Linked Physical and Risk-Neutral SPX/VIX Laws: The Identified Set under a Divergence-Bounded Pricing Kernel

Shaosai Huang

doi:10.2139/ssrn.7404180

Working paper · September 2026

Linked Physical and Risk-Neutral SPX/VIX Laws: The Identified Set under a Divergence-Bounded Pricing Kernel

Shaosai Huang

∗ Working paper. Comments welcome.

Kspectra Research Inc., Toronto, Canada · [email protected]

Abstract

Option prices constrain a pricing law $Q$ ; historical dynamics are statements under a physical law $P$ . We ask for the set of physical laws consistent with a complete SPX/VIX pricing market and a divergence bound on the pricing kernel. Making the terminal likelihood $L=dP/dQ$ a coordinate of the carrier turns every physical row and every divergence bound $\E _Q[\varphi (L)]\le C$ into an affine row on one convex law space. Inside a fixed compact convex adapted-law class, the identified set is then a compact convex fibre with closed graph and measurable selectors; it is nonempty if and only if no finite strict linked separator exists; every continuous affine physical readout has an exact attained interval on it, and conditional-cell forecasts have sharp attained intervals through a separate linear-fractional argument. The bound has three classical instances: an almost-sure density-ratio band with a gain–loss reading, a second-moment band, and a relative-entropy band. Whenever some member of the declared band family restores compatibility, a minimal band width is attained. Under the second-moment band, the two controlled moments minus one are squared Hansen–Jagannathan maximal Sharpe ratios; with the opposite moment cap held fixed and width measured in Sharpe-ratio units, this minimum is the smallest maximal physical Sharpe ratio a compatible kernel must admit, the good-deal bound read in reverse. Worked obstructions and the closed-convex-hull boundary for tests against fixed architectures are included. All results are class-relative: compactness is proved on a finite grid and assumed in continuous time, and we neither point-recover $P$ nor claim empirical adequacy of any band.

Contents

1Introduction
2Canonical market, band, and bridge class
3Physical laws as affine rows
4Geometry of the identified set
5Existence: finite witnesses
6Inference on the identified set
7Scope: architectures and charts
1. 7.1Architecture boundary: the closed convex quote hull
2. 7.2Selected-chart identification is a separate tier
8Conclusion
References
Notes
How to cite

Keywords: SPX/VIX joint calibration; physical and risk-neutral measures; pricing kernel; identified set; set-valued recovery; good-deal bounds; relative entropy; adapted weak topology; finite separator.

MSC 2020: 91G20 (primary); 60G44, 62P05, 90C34, 91G70.

JEL: G12, G13, C14, C58, C61.

1Introduction

A rich SPX option surface describes risk-neutral state prices [9]. A time series of returns and option surfaces describes physical dynamics. Two literatures connect them, and both presume that the object of interest is a single physical law. Recovery theory asks when option prices determine $P$ [46], and its critics show that the determination rests on structure the data do not supply [8]. The pricing-kernel bound literature uses security payoffs and returns to restrict the admissible stochastic discount factors [33]; option-based work derives one-sided or two-sided bounds on a chosen physical quantity, usually the expected market return [12, 40]. This paper takes the set-valued position between them: fix the complete SPX/VIX pricing market, fix a divergence bound on the kernel, and characterize the identified set of linked physical laws, its geometry, and the exact range of any physical readout on it. Point recovery is the degenerate case in which that set is a singleton; the bound literature computes one readout’s range under one particular kernel restriction.

The same object answers a second question. A natural two-stage workflow fits a $Q$ model to option prices, fits a $P$ model to historical data, and connects their parameters afterward. Whether or not that workflow is used in a particular implementation, it leaves a logically prior question:

Can the complete declared SPX/VIX pricing market and the requested physical surface dynamics be carried by one filtered state and one equivalent change of measure?

That is the question of whether the identified set is nonempty, and the answer need not be yes, as Cheridito et al. [14] already show for affine models; Section 5.3 works out four ways it can be no. The pricing law may exclude an event required by the physical target. Equivalent continuous semimartingale laws cannot prescribe different pathwise quadratic variations [37]. A divergence-bounded likelihood may be unable to move enough mass into a physical stress scenario [7, 17]. Reduced-state mimicking can preserve one-time marginals without preserving full-history conditional laws [32], so conditional laws fitted on a reduced state may conflict with the full market filtration. These are compatibility failures, not failures of a numerical optimizer.

The key idea. What makes both questions tractable is one change of viewpoint, and it is the contribution of this paper: a carrier on which the physical law is not a second object but a family of linear rows in the pricing law. We make the terminal likelihood $L=dP/dQ$ , its density process, and the forward-variance budget coordinates of the canonical state, on the same raw filtration as spot and the SPX and VIX surfaces. Then

\begin{equation} \E _P[F]=\E _Q[LF] \label {eq:rn-intro} \end{equation}

for every physical test $F$ , so a physical path law is countably many $L$ -weighted affine rows, a conditional kernel is the vanishing of $L$ -weighted residuals against a countable algebra generating the observed past, and a divergence bound $\E _Q[\varphi (L)]\le C$ is one more affine row; full-history conditioning survives mixing because the conditioning variables are coordinates. Pricing, physics, and the kernel bound are thereby linear constraints on one convex set of laws, and the two questions above become the nonemptiness and the projections of that set. The construction composes two established devices, density weighting under linear constraints [2, 18] and lifting a state so that conditional constraints become linear [6, 29, 42], into a single system in which the physical rows, the complete pricing market, and the kernel bound coexist on one filtration; that system is what every result below is about. The lift is not itself a theorem, since strict equivalence is not weakly closed, weak convergence need not preserve conditional expectations, and separately fitted VIX windows need not share one budget; we therefore fix the bridge class before calibration, with one raw information structure, compactness in a topology preserving full-history rows, and the band.

Main results. After the lift, every question in the paper is a question about one compact convex set: the fibre $\mathfrak F(y)$ of laws in $\cK$ that match every declared pricing and physical row. The results are the four things one can ask of such a set: what it looks like, whether it is empty, what it says about a physical quantity, and what it is a set of. All statements are relative to one fixed compact convex adapted-law class $\cK$ (Assumption 2.5).

