipschitz
property in strike, and the same uniform-integrability estimate extend calibration to every time
and strike. This proves (a)(b).
If the total intersection is empty, compactness produces a finite empty subintersection. The
finite quote image of is compact and convex, so strict finite-dimensional separation gives (36).
Conversely, a calibrated law contradicts that inequality. Thus (a)(c).
For the pathwise strengthening, put . Let
These compact sets decrease to , so their support values at
decrease to : maximizers accumulate, along a weakly convergent subsequence, in , since each is weakly
continuous and the intersection identity is the determination argument of Lemma 3.3. Finite-dimensional
Lagrange separation gives
For each finite structural combination, the concave moment-value argument
used in Theorem 2.2 gives the exact one-moment dual with . Hence
The outer infimum need not
be attained. The strict separation gap allows a feasible tuple within half the gap, proving
(37)–(38).
Finally, Lemma 3.3 says that the calibrated law in (a) is itself the clock model in (d). Conversely,
project an admissible clock model onto . Its stochastic logarithm and squared-minus-bracket are
martingales; the conditional budget has the tower and diagonal log identity; conditioning from any larger
original filtration down to the canonical filtration preserves these rows. The jointly continuous version of
the conditional bracket satisfies the pathwise nonnegativity and additivity constraints almost surely, per
countable tuple and then by continuity. Membership in and calibration then give the calibrated law in
(a). This proves (a)(d).
It remains to consider . The closed sets ∎
have empty total intersection. Compactness of therefore gives
an for which . The compact convex residual image
does not contain zero. Strict separation yields and
such that, with , for every . Hence . Applying the exact one-moment dual with to and
choosing a tuple within half this gap gives and such that on and . This proves the empty-class
branch.
Remark 3.7 (Why the common class cannot vary) .Condition (b) is not the assertion that each
finite panel can be fitted by some increasingly irregular model. Every already lies in the same
compact . If one starts instead from genuinely finite arrays, a valid projective route must impose
uniform integrability, full-history rows, and common tight moduli for spot, bracket, and the
three-parameter budget field. Example 4.4 shows that omitting this uniformity invalidates the
implication.
The finiteness in (b) and (c) is on the quote side only. Each ranges over , which is cut out by
the complete countable family of structural rows together with the class, so (a)(b) is a compact
finite-intersection theorem and not a finite-dimensional market test. What is genuinely finite is the
witness in (c) and its pathwise form: finitely many quotes, finitely many structural rows, and the
class tail bound. Finite-dimensional tests are the grid programs of Theorem 2.2 and Section 5.2.
3.2The clock/calendar-time boundary
Definition 3.8 (Clock and calendar-time realizations) .A clock realization is a positive continuous
true martingale whose stochastic logarithm has continuous integrable quadratic variation . Its
budget is . It has a calendar-time Brownian representation if for a nonnegative predictable process
. This is the weakest sense in which the model is a local–stochastic volatility model: may be
any predictable functional of the stopped history, and no Markov or finite-dimensional structure is
implied.
Theorem 3.9 (Exact threshold for a calendar-time Brownian representation) .Every structural
continuum law is an exact clock realization. It has a weak calendar-time representation
on a Brownian
extension if and only if almost surely. In that case has a predictable Borel version as a functional of time
and the complete stopped canonical history. Smooth coefficients, uniform ellipticity, and a fixed finite
factor dimension require additional assumptions.
Proof.The clock statement is Lemma 3.3. Suppose . Enlarge the space by an independent Brownian
motion and set
The summands are orthogonal continuous local martingales and ; Lévy’s
characterization makes Brownian. The integral of over has zero bracket, hence vanishes, giving
(40). Conversely, that SDE implies , so absolute continuity is necessary.
The completed raw canonical filtration is generated, modulo null sets, by the stopped histories
of . Concretely, is raw-predictable and is a version of the density by Lebesgue differentiation;
standard-Borel factorization makes it a Borel nonanticipative functional of the stopped history. This
history is a Markov state in the tautological transition-kernel sense, but can be infinite-dimensional.