(i): Geometry. The fibre is compact and convex, the map from targets to fibres has closed graph and one-sided solution-set stability, and it admits Borel and variational selectors (Theorem 4.1). The set of feasible targets is itself compact and convex. Compactness and convexity are not decoration: the separation argument of (ii) and the support-function representations of (iii) both live on them, and both are supplied by the lift rather than assumed of the market.
(ii): Existence. The fibre is nonempty if and only if every finite determining panel is feasible, if and only if no finite strict linked separator exists, the physical terms of the separator being $L$ -weighted expectations under the same pricing law as its pricing terms (Theorem 5.2). Emptiness therefore always has a finite witness. On a finite panel the witness is a violated support inequality (Proposition 5.3); on a finite grid, where compactness is proved rather than assumed, it is a pathwise certificate with explicit gain, moment, and band multipliers (Proposition 5.4). Section 5.3 shows what witnesses look like: a band that cannot move enough mass, a pricing support that excludes a physical event, or covariance characteristics prescribed twice. The third is bracket invariance under equivalent change of measure (Proposition 5.7), and Remark 5.9 records that it is at once the hypothesis licensing current joint $P/Q$ work and the constraint on it.
(iii): Inference. On a nonempty fibre, every continuous affine physical readout has an exact attained identified interval, with dual slice-endpoint formulas on a finite panel (Proposition 6.2), and every conditional cell forecast has a sharp linear-fractional range whose endpoints are zeros of affine support-oracle problems (Proposition 6.3). The answer can be an interval rather than a point because, on an atomless carrier and without further class restrictions, finitely many physical moments need not identify the likelihood (Proposition 6.1). Such rows carry sensitivity analysis under a bounded density ratio onto a martingale-constrained law class, where each bound is a constrained optimization rather than a sort. The band is the hinge between (ii) and (iii): widen it and the fibre appears, narrow it and the intervals tighten. Whenever the declared band family contains a compatible member, the smallest width at which a compatible bridge exists is attained, also for a compact target uncertainty set (Proposition 6.5). Under the second-moment band, the two controlled moments minus one are squared Hansen–Jagannathan maximal Sharpe ratios (Lemma 2.7); holding the opposite moment cap fixed and measuring width in Sharpe-ratio units makes this threshold the smallest maximal physical Sharpe ratio any compatible kernel must admit, the good-deal bound read in reverse. In three states the three bands select three different fibre points (Example 6.9).
(iv): Scope. The fibre is a set of laws in $\cK$ , not of models in a chosen family $\mathfrak M\subset \cK$ . A linear support test against a fixed family is a sound rejection test, but nonrejection certifies only membership in the closed convex hull of the family’s quote image (Proposition 7.1), so a good or bad fit of one architecture settles neither compatibility nor market-level conflict. Whether a selected finite model is locally identified is a separate, chart-level question (Proposition 7.5) that does not make the law-level fibre a singleton.

The band and its readings. The band that runs through these results is where the class meets practice. Since $R\mapsto \E _R[\varphi (L)]$ is affine for any $\varphi$ , the bound is convex automatically and weakly closed when $\varphi$ is lower semicontinuous, and the compatibility theorem holds for any superlinear $\varphi$ . Three choices are classical. An almost-sure band $\ell _-\le L\le \ell _+$ implies the gain–loss restriction of Bernardo and Ledoit [7]; the ratio-only restriction is recovered by allowing the normalized endpoints to vary at a fixed ratio. A second-moment band is the Hansen–Jagannathan class: $\E _Q[L^{-1}]-1$ is the squared maximal physical Sharpe ratio, so capping it is the good-deal restriction of Cochrane and Saá-Requejo [17]. A relative-entropy band bounds $H(P\,|\,Q)$ on one side and $H(Q\,|\,P)$ on the other. With $P$ as benchmark and $Q$ as the distorted law, the second has the robust-control orientation of Hansen and Sargent [34]; the first is the reverse-KL mirror. The second is also the objective of entropic joint calibration [29, 30]. The almost-sure band is the sharpest but is violated by every model whose likelihood is unbounded or has essential infimum zero, as with Gaussian-mixture innovations and a shifted drift; any model whose likelihood has finite moments of the required orders lies inside the moment and entropy bands at sufficiently large levels. We state results for a general band and record the three readings where they differ.

What is not claimed. The generality of the band does not extend to the class: every result is class-relative. Compactness and adapted closure are proved on a finite grid in Section 5.2 and assumed in continuous time; they are not consequences of the Radon–Nikodym identity, and Assumption 2.5 is the main gate for any concrete continuous carrier. We do not point-recover $P$ from option prices, establish that historical SPX/VIX data satisfy any band, turn every separator into a traded arbitrage, identify a bridge from finitely many moments, or compute $\kappa _*$ on data. The alternative theorem is short by design: once the carrier and rows are built, existence is finite intersection plus finite-dimensional separation, and the contribution is the carrier that makes that argument available and the readings it supports.

Relation to the literature. That contribution is best located against three literatures that address this question or a neighbour of it. Recovery theory asks option prices to determine $P$ [35, 46]; Borovička et al. [8] show that the determination rests on structure the data do not supply, and its empirical record is weak [36]. We give the set that remains when that structure is dropped. The Hansen–Jagannathan bound restricts the kernel using security-market payoffs and returns [33]; option-based bounds target physical quantities such as the expected market return [12, 40]. We bound any continuous affine readout, two-sidedly and sharply, on the full SPX/VIX carrier and under a divergence ball rather than a covariance sign; the price is that our intervals are sharp only inside a declared class, whereas the negative correlation condition of Martin [40] needs almost nothing. Joint $P/Q$ modelling asks when one state carries both dynamics, parametrically in Cheridito et al. [14] and in the estimation literature [5, 15, 26, 43], and most recently for SPX/VIX in Guyon and Parent [31], who compare option values generated from a $P$ -estimated model with market prices and report structured residuals that they read as a possible tail risk premium. We ask the same question nonparametrically on a declared filtration, and $\kappa _*$ is the scalar such a residual calls for. Kernel estimation [20, 48] selects one bridge or one projection; our fibre may remain set-valued. The pricing-kernel puzzle [19] is the documented tension that motivates asking whether a linked pair exists at all.

The paper builds on two lines. The alternative theorem is a carrier-level convex-feasibility analogue in the lineage of the good-deal fundamental theorem of Staum [51], anticipated by Černý and Hodges [53]: it uses the same separation logic on a carrier that holds the complete SPX/VIX pricing market, the likelihood and its density process as coordinates, and full-history physical rows. This is not a specialization of the traded-payoff theorem, because some of the present separator’s rows are not traded. The identified intervals carry the marginal sensitivity model of Tan [52] and Dorn and Guo [24], with its sharp conditional bounds by fractional programming, from covariate-wise reweighting to a martingale-constrained law class, where each bound is a constrained optimization rather than a sort. The complete pricing market itself is the object built by entropic joint SPX/VIX calibration [29, 30], which we take as given on the pricing side and extend to the physical side.

Organization. The paper follows the order of the main results. Section 2 builds the canonical market, the band with its three readings, and the bridge class; Section 3 encodes physical laws as affine rows and defines the fibre. Sections 4 to 7 then answer the four questions of the main results in order: the geometry of the fibre (Section 4); its nonemptiness, with computable certificates and the three obstructions (Section 5); inference on it, from nonidentification through exact intervals to the minimal divergence, with a finite-grid illustration (Section 6); and its scope against fixed architectures and selected charts (Section 7). Section 8 concludes.

2Canonical market, band, and bridge class

2.1Carrier, structural law, and linked pair

Fix a horizon $H>0$ and one VIX tenor $\Delta \in (0,H)$ , and put $H_\Delta =H-\Delta$ . Let $\Omega$ be a Polish enhanced path space with coordinates

\begin{equation} Z=(X,A,B,\mathcal C^S,\mathcal C^V,L,D). \label {eq:coordinates} \end{equation}

Here $X$ is continuous log spot, $A$ is a continuous nondecreasing clock normalized by $A_0=0$ , $B(r;u,v)$ is a continuous nonnegative forward-budget field for $0\le r\le u\le v\le H$ , $\mathcal C^S$ and $\mathcal C^V$ are cadlag raw price-surface processes with maturity domains $[0,H]$ and $[0,H_\Delta ]$ , respectively, and are frozen after maturity. The coordinate $L$ is a terminal likelihood and $D$ is a cadlag density process. All roots are deterministic. The raw filtration $\cF _t^0$ contains exactly the coordinates revealed through $t$ ; $L$ is revealed only at $H$ . The surface coordinates are raw price processes as in dynamic option-surface models [11, 39, 49].

Set

\begin{equation} S_t=e^{X_t},\qquad N_t=X_t-X_0+\frac 12A_t,\qquad V_t^\Delta =\sqrt {B(t;t,t+\Delta )/\Delta },\qquad 0\le t\le H_\Delta . \label {eq:spot-clock-vix} \end{equation}

Definition 2.1 (Divergence band) . A band is a pair of nonnegative lower semicontinuous functions $\varphi ,\psi :[0,\infty )\to [0,\infty ]$ and levels $C_+,C_-\in [0,\infty )$ such that

\begin{equation} \lim _{x\to \infty }\frac {\varphi (x)}{x}=\infty , \qquad \psi (0)=\lim _{x\downarrow 0}\psi (x)=\infty . \label {eq:band-growth} \end{equation}

A law

R

\Omega

satisfies the band when

\begin{equation} \E _R[\varphi (L)]\le C_+,\qquad \E _R[\psi (L)]\le C_-. \label {eq:band} \end{equation}

Three instances are used below. The almost-sure band takes

\varphi =\psi

equal to the convex indicator of

[\ell _-,\ell _+]

, with

0<\ell _-<1<\ell _+<\infty

and

C_\pm =0

, so that (5) reads

\ell _-\le L\le \ell _+

R

-a.s. The moment band takes

\varphi (x)=x^p

and

\psi (x)=x^{-r}

with

p>1

and

r>0

. The entropy band takes

\varphi (x)=x\log x-x+1

and

\psi (x)=x-1-\log x

, both nonnegative and convex; when

\E _R[L]=1

these rows are the relative entropies

H(P_R\,|\,Q_R)

and

H(Q_R\,|\,P_R)

of Lemma 2.4.

For compact minimal-width families we also use the closed extension $0<\ell _-\le 1\le \ell _+<\infty$ of the almost-sure instance. The boundary $\ell _- = \ell _+ =1$ is the singleton band $L=1$ ; strict inequalities are what give slack around that likelihood.

Since $L$ is a coordinate, $R\mapsto \E _R[\varphi (L)]$ and $R\mapsto \E _R[\psi (L)]$ are affine on the space of laws, so (5) consists of two affine inequality rows. Lower semicontinuity and nonnegativity make both rows weakly closed by the Portmanteau theorem; the growth of $\varphi$ makes $L$ uniformly integrable over any family satisfying (5) by the de la Vallée-Poussin criterion; and the growth of $\psi$ forces $L>0$ almost surely in every weak limit. These three properties are the only ones the compatibility proofs use, so the almost-sure band is a special case rather than the general object.

Definition 2.2 (Structural pricing law) . Fix a band. A law $R$ on $\Omega$ is structural when, in the completed raw filtration:

(i): $N$ , $N^2-A$ , and $S$ are true martingales;
(ii): $B$ is interval additive and
$\begin{equation} B(r;u,v)=\E _R[A_v-A_u\mid \cF _r^0],\qquad r\le u\le v; \label {eq:budget} \end{equation}$
(iii): for $T\in [0,H]$ and $T'\in [0,H_\Delta ]$ , respectively, the frozen processes $\mathcal C^S_\cdot (T,K)$ and $\mathcal C^V_\cdot (T',K)$ are true martingales with terminal values
$\begin{align} \mathcal C^S_T(T,K)&=(S_T-K)^+, &\mathcal C^V_{T'}(T',K)&=(V_{T'}^\Delta -K)^+; \label {eq:settlement} \end{align}$
(iv): $D_0=1$ , $D$ is a true martingale, $D_H=L$ , and $R$ satisfies the band (5).

All conditional statements are tested against bounded functions of the full raw past, not merely the current spot.

Proposition 2.3 (Self-contained pricing-side interface) . If $R$ is structural and $Q=R$ , then $S$ is a positive true $Q$ -martingale, $A=\langle N\rangle ^Q$ , and every declared VIX window is generated by the same additive bracket budget. Moreover, for $t\le T$ and $t\le T'\le H_\Delta$ ,

\begin{align} \mathcal C^S_t(T,K)&=\E _Q[(S_T-K)^+\mid \cF _t^0],\nonumber \\ \mathcal C^V_t(T',K)&=\E _Q[(V_{T'}^\Delta -K)^+\mid \cF _t^0], \label {eq:pricing-interface} \end{align}

and (6) supplies the corresponding conditional bracket increments. Thus Definition 2.2 is the complete pricing-side input used below; no separate pricing theorem is invoked in the compatibility proofs.

Proof.Since $A$ is continuous, adapted, increasing, and starts at zero, it is predictable. Continuity of $N$ , together with the local martingales $N$ and $N^2-A$ , therefore identifies $A$ with $\langle N\rangle ^Q$ by uniqueness of predictable quadratic variation. Moreover, $S=S_0\exp (N-\frac 12\langle N\rangle ^Q)=S_0\mathcal E(N)$ ; the separate $S$ row makes this stochastic exponential a true rather than merely local martingale. Each raw surface process is a true martingale stopped at its maturity and has the terminal value in (7); the defining property of conditional expectation gives (8). Interval additivity and (6) give the common budget assertion.∎

Lemma 2.4 (Linked pair) . For structural $R$ , define

\begin{equation} Q_R:=R,\qquad P_R(G):=\E _R[L\1_G],\quad G\in \cF _H^0. \label {eq:pair} \end{equation}

Then

P_R

is a probability law,

P_R\sim Q_R

, and

\begin{equation} D_t=\E _{Q_R}[L\mid \cF _t^0] =\left .\frac {dP_R}{dQ_R}\right |_{\cF _t^0}. \label {eq:density-process} \end{equation}

Their completions of

\cF _t^0

coincide, and the pathwise quadratic variation of

N

remains

A

under

P_R

Proof.The martingale property gives $\E _RL=\E _RD_0=1$ . For $G\in \cF _t^0$ , $P_R(G)=\E _R[L\1_G]=\E _R[D_t\1_G]$ , proving (10). Finiteness of $\E _R[\varphi (L)]$ gives $L<\infty$ a.s., and finiteness of $\E _R[\psi (L)]$ with $\psi (0)=\infty$ gives $L>0$ a.s.; hence $P_R\sim Q_R$ with common null sets. Quadratic variation is pathwise and invariant under an equivalent change of measure.∎