Nothing in the argument supplies smoothness, nondegeneracy, or finite-dimensional sufficiency.∎
Remark 3.10 (Identification inside the subclass) .A complementary gauge freedom holds inside
the absolutely continuous subclass: in deterministic one-factor operational-time models the surface
determines only the projected clock–variance product, not its factorization (Angstmann and
Gebbie, 2026). The boundary above is orthogonal: it characterizes when a calendar-time density
exists at all, and claims no identification.
3.3Closedness and information-sensitive stability
Let be the product of the two continuous call-surface spaces, with locally uniform convergence in time
and strike, and write
Theorem 3.11 (Closed feasible surface set) .The map is continuous. Its image is compact and
convex. Hence locally uniform limits of feasible surface pairs are feasible, calibration fibers are
compact with a closed graph, and in implies weak convergence of the complete readout paths . This
conclusion concerns the enhanced topology; ordinary weak convergence, and even equality in law,
of the unaugmented paths does not by itself determine conditional VIX when the filtrations differ.
Proof.On each compact -sublevel the canonical coordinate family is bounded and equicontinuous.
Moving-time SPX payoffs and ∎
therefore converge uniformly there. The sublinear envelope and
uniform integrability remove the complement uniformly. This proves joint continuity of expectation
in , and compactness of the parameter sets gives locally uniform surface convergence. A continuous
affine image of compact convex is compact and convex; closed graph and compact fibers follow.
Finally, the map
is continuous in the uniform topology, using . The continuous-mapping theorem
proves the path statement.
Remark 3.12 (Why the filtration coordinate cannot be dropped) . Let a fair coin choose
between two deterministic variance profiles that agree through but have different integrals over ,
and set
Give this same pair of paths either a filtration that reveals at or the natural filtration
of , which cannot yet distinguish the profiles. The full unaugmented path law is identical in the
two models, but at the conditional increment is branch-specific in the first model and the mixture
average in the second. Their conditional VIX laws therefore differ.
Theorem 3.11 avoids this loss by fixing the raw canonical filtration and carrying
, the relevant prediction coordinate, in the state, in the spirit of adapted topologies
(Backhoff-Veraguas et al., 2020). The moment, uniform-integrability, and modulus controls
encoded by Assumption 3.1 then make that enhanced state compact enough for closure. This is a qualitative closedness statement; quantitative marked-Bass, budget, and fixed-tenor VIX error
rates are not asserted here.
4Analytic examples and sharp boundaries
Each example isolates one hypothesis or boundary of the theory. Example 4.1 shows that pairwise
consistency does not glue and exhibits the finite witness; Example 4.2 shows the same failure with
complete vanilla smiles alone on genuinely overlapping windows, through the overlap inequality of
Proposition 3.5; Remark 4.3 records why this is not a marginal-problem phenomenon; Example 4.4
shows that the common-class qualifier in Theorem 3.6(b) cannot be dropped; Examples 4.5 and 4.6
inhabit the two sides of the clock boundary; Example 4.7 computes a separator and shows it need not
be a traded arbitrage; and Example 4.8 separates weak realization from positive-variance
mixtures.
Example 4.1 (A four-date panel: pairwise-consistent fixings, no common law) . Take the
grid with unit cells, tenor , spot frozen on , and write for the fixings . The panel consists
of:
(i)
VIX calls at all strikes , , so each is Bernoulli on ;
(ii)
three pairwise dispersion claims with payoff and price zero, : since by (21), each claim forces almost surely; and
(iii)
SPX calls at dates generated by , , and , where is the unit-mean lognormal law with log-variance .
Deleting any one claim in (ii) leaves an exactly calibratable market. Reveal a regime on through the budget
coordinates by a continuous -valued martingale absorbed at , independent of the spot driver; set , , or
according to whether the deleted pair is , , or ; and on each cell run a unit lognormal edge if its budget is
one, freezing the spot otherwise. Each model is a positive continuous martingale generating its own
budgets and matching (i), (iii), and both retained claims. All three budgets are then -measurable, which
no quote or structural row forbids.