2.2The bridge class

Assumption 2.5 (Adapted-compact bridge class) . There is a metrizable adapted-law space $(\mathfrak X_{\rm ad},\tau _{\rm ad})$ and a nonempty compact convex subset $\cK \subset \mathfrak X_{\rm ad}$ such that:

(a): every $R\in \cK$ is structural on the same raw canonical filtration, with the same deterministic roots and the same band $(\varphi ,\psi ,C_+,C_-)$ ;
(b): $\tau _{\rm ad}$ dominates weak convergence and preserves all declared full-history conditional rows;
(c): an inf-compact envelope $\Psi \ge 1$ satisfies $\Psi \ge 1+\varphi (L)+\psi (L)$ almost surely under every $R\in \cK$ and $\sup _{R\in \cK }\E _R\Psi <\infty$ , and each unbounded row $G$ below, pricing or $L$ -weighted physical, obeys $|G|\le c_G\Psi ^{\alpha _G}$ for some $\alpha _G<1$ ; for the almost-sure band the first requirement is vacuous, since $\varphi (L)=\psi (L)=0$ almost surely under every law of the class;
(d): every target-row expectation below is continuous on $\cK$ .

The band rows contribute to (c) and (d) but do not deliver them. Coercivity of $\varphi$ at infinity and of $\psi$ at zero makes the marginal law of $L$ tight and uniformly integrable, so every physical row with a bounded continuous integrand is continuous under weak convergence with no further hypothesis; for unbounded integrands the envelope condition in (c) does the same work. What the band cannot supply is tightness of the path coordinates or closure of the full-history conditional rows. Coordinate tightness needs modulus estimates, and preservation of conditioning on the full past needs an adapted or extended-weak closure argument. Assumption 2.5 is the main class-verification gate. In finite discrete time the several natural adapted topologies agree [3], adapted distances give a useful stability notion for finance [4], and compactness has an Arzelà–Ascoli-type characterization rather than an ordinary Prokhorov criterion [25]. Those results do not supply a compactness theorem for the present continuous-path carrier. In continuous time, adapted compactness and preservation of the declared conditional rows remain the topology-specific obligations assumed in Assumption 2.5; marginal tightness alone is not sufficient. On the finite grid of Section 5.2, compactness is instead proved directly in the ordinary weak topology.

Remark 2.6 (Nonemptiness and the pricing-side interface) . Nonemptiness of $\cK$ means that at least one structural pricing carrier exists before the additional physical targets are imposed. The interface is explicit in Definition 2.2 and Proposition 2.3. An external construction, such as a joint SPX/VIX calibration [28–30], may be used to establish nonemptiness and adapted compactness only after matching every item of Assumption 2.5; those properties are not inferred from individual option surfaces, the divergence band, or the Radon–Nikodym identity.

2.3Economic meaning of the band

The band is a declared overlap class, not an assertion that the true market likelihood satisfies it. Its economic content depends on which band is chosen, and the three instances of Definition 2.1 have three classical readings.

Almost-sure band. For every event $A$ ,

\begin{equation} \ell _-Q(A)\le P(A)\le \ell _+Q(A). \label {eq:event-bounds} \end{equation}

For a future event and current history,

\begin{equation} P(A\mid \cF _t^0)=\frac {\E _Q[L\1_A\mid \cF _t^0]}{D_t}, \label {eq:bayes} \end{equation}

\begin{equation} \frac {\ell _-}{\ell _+}Q(A\mid \cF _t^0) \le P(A\mid \cF _t^0) \le \frac {\ell _+}{\ell _-}Q(A\mid \cF _t^0), \label {eq:conditional-bounds} \end{equation}

with sharper history-dependent denominators $D_t$ in place of the outer band. The class imposes common scenario support and caps every probability distortion pointwise. The fixed envelope implies a gain–loss cap $\ell _+/\ell _-$ : Bernardo and Ledoit [7] bound the ratio of the essential extrema of the normalized pricing kernel $dQ/dP=1/L$ . Their ratio-only restriction does not fix $\ell _-$ and $\ell _+$ separately; conversely, it is the union of normalized envelopes $a\le L\le b$ with $b/a$ below the cap. The variable-endpoint family in Proposition 6.5 implements that union. These algebraic gain–loss statements concern the stipulated class of claims valued by $Q$ ; a market interpretation for a smaller attainable payoff span requires a separate trading-space argument. The fixed envelope is the most restrictive of the three bands for the tail-support reason recorded in Remark 5.9.

Moment band. With zero rates the stochastic discount factor is $m=1/L$ up to normalization. For the moment band with $(p,r)=(2,1)$ , the two controlled moments minus one are the two Hansen–Jagannathan variances, equivalently the squared maximal Sharpe ratios in (14).

Lemma 2.7 (Maximal Sharpe ratios from likelihood second moments) . Let $P\sim Q$ with $L=dP/dQ$ , $\E _Q[L^2]<\infty$ , and $\E _Q[L^{-1}]<\infty$ . Over bounded payoffs $X$ with positive variance, with the convention that the supremum is zero if this class is empty,

\begin{equation} \sup _X\frac {\E _P[X]-\E _Q[X]}{\sigma _P(X)} =\sqrt {\E _Q[L^{-1}]-1}, \qquad \sup _X\frac {\E _Q[X]-\E _P[X]}{\sigma _Q(X)} =\sqrt {\E _Q[L^{2}]-1}. \label {eq:sharpe-identities} \end{equation}

Proof.Since $\E _P[L^{-1}]=\E _Q[1]=1$ , the numerator of the first ratio is ∎

\E _P[X]-\E _Q[X]=\E _P[X(1-L^{-1})]=\operatorname {Cov}_P(X,-L^{-1}),

which Cauchy–Schwarz bounds by

\sigma _P(X)\,\sigma _P(L^{-1})

; unless the variance of

L^{-1}

is zero, the bound is approached by the bounded truncations

X_n=-\min \{L^{-1},n\}

, which converge to

-L^{-1}

L^2(P)

. If that variance is zero,

L^{-1}=1

P

-a.s., hence

P=Q

and every numerator is zero; the supremum is zero whether the admissible class is nonempty or is empty under the stated convention. Now

\operatorname {Var}_P(L^{-1})=\E _P[L^{-2}]-1

, and

\E _P[L^{-2}]=\E _Q[L^{-1}]

. The second identity is the same argument with the roles of

P

and

Q

exchanged, using

\E _Q[L]=1

and

\operatorname {Var}_Q(L)=\E _Q[L^2]-1

The first identity is the Hansen–Jagannathan bound [33]: on the stipulated space of all bounded claims valued by $\E _Q$ , the cap $\E _Q[L^{-1}]\le 1+h^2$ is exactly the restriction that no such claim has physical Sharpe ratio above $h$ . It yields the good-deal restriction of Cochrane and Saá-Requejo [17]; on a smaller traded payoff span it is sufficient but need not be necessary. The second identity is its mirror under the pricing measure. Under the moment band the pointwise bounds (11) are replaced by Hölder bounds: for every event $A$ ,

\begin{equation} \frac {Q(A)^{(r+1)/r}}{C_-^{1/r}} \le P(A) \le C_+^{1/p}\,Q(A)^{1-1/p}. \label {eq:holder-event-bounds} \end{equation}

The upper bound is Hölder applied to $\E _Q[L\1_A]$ . For the lower bound, write

Q(A)=\E _Q\bigl [(L\1_A)^{r/(r+1)}\,(L^{-r}\1_A)^{1/(r+1)}\bigr ]

and apply Hölder with exponents $(r+1)/r$ and $r+1$ . These are weaker per event than (11), which is the price of a class that admits models with unbounded innovations at large enough levels; the exact interval of Proposition 6.2 below is sharp under either band.