No law matches the whole panel: by (i) the events have probability each, and by (ii) they are
pairwise disjoint, so their probabilities would sum to . The witness is a finite portfolio. For put , a call
spread at strikes and ; then pathwise, since on , and the quoted curve prices it at . On the whole state space,
where the second inequality is multilinear on , hence holds by checking its vertices, and extends off
the cube because whenever . The panel prices the left side at and the right side at : six calls, three pair
claims, and cash realize the witness of Theorem 2.2 with , , , and a strict gap of . The obstruction is
joint: every smile is matched and each pairwise test passes in some model, yet the three pairwise tests
cannot pass together. The three windows here are the abutting cells , , and , and the contradiction uses
the nonvanilla pairwise dispersion claims in (ii). Accordingly, the example does not establish
failure of joint calibration for a panel consisting only of listed vanilla weekly SPX and VIX
options.
Example 4.2 (Overlapping vanilla windows with no common law) . Take the grid with unit cells
and tenor , so that the window of the fixing at and the window of the fixing at share the cell .
Quote only complete vanilla smiles: SPX smiles and , and VIX smiles and .
Each sub-panel with one VIX smile deleted is calibratable. Without the fixing at : reveal
through the budget coordinates on as in Example 4.1, freeze the spot to date , run a lognormal
edge of log-variance on , and freeze again. Without the fixing at : freeze the spot to date , reveal the
mixing variable of the cell after the date- fixing—for some fixed , continuously through the budget
coordinates on by the absorbed martingale of Example 4.1 started at , with bracket (variance)
rate meanwhile and on , so that and —and freeze on ; an instantaneous draw at would be
-measurable under the usual augmentation and would break continuity of in valuation time. Both
models satisfy every row and every retained smile. For the full panel the mean rows of Corollary 3.4
hold with , and each window is feasible as a one-window problem.
No law matches the full panel. The frozen spot forces , and (33) at , reads , which fails on ;
equivalently (34) at reads . The obstruction is conditional: the spot is frozen on and, since and
is a martingale, on as well, so the tower gives almost surely—a convex-order violation between
the two squared-VIX laws, invisible to the mean rows and to each window separately. The frozen
spot is a normalization: with both VIX smiles held fixed, (34) at shows that any SPX smiles with
produce the same contradiction, and there the positivity of carries the argument.
Remark 4.3 (Why ordinary marginal combinatorics does not decide the problem) . It is worth
recording why the classical marginal problem does not decide this question. Write the quoted
intervals as and order them by nondecreasing left endpoint. For , if , then, whenever the intersection is nonempty,
Thus the interval hypergraph has the running intersection property and is acyclic,
so Vorob’ev’s theorem (Vorob’ev, 1962) would grant extension were the data joint laws of the
cell budgets. The cited theorem is stated for finite coordinate spaces; for real-valued budgets
the positive direction follows by disintegrating each hyperedge law along its running-intersection
separator and adjoining the new coordinates through the resulting regular conditional kernels in
running-intersection order, which preserves every earlier marginal. They are not: a VIX quote fixes
the law of their sum, a pushforward that neither determines the joint law nor restricts canonically
to an intersection, so neither direction of the dichotomy transfers. Moreover, admissibility here
includes the martingale tower (26), positivity, the diagonal log-budget identity, and any joint
structural or quoted rows declared by the panel; these constraints have no counterpart in the
ordinary marginal-extension theorem. Failure of that theorem to transfer is therefore not a claim
that the obstruction is purely conditional. Example 4.1 shows both: its windows abut without
overlapping, and its bite comes from joint dispersion claims that are not vanilla; Example 4.2
shows that the conditional route bites with complete vanilla smiles alone once the windows overlap.
Whether some Vorob’ev-type dichotomy holds for sum-functional data under a martingale tower,
and what the right acyclicity notion for such data would be, we leave open.
Example 4.4 (Finite feasibility with a changing class) . Fix and . Prescribe SPX marginals
Every finite ordered subpanel has a positive continuous martingale realization: keep spot at one
until the last declared date before , run a time-scaled Bass martingale to the two-point law before
the first declared date after , and keep it constant thereafter. But no continuous process has
the entire marginal family. Continuity and force almost surely, contradicting the nondegenerate
prescribed law. The finite realizations compress variation into shrinking intervals and cannot satisfy
a common compact modulus. Thus “all finite panels fit” is false without the same-class qualifier in
Theorem 3.6.
Example 4.5 (An explicit compatible rolling market with a stochastic clock) . Fix and with ,
independent Brownian motions , and the CIR variance . Let , , , and
Then : it is jointly continuous,
nonnegative (each is a convex combination of and ), additive, and a martingale in the valuation
time because the mean flow composes; Fubini gives , so (26) holds. Conditionally on the variance
path is a Gaussian exponential, hence a positive continuous true martingale, and (24)–(27) follow.
The rolling VIX is affine in the variance state,
a nondegenerate noncentral- smile at every fixing ,
while SPX smiles are the corresponding integrated-variance lognormal mixtures.
For the class take and
with -Hölder seminorms taken over rational coordinates, so is an
increasing supremum of bounded continuous cylinder functions (finite rational maxima truncated
at level ) and its sublevels are compact by Arzelà–Ascoli. Every polynomial coordinate has finite
expectation for all parameters, by Burkholder–Davis–Gundy bounds and the moment form of the
Kolmogorov–Chentsov criterion, reducing to polynomial moments of ; in particular is covered.
The exponential coordinate is controlled by conditioning on the variance path: , which is finite
for every horizon precisely when , the Riccati explosion threshold for integrated CIR exponential
moments (Andersen and Piterbarg, 2007); Doob’s maximal inequality applied to then bounds
. Hence the law lies in for any : the structural class contains an explicit market with genuinely
stochastic budgets, nondegenerate rolling VIX smiles, and the absolutely continuous clock . Together
with Example 4.6 below, both sides of the clock boundary of Theorem 3.9 are inhabited. Nonzero
spot–variance correlation is possible under the corresponding moment-explosion conditions and is
not needed here.
Example 4.6 (A strict clock model without a volatility) . Let be the Cantor function and
Brownian motion. Set
with filtration . Then is a positive continuous true martingale with moments
of every order, and
is a continuous deterministic additive budget. Hence the model produces a
complete compatible SPX/VIX surface and is an exact clock realization. The measure is singular
with respect to Lebesgue measure, so no predictable can satisfy . The subclass is therefore strictly
smaller than the clock class.
Example 4.7 (A computed finite structural separator) . Take one period of length one, , and restrict the
carrier to , enforced by zero-priced calls above and puts below as in Example 4.1, or by
declaring the compact spot range in . Let denote the one-cell budget. On this interval, Taylor’s
theorem and give
Therefore, pathwise,
The first term in is the diagonal structural residual
and the second is a constant-position SPX gain. Every admissible law has , so
A quoted
pair and has strict separation margin one. If the squared-return payoff is synthesized from
calls and is traded as a variance/log-contract claim, the certificate is executable. Without
the budget instrument it remains an exact class-relative obstruction, not a claim of traded
arbitrage.
Example 4.8 (Weak realization versus positive-variance mixtures) . Let and . This is a
nonconstant positive martingale with budget
The strike- call has price zero. Theorem 2.9
gives an exact continuous Bass realization. Every finite mixture of lognormal laws with strictly
positive component variances has full support on and prices that call strictly positively. Thus an
exact weak realization exists while an exact positive-variance finite-mixture realization does not.
Relative-interior hypotheses for stronger regular architectures are substantive.
5Implications and limitations
5.1From marginal calibration to common-coupling certification
The practical distinction is between fitting marginal surfaces and certifying one common filtered
coupling. In the complete-surface idealization, SPX calls determine each law and VIX calls
determine each law ; finitely many strikes with bid–ask bands only constrain those laws. Neither
observation determines their joint law, the filtration, or the allocation of variance across overlapping
windows.
Fix one ambient carrier and declared class. For each fixing , let be the one-window laws that fit the
corresponding quote block and local martingale and diagonal rows, and let denote projection from a full
law to that window. Separate calibration establishes only for every . Let be the full laws
on that carrier satisfying every full-history structural row: the spot martingale, diagonal,
positivity, additivity, and budget towers, including in particular the shared-cell and cross-window
conditions. Rolling calibration requires
so that every window budget is the projection of one
nonnegative additive field
This is the common-coupling problem; it is stronger than selecting
an arbitrary copula between the VIX marginals. A parametric full-process model chooses a
coupling by construction, while the architecture-free question is whether any such coupling exists.