Entropy band. Under $\E _R[L]=1$ the $\varphi$ -row is $H(P\,|\,Q)$ and the $\psi$ -row is $H(Q\,|\,P)$ . With $P$ as benchmark and $Q$ as the distorted law, the latter has the robust-control orientation of Hansen and Sargent [34]; the former is its reverse-KL mirror. The $\psi$ -row is the one calibration practice minimizes: entropic calibration of a candidate $Q$ to the option surface with $P$ as prior minimizes $H(Q\,|\,P)$ , in weighted Monte Carlo [2] and in martingale Schrödinger bridges [29, 30, 42]. In those problems the prior is fixed and the entropy is minimized; here both sides are bounded across the bridge, and the minimal bound over the compatible fibre, defined in Proposition 6.5, is a Schrödinger-type value for the linked pair.

In every case the reciprocal change-of-measure component is related to a stochastic discount factor only after a numeraire and normalization are fixed, and no band is a preference model.

Strict equivalence is not weakly closed without uniform overlap. On $\{0,1\}$ , for $n\ge 2$ let

\begin{equation} Q_n=(1-n^{-1})\delta _0+n^{-1}\delta _1,\qquad P_n=n^{-1}\delta _0+(1-n^{-1})\delta _1. \end{equation}

Then $P_n\sim Q_n$ but $(P_n,Q_n)\Rightarrow (\delta _1,\delta _0)$ , a singular pair; the likelihood is $n-1$ at state $1$ and $1/(n-1)$ at state $0$ , so $\E _{Q_n}[L^2]$ and $\E _{Q_n}[L^{-1}]$ both diverge and the sequence leaves every band of Definition 2.1. This is what the two band rows are for: the $\varphi$ -row keeps $L$ uniformly integrable, and the $\psi$ -row keeps it away from zero in the limit. Sending the levels to infinity through separately solved problems does not prove attainment in an unrestricted equivalent-law class.

3Physical laws as affine rows

Let $Y$ be a declared observable path coordinate, such as the pair of raw SPX and VIX surface processes, taking values in a Polish space $\mathsf Y$ . Fix bounded convergence-determining functions $(f_k)_{k\ge 1}\subset C_b(\mathsf Y)$ . A target physical path law $\nu ^P$ is encoded by

\begin{equation} \E _R[Lf_k(Y)]=\int f_k\,d\nu ^P,\qquad k\ge 1. \label {eq:path-rows} \end{equation}

For a conditional target, let $U$ be a Polish-valued future coordinate and $\kappa _s(H_s,du)$ a Borel kernel from the stopped raw history. Take a bounded convergence-determining family $(g_j)$ and a countable multiplicative algebra $\cH _s^\circ$ of bounded full-history functions containing constants and generating $\cF _s^0$ . Impose

\begin{equation} \E _R\!\left [Lh(H_s)\left \{g_j(U)- \int g_j(u)\kappa _s(H_s,du)\right \}\right ]=0 \label {eq:kernel-rows} \end{equation}

for every $h\in \cH _s^\circ$ and $j$ .

Lemma 3.1 (Meaning of the physical rows) . For $R\in \cK$ , equations (17) for all $k$ are equivalent to $\Law _{P_R}(Y)=\nu ^P$ . Equations (18) for all $h,j$ are equivalent to

\begin{equation} \Law _{P_R}(U\mid \cF _s^0)=\kappa _s(H_s,\cdot ) \quad P_R\text {-a.s.} \label {eq:kernel-meaning} \end{equation}

Proof.The identity $\E _{P_R}F=\E _R[LF]$ turns (17) into equality of integrals under two probability laws. The convergence-determining family separates them. It turns (18) into $\E _{P_R}[h(H_s)Z_j]=0$ , where $Z_j=g_j(U)-\int g_jd\kappa _s(H_s)$ . The functional monotone-class theorem extends from $\cH _s^\circ$ to every bounded $\cF _s^0$ -measurable multiplier, so $\E _{P_R}[Z_j\mid \cF _s^0]=0$ . Countable determination identifies the kernel. The converses follow by conditioning.∎

Testing only functions of $(S_s,V_s)$ would identify conditioning on their sigma-field, not on the full market history. Equality requires a separately proved sufficient-state property. Reweighting a simulated ensemble by $L$ to match finitely many price rows is the operational ancestor of these rows [2].

Let $(\Gamma _k)_{k\ge 1}$ enumerate countably many continuous affine rows on $\cK$ : dense SPX/VIX pricing rows and the $L$ -weighted physical rows above. Let $y=(y_k)_{k\ge 1}$ be their targets.

Assumption 3.2 (Countable determination) . The enumeration determines the complete declared SPX/VIX quote surfaces, the physical path law, and each physical conditional kernel. Parameter continuity extends dense maturity, strike, and time rows to their full domains.

Definition 3.3 (Compatible-law fibre) .

\begin{equation} \mathfrak F(y):=\{R\in \cK :\Gamma _k(R)=y_k\text { for every }k\ge 1\}. \label {eq:fibre} \end{equation}

Each certificate

R\in \mathfrak F(y)

induces the linked pair

(P_R,Q_R)

4Geometry of the identified set

This section answers the first of the four questions of the introduction: what the identified set looks like. The answer is that it is a compact convex fibre with a closed graph, and that this geometry is inherited from the lift rather than assumed of the market.

Equip $\R ^\N$ with its product metric, write $\Gamma (R)=(\Gamma _k(R))_{k\ge 1}$ , and set $\cY _{\rm feas}=\Gamma (\cK )$ .

Theorem 4.1 (Stable compatible-law fibres) . $\cY _{\rm feas}$ is compact and convex. The graph of $y\mapsto \mathfrak F(y)$ is closed. If $y_n\to y$ , $R_n\in \mathfrak F(y_n)$ , and $d_{\rm ad}$ metrizes $\tau _{\rm ad}$ on $\cK$ , then

\begin{equation} \dist _{d_{\rm ad}}(R_n,\mathfrak F(y))\longrightarrow 0. \label {eq:set-stability} \end{equation}

There is a Borel selector

s:\cY _{\rm feas}\to \cK

with

s(y)\in \mathfrak F(y)

Proof. $\Gamma$ is continuous and affine, hence has compact convex image. Its inverse fibres have a closed graph. If (21) failed, a subsequence would remain at positive distance from $\mathfrak F(y)$ . Compactness gives a further subsequence converging to $R_*$ , and graph closedness gives $R_*\in \mathfrak F(y)$ , a contradiction. The inverse correspondence has nonempty closed values. If $O\subset \cK$ is open, write $O=\bigcup _mK_m$ with $K_m$ compact. Then ∎

\{y:\mathfrak F(y)\cap O\ne \varnothing \} =\Gamma (O)=\bigcup _m\Gamma (K_m)

is Borel because every

\Gamma (K_m)

is compact. Thus the correspondence is weakly measurable, and the Kuratowski–Ryll-Nardzewski theorem supplies a Borel selector.