Example 4.2 shows that every one-window problem may be feasible while the rolling problem is
not.
Theorems 2.2 and 3.6 therefore change the validation question from whether a selected calibrator fits
chosen panels to whether any law in the declared class generates the entire rolling market. A rejection is
localized to finitely many coordinates.
5.2Practitioner coupling audit
The logical implications are asymmetric. Failure of a marginal/root row in Corollary 3.4, or exact
violation of the overlap inequality in Proposition 3.5, rules out every common law satisfying
the maintained idealization. Passing either test says only that the corresponding necessary
obstruction is absent; it does not construct a coupling. At finite resolution, a validated primal
common-law witness certifies existence for the stated panel, carrier, and class. A verified strict
separator certifies nonexistence there. Bare numerical infeasibility of an inner discretization is
not such a certificate unless it is supported by a valid dual separator or by a sound outer
relaxation.
Table 1 separates data admissibility, marginal implications, local feasibility, a cross-window projection
screen, finite common coupling, and the continuum lift. These stages are not interchangeable notions of
“fit.”
Table 1. Practitioner coupling audit. Vanilla prices constrain marginal information; the later gates
ask whether those marginals admit one filtered coupling. Passing P–O does not certify G.
Code
Question
Evidence or test
If it fails
If it passes
P Data
Is the quoted panel a valid input?
Bid–ask, static-arbitrage, settlement, tenor, interpolation, and tail checks.
Repair or reject the panel; failure does not specifically diagnose coupling.
The inputs are usable; no mathematical feasibility conclusion.
M Marginal
Does one root-budget curve fit all marginals?
SPX convex order (the spot-martingale row), plus log-strip/root and VIX-square mean identities (Corollary 3.4).
No common rolling law can fit the panel.
Necessary unconditional rows only; no conditional coupling.
L Local
Does every window admit a one-window law?
Separate feasibility for on the declared carrier (the one-window problem; finite formulation via Theorem 2.2
and Remark 2.7).
No common rolling law can fit the panel in that class.
Possibly different laws fit different windows; no gluing conclusion.
O Screen
Does the observable overlap projection hold?
Complete VIX strips at , the SPX log strip on , and (34)
for all (Proposition 3.5).
No common filtered coupling exists.
This obstruction is absent only; there is no existence witness.
G Common
Does one finite rolling coupling exist?
One satisfies every quote, martingale, diagonal, positivity, additivity, and tower row (Theorem 2.2; Proposition 2.5).
A verified strict separator excludes every law in the declared finite class.
A common finite-panel law exists; Theorem 2.9
gives an exact positive continuous realization.
C Cont.
Does one continuum common law exist?
Every determining finite quote restriction is feasible in the same fixed class (Theorem 3.6).
A finite quote, structural, or class witness excludes the continuum law.
One common-budget stochastic-clock law exists; uniqueness is not implied.
Stages P, M, and O use observable marginal information. Stages L and G introduce latent joint
variables and structural rows. Stage C requires complete continuous quote surfaces as a market
idealization or interpolation assumption and, separately, the compactness, moment bound, and
path/budget regularity supplied by the declared class. The continuum alternative is therefore not itself a
finite-dimensional price test: each finite quote restriction remains tested against the same complete
structural class.
5.3What the current computations establish
The calculations concern different projections of Table 1; none identifies the market’s actual copula or
filtration. On the committed 27 August 2026 delayed-quote snapshot, the date-collapsed transformed-band
surrogate for Stage M is feasible and has a maximum common inward margin of basis points of total
variance. Its Stage-O screen reports no flag on ten date-overlapping consecutive pairs; four use
same-calendar-date SPX expiries as timing proxies and six use maturity brackets. Because those
calculations collapse AM/PM settlement times, hard-code a thirty-day tenor, and use declared tail, parity,
and quadrature conventions, neither the inward margin nor a nonpositive overlap residual is a
settlement-aware feasibility certificate.
A separate settlement-aware, quote-constrained experiment on four proprietary ORATS panels tests a
projected budget/tower subsystem. On 16 March 2020, 8 April 2025, and 30 June 2026 it constructs
respectively 31-, 39-, and 19-atom joint VIX/budget witnesses fitting every retained raw SPX and VIX
call/put band and the common-forward mean rows. A fail-closed checker independent of the generating
program reconstructs every retained row; the largest equality, quote-band, or nonnegativity residual is .
These are floating-point witnesses at tolerance , not exact-rational proofs. On 5 August 2024 both selected
VIX smiles fail butterfly convexity inside their raw snapshot bands. Exact calibration to that unmodified
panel is impossible, but the Stage-P failure does not diagnose a specifically cross-window coupling
defect.
The positive witnesses establish nonemptiness only of the projected budget/tower subsystem on the
declared finite supports. The experiment has no exact AM SPX marginal at both VIX fixing
times and does not impose the joint laws, conditional spot-martingale rows, or conditional
diagonal log rows. Only the 2026 panel has equal window tenors and directly matches the paper’s
fixed- setting; the 2020 and 2025 panels are unequal-tenor extensions of the same reduced
subsystem. Thus these are not Stage-G certificates, and three positive witnesses are not a
historical-frequency estimate. They show that the coupling question can be posed against
actual quote bands and that no reduced budget–tower obstruction appears on those panels and
supports; they do not establish the full SPX/VIX filtered coupling or the prevalence of market
inconsistency.
5.4Missing instruments and post-feasibility model checks
The principal observational gap is the conditional law. Listed VIX options give the marginal law of each
fixing separately and never the -conditional law of the next one. Further instruments narrow this gap; none closes it. A mid-curve VIX option, expiring at fixing on the VIX future for fixing , observes the law
of the scalar , which by conditional Jensen satisfies pathwise; an option on the forward variance level,
that is on the VIX-squared future, observes the law of , and hence the law of the budget up to
the known tenor factor. Only the latter adds a quote row on the marginal law of a forward
budget; the mid-curve adds the law of a conditional mean and the Jensen bound, and no more.
Two models can share complete VIX and mid-curve strips while their forward-budget laws
differ: with and , revealing nothing at gives and forward budget , whereas revealing at
a fair coin that selects or gives conditional mean on both branches and forward budget
or . The tower row is a conditional identity between that budget and the later fixing; its
verification would require the joint law with the conditioning history, so it remains a structural row
imposed in the program. VIX calendar-spread options likewise observe the law of the scalar
spread between two futures, not the pairwise joint law of the fixings; the dispersion claims of
Example 4.1 are a different object. Realized-variance options on paired with VIX options at
price the unrevealed variance of the budget, , which bounds the revelation defect of any
mark for the budget row. Making every history-conditioned row observable would require a
spanning family of path-dependent claims that no listed market supplies. These instruments
therefore enrich the quote panel against which the structural rows are tested; they do not convert
the structural rows into quote rows, and the continuum class remains a declaration in every
case.
Panel-safety (Definition 2.13) and the early-revelation diagnostic apply after a candidate construction
has been specified. They test whether marks introduced by an algorithm preserve the intended conditional
rows; they are not additional market coupling tests. The clock/Brownian distinction is later still. Once a
common continuum law exists, Theorem 3.9 asks whether its bracket is absolutely continuous and hence
admits a calendar-time volatility density. Passing the coupling audit does not imply that regularity,
low-dimensionality, or parsimony.
5.5Limitations
The class-relative nature is unavoidable, and Section 1.3 records what that excludes. What
option prices leave undetermined is specific: they supply neither path tightness, nor budget
continuity, nor logarithmic uniform integrability, nor a volatility density, so each is a declaration
rather than a quote row. The calendar-time representation is likewise an existence category
and not a parsimony statement: a regular low-dimensional diffusion requires in addition a
sufficient finite-dimensional state and a regular factorization of the bracket density through that
state.