The stability is one-sided solution-set stability, not Hausdorff continuity. It gives no inverse rate. It is also weaker than what is available in the pricing-only problem: in their one-step real-line setting, for martingale measures with fixed marginals, Neufeld and Sester [41] prove full lower and upper hemicontinuity in the marginals by a set-valued map argument. We do not claim the two-sided statement here, because the $L$ -weighted physical rows and the full-history conditional rows are not marginal restrictions and the corresponding lower hemicontinuity is not established for the class of Assumption 2.5. If $I:\cK \to (-\infty ,\infty ]$ is proper and lower semicontinuous and finite somewhere on a feasible fibre, compactness gives an $I$ -minimizer. A continuous strictly convex $I$ gives a unique Borel selector. Entropy is one possible criterion under its own domain and lower-semicontinuity conditions [18, 42]; the raw market does not identify that choice.

5Existence: finite witnesses

This section answers the second question: when the fibre is nonempty, and what a witness of emptiness looks like. Theorem 5.2 gives the finite alternative; Sections 5.1 and 5.2 give computable certificates on a finite panel and on a finite grid; and Section 5.3 exhibits the three obstructions.

For finite $J\subset \N$ , let

\begin{equation} F_J(y):=\{R\in \cK :\Gamma _k(R)=y_k, k\in J\}. \label {eq:finite-fibre} \end{equation}

Definition 5.1 (Finite strict linked separator) . A finite strict linked separator is a finite set $J$ and coefficients $(a_k)_{k\in J}$ such that

\begin{equation} \sup _{R\in \cK }\sum _{k\in J}a_k\Gamma _k(R) <\sum _{k\in J}a_ky_k. \label {eq:separator} \end{equation}

Physical terms on the left are

L

-weighted expectations under the same canonical pricing law as the pricing terms.

Theorem 5.2 (Complete linked alternative) . Under Assumptions 2.5 and 3.2, the following are equivalent:

(a): $\mathfrak F(y)\ne \varnothing$ ;
(b): one equivalent pair $(P,Q)$ on the declared filtered state, whose enhanced $Q$ -law belongs to $\cK$ , carries the complete structural SPX/VIX market under $Q$ , satisfies all physical targets under $P$ , and whose likelihood $dP/dQ$ satisfies the band (5);
(c): $F_J(y)\ne \varnothing$ for every finite $J\subset \N$ ;
(d): there are $R_n\in \cK$ satisfying $\Gamma _k(R_n)=y_k$ for $k\le n$ ;
(e): no finite strict linked separator exists.

When feasible, $\mathfrak F(y)$ is compact and convex. When infeasible, a finite collection of declared pricing and physical rows witnesses failure.

Proof.Lemmas 2.4 and 3.1, the structural settlements, and countable determination prove (a) $\Leftrightarrow$ (b). Clearly (a) implies (c), and (c) implies (d). Under (d), compact metrizability gives a convergent subsequence $R_{n_j}\to R$ . For fixed $k$ , all sufficiently large terms fit row $k$ ; continuity gives $\Gamma _k(R)=y_k$ . Thus $R\in \mathfrak F(y)$ .

If (a) holds, no separator can integrate to a strict inequality. If (a) fails, the finite-intersection property for the closed sets $F_{\{k\}}(y)$ gives finite $J$ with $F_J(y)=\varnothing$ . The image ∎

C_J=\{(\Gamma _k(R))_{k\in J}:R\in \cK \}

is compact and convex, while

y_J\notin C_J

. Strict finite-dimensional separation yields (23). The fibre is a closed affine section of compact convex

\cK

Once the carrier and rows are built, the proof is short by design: the work has been moved into the lifted formulation, whose adapted compactness and row closure are the substantive obligations. The argument places the theorem in the lineage of full-versus-finite-subsystem consistency in linear semi-infinite optimization [27] and of the good-deal fundamental theorems [1, 51], the martingale selection problem [10, 44], and finite-panel realizability [21], now on a carrier that holds both laws.

5.1Finite-panel support criterion

For a finite panel, separate the pricing and physical integrands as $\Phi _J=(\Phi _i^Q)$ and $\Psi _J=(\Phi _j^P)$ , so

\begin{equation} T_J(R)=\bigl (\E _R\Phi _J,\E _R[L\Psi _J]\bigr ). \label {eq:finite-map} \end{equation}

For $a=(a_Q,a_P)$ define

\begin{equation} h_J(a)=\max _{R\in \cK } \E _R[a_Q^\top \Phi _J+La_P^\top \Psi _J]. \label {eq:support-function} \end{equation}

Proposition 5.3 (Finite support criterion) . The finite target $(q_J,p_J)$ is attainable if and only if

\begin{equation} a_Q^\top q_J+a_P^\top p_J\le h_J(a) \quad \text {for every }a. \label {eq:support-test} \end{equation}

The maximum is attained. If the target is infeasible, one

a

violates (26) strictly.

Proof. $T_J(\cK )$ is a compact convex subset of finite-dimensional Euclidean space. The assertion is its support-function representation and strict separation from an exterior point.∎

5.2Pathwise certificate on a finite grid

For a pathwise refinement, fix dates $0=t_0<\cdots <t_m=H$ and a closed Polish finite-grid carrier $\widehat \Omega$ . Its histories include $D_0=1$ and $D_j\in (0,\infty )$ . Let $R_j^Q$ and $R_j^P$ be continuous next-step residual vectors. Let $\mathcal W$ consist of finite sums

\begin{equation} W=\sum _{j<m}\{f_j(H_j)^\top R_j^Q+c_j(H_j)(D_{j+1}-D_j) +g_j(H_j)^\top D_HR_j^P\}, \label {eq:gain} \end{equation}

with bounded continuous multipliers. Take a band $(\varphi ,\psi ,C_+,C_-)$ with $\varphi ,\psi$ finite on $(0,\infty )$ , as for the moment and entropy bands. For continuous inf-compact $\Upsilon \ge 1$ and $r>1$ , define

\begin{align} \mathcal P_M&=\{R\in \mathcal P(\widehat \Omega ):\E _R\Upsilon ^r\le M,\ \E _R\varphi (D_H)\le C_+,\ \E _R\psi (D_H)\le C_-\},\nonumber \\ \mathcal A_M&=\{R\in \mathcal P_M:\E _RW=0\ \forall W\in \mathcal W\}. \label {eq:finite-classes} \end{align}