6Conclusion
Within one fixed coercive continuous-path class, a complete SPX surface and a rolling fixed-tenor
VIX surface are compatible precisely when their dense finite restrictions share one common
martingale–bracket–budget law. Failure has a finite strict witness. At finite resolution every feasible
structural law admits an exact positive continuous conditional-Bass realization; at continuum resolution
every calibrated structural law is itself a stochastic-clock model. Absolute continuity of the clock is the exact additional gate to a calendar-time Brownian representation. The examples show why none of
common-window gluing, common compactness, filtration augmentation, or clock regularity can be
suppressed. The CIR and Cantor-clock examples explicitly inhabit the absolutely continuous and singular
sides of the clock boundary. Throughout, the decisive coordinate is informational: what a
model reveals, and when, is part of its identity, which is why the state carries the budget
field and why closedness is stated in the enhanced topology. The main question left open is
whether a Vorob’ev-type dichotomy exists for sum-functional data under a martingale tower
(Remark 4.3). The overlap inequality shows that the tower bites on complete vanilla smiles
(Example 4.2); whether it and its conditional refinements exhaust the vanilla obstructions is
open.
Code and data availability. The public reproducibility package contains the linear-programming scripts, the 27 August 2026 CBOE
delayed-quote snapshot, and its outputs. Archived separately with the working paper are the
reduced-coupling program, aggregate output, compressed numerical witnesses, frozen hashes, and a
checker independent of the generating program. The underlying ORATS chains are proprietary and are
not redistributed; reproducing those calculations requires local access to the same raw files. The
documentation records the settlement, support, tail, and tolerance conventions behind the numerical
statements in Section 5.3.
AI-use disclosure. The author used Anthropic Claude Code and OpenAI Codex as interactive research and writing tools.
They assisted with exploratory discussion, testing and refinement of ideas, literature and source
organization, code development and verification, mathematical error checking, and editorial
revision.
AExact finite structural cubature
This appendix supplies the finite-tree reduction invoked after Theorem 2.9. It preserves a chosen
finite payoff panel and the coercive moment exactly; it makes no positive-variance mixture
claim.
Lemma A.1 (Finite barycentric cubature) .Let be a probability measure on a measurable space , let be integrable, and let have full -measure. There are , points , and positive weights summing
to one such that
Proof.Put and . Separation gives . If is on the relative boundary of , a supporting affine functional
through has a nonnegative gap on with zero expectation. The gap therefore vanishes almost
surely. Restrict to that full-measure hyperplane section and repeat. After at most steps, lies in
the relative interior of the closed convex hull of the remaining image. A convex set and its closure
have the same relative interior, so lies in the convex hull itself. Carathéodory’s theorem then gives
the representation with at most points; zero weights are discarded. This is the finite-dimensional
cubature argument used in Bayer and Teichmann (2006).∎
For , define the stopped budget coordinate
and the adapted vector
Lemma A.2 (The structural rows form one vector martingale) .For a law with finite first
-moment, the conditional identities in (7) hold if and only if in (44) is a vector martingale in the
full-history filtration.
Proof.The first coordinate gives the spot row. The increment of is . For fixed , the increment of
is when and zero afterward. These are exactly the three residual families.∎
Lemma A.3 (Finite structural cubature) .Let and let be a finite -dimensional payoff vector with
the growth imposed in Section 2. There is a finite nonrecombining atomic tree law , supported on
paths from an -full carrier, such that
Each selected node has at most successors.
Proof.Set and choose Borel versions . Regular conditional successor laws exist because all history
spaces are standard Borel. Starting from a Borel full-measure carrier on which , choose carriers
backward so that the conditional successor law at every is concentrated on prefix extensions in and
The finitely many conditional identities and the conditional concentration property hold almost
surely; intersecting their Borel full-measure sets at each date gives these carriers. The deterministic
root is retained in .
At a selected history , apply Lemma A.1 to its conditional successor law on and the coordinates
. At most successors reproduce both displayed conditional means. Recurse at every selected
successor through date . The resulting finite tree makes a martingale, so all structural rows hold
by Lemma A.2. Backward induction on the tree preserves ; at the root this preserves both and .
The latter identity retains the moment cap. This is the marked full-history form of the martingale
cubature reduction in Beiglböck and Nutz (2014).∎
The reduction preserves the selected finite panel, not an entire marginal law, a continuum of strikes, or
the original conditional kernels.
BDetails for the conditional Bass realization
This appendix expands the four compressed steps behind Lemma 2.8 and Theorem 2.9. The statements
are not repeated; notation is that of Section 2, and denote the objects of (14) built from the
successor-spot kernel below.
(a) Parameterized disintegration and randomization
Two classical facts about standard Borel spaces are used, in the following exact forms. Measurable
disintegration (Kallenberg, 2002, Theorem 6.4): if is a probability kernel between standard Borel spaces
and is a Borel map on the target, there exist a kernel , the -image of , and a kernel concentrated on the
fiber , with . Randomization (Kallenberg, 2002, Lemma 3.22): for a probability kernel between
standard Borel spaces there is a Borel map such that for every whenever is uniform on .
Identify the successor state with through , and apply both facts with the date- history and : this
yields the successor-spot kernel and a mark kernel on , realized by a Borel map with for uniform . By
Lemma 2.1 and , for -a.e. the kernel has mean equal to the current spot and satisfies (13). On the
exceptional -null set redefine as the law of with standard normal and read from , and let
freeze the marks, . The patched kernels satisfy (13) at every history, are jointly Borel, and
agree with the disintegration of almost everywhere, so the skeleton law of the construction is
unchanged.
(b) Measurable quantile and convolution regularity
The set identity displayed in the proof of Lemma 2.8 exhibits as a countable union of measurable sets, so
is jointly measurable, and is nondecreasing in ; hence is jointly measurable, nondecreasing in , and
strictly positive for a.e. , and in (14) is jointly measurable. Write with the density. Fix , , . For , since ,
and the -derivative reduces to two -derivatives through the heat equation for . Hölder’s inequality against
with exponents and dominate every derivative of the integrand, uniformly on compacts of , by one
integrable function. Hence is there, strictly positive, and solves ; Itô’s formula on gives
.
(c) The no-early-revelation identity
On a space carrying independent Brownian motions set, for cell , , , , frozen after the cell. The model
filtration is generated by the memory factor, i.e., by all cell coordinates run up to time (the root is
deterministic). Let . The recursion makes a Borel function of the frozen endpoints, so is
-measurable and , while is independent of . The successor state is with
a successor state in
the coordinates . For bounded Borel , the freezing lemma for conditional expectations under
independence gives with for independent standard normals . Since is uniform, ; since is an
independent uniform, the randomization property of and the product structure give
the
last equality being the disintegration of along . Conditioning from down to yields (18): no
functional of the past drivers beyond biases the successor kernel. The same independence
gives cell martingality in the full filtration: for the increments of after are independent of
, which adds to only the stopped and the -path, itself independent of ; hence . With the
boundary identity and the tower property across the finitely many cells, is a true martingale on
.
(d) Closure and uniform integrability for (13)
Fix . Since , , the martingale is uniformly integrable and closed at ; Lévy’s upward theorem gives a.s., so
the version is a.s. continuous on all of . As charges , a.s.; a nonnegative martingale is absorbed at zero,
so a.s. With , put ; these are nested and increase to a.s. because is finite on . On the integrand is
bounded, so the stopped Itô expansion of is valid, the stochastic integral has zero mean, and . Uniform
integrability of holds by splitting: optional sampling gives , so is dominated by the uniformly integrable
family ; and since is convex, conditional Jensen gives , uniformly integrable because by (13). A family dominated in absolute value by uniformly integrable families is uniformly integrable; with the
a.s. convergence this gives , finite by (13). Monotone convergence along the nested gives . Combining the
two limits proves (15), with both sides finite; in particular a.s. Taking , the diagonal row of (7)
transported through (18) gives the cell identity, and summation with the tower rows gives
(17).
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How to cite
Shaosai Huang (2026). Common-Budget Certification of Rolling SPX/VIX Markets: Finite Witnesses, Exact Continuous Realization, and the Clock Boundary. Working paper, version of September 2026. Kspectra Research. SSRN 7399598 (doi:10.2139/ssrn.7399598). https://kspectra.ai/papers/common-budget-certification-spx-vix/
@misc{huang2026common,
author = {Huang, Shaosai},
title = {{Common-Budget Certification of Rolling SPX/VIX Markets: Finite Witnesses, Exact Continuous Realization, and the Clock Boundary}},
year = {2026},
month = sep,
note = {Working paper, version of September 2026},
doi = {10.2139/ssrn.7399598},
url = {https://kspectra.ai/papers/common-budget-certification-spx-vix/}
}