Inf-compactness of $\Upsilon$ on $\widehat \Omega$ already forces $\Upsilon \to \infty$ as any $D_j$ tends to $0$ or $\infty$ ; the two band rows are separate affine inequalities with their own levels. For a law on this carrier, set

\begin{equation} \widehat T_J(R):=\bigl (\E _R\Phi _J,\E _R[D_H\Psi _J]\bigr ). \label {eq:pathwise-map} \end{equation}

Assume that some $R\in \mathcal A_M$ satisfies all three caps in (28) strictly, and that, for some $C<\infty$ ,

\begin{equation} \|\Phi _J\|+D_H\|\Psi _J\|+ \sum _{j<m}\bigl (\|R_j^Q\|+|D_{j+1}-D_j| +D_H\|R_j^P\|\bigr )\le C\Upsilon . \label {eq:pathwise-envelope} \end{equation}

The likelihood weight now appears explicitly because $D_H$ is no longer bounded. For example, by Young’s inequality, (30) follows from separate power dominations $D_H^{p}\le c\Upsilon$ and $\|\Psi _J\|^{q}+\|R_j^P\|^{q}\le c\Upsilon$ with $1/p+1/q=1$ , together with the analogous $\Upsilon$ -domination of $\Phi _J$ , $R_j^Q$ , and the density increments $D_{j+1}-D_j$ .

Proposition 5.4 (Pathwise finite certificate) . If $y=(q_J,p_J)$ lies outside $\widehat T_J(\mathcal A_M)$ , there are $a=(a_Q,a_P)$ , multipliers $\eta ,\eta _+,\eta _-\ge 0$ , $b\in \R$ , and $W\in \mathcal W$ such that

\begin{align} b+\eta \Upsilon ^r+\eta _+\varphi (D_H)+\eta _-\psi (D_H)+W &\ge a_Q^\top \Phi _J+D_Ha_P^\top \Psi _J &&\text {on }\widehat \Omega , \label {eq:domination}\\ b+\eta M+\eta _+C_++\eta _-C_- &<a_Q^\top q_J+a_P^\top p_J. \label {eq:gap} \end{align}

Conversely, such a tuple proves infeasibility. Only finitely many conditional rows occur in $W$ . For the almost-sure band, restrict the carrier to $D_j\in [\ell _-,\ell _+]$ , drop the two band rows, and set $\eta _\pm =0$ ; the statement and proof are then those of the bounded case.

Proof.Inf-compactness and $r>1$ make $\{\E _R\Upsilon ^r\le M\}$ weakly compact; the two band rows are weakly closed because $\varphi \circ D_H$ and $\psi \circ D_H$ are nonnegative and lower semicontinuous, so $\mathcal P_M$ is compact and convex. Condition (30) makes every displayed expectation, including the $D_H$ -weighted ones, uniformly integrable and hence continuous, so $\mathcal A_M$ is compact and convex. Thus $\widehat T_J(\mathcal A_M)$ is compact and convex, and strict separation gives $a$ with a positive support gap. With $G=a_Q^\top \Phi _J+D_Ha_P^\top \Psi _J$ , let

\begin{equation*} \mathcal N_M:=\{W\in \mathcal W:\E _RW=0\text { for every } R\in \mathcal P_M\}. \end{equation*}

Give the Hausdorff quotient

\mathcal W/\mathcal N_M

the locally convex topology generated by the evaluation maps

[W]\mapsto \E _RW

R\in \mathcal P_M

. Sion’s theorem, with the compact convex class

\mathcal P_M

as its compact argument, gives

\begin{equation} \sup _{R\in \mathcal A_M}\E _RG =\inf _{W\in \mathcal W}\sup _{R\in \mathcal P_M}\E _R(G-W). \label {eq:minimax} \end{equation}

For an inadmissible law, scaling a violated gain makes the inner infimum

-\infty

; an admissible law annihilates every gain. Sion’s theorem applies because every expectation is finite and continuous under (30) [50].

For fixed $W$ , put $F=G-W$ . The envelope gives $|F|\le C_W\Upsilon$ . For caps $(s,s_+,s_-)$ , define ∎

\begin{equation*} H_W(s,s_+,s_-):=\max \{\E _RF:R\in \mathcal P(\widehat \Omega ), \ \E _R\Upsilon ^r\le s,\ \E _R\varphi (D_H)\le s_+,\ \E _R\psi (D_H)\le s_-\}. \end{equation*}

The three constraints are affine in

R

, so

H_W

is concave and nondecreasing in each argument, and it is finite and attained wherever its feasible set is nonempty, by the compactness just established. The Slater law places

(M,C_+,C_-)

in the interior of the domain, so

H_W

has a supergradient

(\eta ,\eta _+,\eta _-)\ge 0

there. For any

\widehat \omega \in \widehat \Omega

the Dirac law is feasible at the caps

(\Upsilon ^r,\varphi (D_H),\psi (D_H))(\widehat \omega )

, so the supergradient inequality gives

\begin{equation*} F(\widehat \omega )\le H_W(M,C_+,C_-) +\eta \bigl (\Upsilon ^r(\widehat \omega )-M\bigr ) +\eta _+\bigl (\varphi (D_H(\widehat \omega ))-C_+\bigr ) +\eta _-\bigl (\psi (D_H(\widehat \omega ))-C_-\bigr ). \end{equation*}

Taking the supremum over

\widehat \omega

yields the reverse of weak duality, hence

\begin{equation} \sup _{R\in \mathcal P_M}\E _R(G-W) =\inf _{\eta ,\eta _\pm \ge 0}\Bigl \{\eta M+\eta _+C_++\eta _-C_-+ \sup _{\widehat \omega } \bigl [G-W-\eta \Upsilon ^r-\eta _+\varphi (D_H)-\eta _-\psi (D_H)\bigr ] (\widehat \omega )\Bigr \}. \label {eq:moment-dual} \end{equation}

The multiplier infimum in (34) is attained by the supergradient just constructed. The outer infimum over

W

in (33) need not be attained; nevertheless the strict support gap allows a choice of

W

, followed by its attaining multipliers, whose value lies inside that gap. Taking

b

equal to the inner supremum gives (31)–(32). Conversely, integrating (31) under a calibrating law annihilates

W

, uses the three caps, and contradicts (32).

The band multipliers $\eta _\pm$ are the finite-grid form of the statement that the band is itself a row: a certificate may charge part of the violation to the divergence constraint, and the pair $(\eta _+,\eta _-)$ records how much. When both vanish, the target is infeasible for reasons that no widening of the band can repair.

The witness is mathematical, not automatically a traded arbitrage: physical rows and the coercive penalty need not be traded claims. A literal arbitrage interpretation requires a separate superhedging implementation and, with it, the standard no-free-lunch machinery relating admissible strategies to equivalent martingale measures [22]. This is a substantive difference from the pricing-side certificates of Guyon [29], which are traded portfolios.

5.3What witnesses look like

Example 5.5 (Two-state band obstruction) . Let $\Omega _0=\{u,d\}$ , with pricing target $Q\{u\}=q=1/2$ and physical target $P\{u\}=p=0.9$ . Both are valid probability laws and are equivalent, but their unique density is

\begin{equation} L(u)=p/q=1.8,\qquad L(d)=(1-p)/(1-q)=0.2. \label {eq:toy-density} \end{equation}

They are incompatible with

\ell _-=0.5

and

\ell _+=1.5

. The explicit separator is

\begin{equation} (L-\ell _+)\1_{\{u\}}\le 0,\qquad p-\ell _+q=0.15>0. \label {eq:toy-separator} \end{equation}

Compatibility is restored exactly when

\ell _+\ge 1.8

and

\ell _-\le 0.2

. The obstruction is joint: neither marginal law is invalid.

The same unique density has $\E _Q[L^2]=41/25$ and $\E _Q[L^{-1}]=25/9$ . By Lemma 2.7, the maximal risk-neutral and physical Sharpe ratios of the only candidate bridge are therefore $4/5$ and $4/3$ ; the latter can be read directly from the single payoff $\1_{\{u\}}$ , whose physical excess mean is $0.4$ and physical standard deviation $0.3$ . A good-deal cap on the physical Sharpe ratio below $4/3$ is an obstruction of the same kind under the moment band, with the band row itself as the violated row.

Example 5.6 (Support failure) .Let $A$ be a stress event. Suppose the pricing carrier forces $Q(A)=0$ , whereas a separately fitted physical law requires $P(A)=\varepsilon >0$ . Then

\begin{equation} \E _Q[L\1_A]=0<\varepsilon , \label {eq:support-witness} \end{equation}

because

L\1_A=0

Q

-a.s. No

P\ll Q

can link the two, under any band. Enlarging a parameterization without enlarging the pricing support cannot fix the problem.

Proposition 5.7 (Bracket and covariance-characteristic invariance) . Let $P\sim Q$ on a common completed filtration and let $Y$ be a continuous vector semimartingale under both laws. Its matrix quadratic covariation $[Y]$ is the same under $P$ and $Q$ , up to indistinguishability under either law. In particular, if predictable positive-semidefinite processes $a^P$ and $a^Q$ satisfy

\begin{equation} [Y]_t=\int _0^t a_s^P\,ds\quad (P), \qquad [Y]_t=\int _0^t a_s^Q\,ds\quad (Q), \label {eq:covariance-characteristic} \end{equation}

then

a^P=a^Q

dt\otimes P

-a.e. and

dt\otimes Q

-a.e. Diffusion factorizations need only obey

\sigma _t^P(\sigma _t^P)^\top =\sigma _t^Q(\sigma _t^Q)^\top

; the factors themselves need not coincide.

Proof.Quadratic covariation is determined pathwise by the coordinate trajectories, with cross terms obtained by polarization. Equivalent laws have the same null sets, so their quadratic-covariation versions coincide. Uniqueness of the Lebesgue density of the resulting finite-variation matrix measure gives the equality of $a^P$ and $a^Q$ . Equality of diffusion covariances follows from the quadratic-covariation formula; predictable orthogonal rotations show why equality of chosen square roots is unnecessary.∎

Lemma 5.8 (Finite Gaussian-mixture drift shifts) . Let

f(z)=\sum _{k=1}^K w_k\frac {1}{\sigma _k\sqrt {2\pi }} \exp \!\left (-\frac {z^2}{2\sigma _k^2}\right ), \qquad w_k>0,\quad \sigma _k>0,\quad \sum _kw_k=1,

and, for a fixed shift

\delta

, put

\ell _\delta (z)=f(z-\delta )/f(z)

. If

Z

has density

f

, then

\E [\ell _\delta (Z)]=1

and

\begin{equation} \E [\ell _\delta (Z)^\alpha ]<\infty \quad \text {for every }\alpha \in \R . \label {eq:mixture-all-moments} \end{equation}

\delta \ne 0

, the essential infimum of

\ell _\delta (Z)

is zero and its essential supremum is infinite. Consequently, for independent innovations and any fixed finite sequence of shifts, the product likelihood has moments of every real order; if at least one shift is nonzero, it has the same essential-extrema property. The moment conclusion also holds for predictable history-dependent shifts whose absolute values are uniformly bounded over the finite horizon.

Proof.The likelihood normalization follows by translation. Since $(\log f)'(z)$ is a convex combination of the numbers $-z/\sigma _k^2$ , the mean-value theorem gives, uniformly for $|\delta |\le d$ , ∎

|\log f(z-\delta )-\log f(z)|\le C_d(1+|z|).

A finite Gaussian mixture has every exponential moment of

|Z|

, proving (39), uniformly for

|\delta |\le d

. If

\sigma _*:=\max _k\sigma _k

, Gaussian tail dominance yields

\log \ell _\delta (z)=\delta z/\sigma _*^2+O(1)

z\to \pm \infty

; the two tails therefore give zero and infinity when

\delta \ne 0

. Independence proves the fixed-shift product statements, and conditioning backwards with the uniform one-step moment bound proves the last assertion.

Remark 5.9 (The same invariance is a bridge and an obstruction) . Proposition 5.7 is not a technicality peripheral to applied practice; it is the hypothesis on which the leading empirical attempt in this market rests. Guyon and Parent [31] justify comparing a $P$ -estimated path-dependent volatility model with option data precisely by observing that, in a continuous-time diffusive setting, an equivalent change of measure does not alter the volatility coefficient, so a model estimated from returns should already approximate the pricing dynamics up to drift. That is the present proposition, read forwards.

Read backwards it is a constraint. The invariance that licenses the comparison also forbids independently prescribed $P$ - and $Q$ -covariance characteristics on one carrier: a two-stage workflow that fits a pricing model with characteristic $a^Q$ and then, separately, a physical model with $a^P\ne a^Q$ has produced an infeasible pair, whatever residual its optimizer reports. The two readings are the same theorem. Which one is operative depends on whether the volatility coefficient is estimated once and shared, or twice and reconciled afterwards.

The empirical comparison in Guyon and Parent [31] estimates model parameters separately and does not itself construct an equivalent linked pair or its Radon–Nikodym derivative. A deliberately linked, fixed-parameter drift-shift bridge built with their finite Gaussian-mixture innovation would instead fall under Lemma 5.8, provided the standardized shifts satisfy its hypotheses. Such a bridge is excluded from every almost-sure band when a shift is nonzero, while it belongs to the moment and entropy bands at sufficiently large levels. The two properties do different work: unbounded support of $\log L$ rules out an almost-sure band, whereas moment or entropy membership additionally requires the stated integrability. Neither property of an actual empirical $P/Q$ pair may be inferred before that pair and its common innovation law have been constructed.

Example 5.10 (Quadratic-variation obstruction) .Suppose the pricing carrier requires $A_H=\alpha _QH$ almost surely for a scalar covariance rate $\alpha _Q\ge 0$ , while a continuous-path physical target requires

\begin{equation} \E _P[(A_H-\alpha _QH)^2]=\delta >0. \end{equation}

Every equivalent bridge on the carrier instead has

\E _P[(A_H-\alpha _QH)^2] =\E _Q[L(A_H-\alpha _QH)^2]=0

. More generally, Proposition 5.7 rules out independently prescribed

P

- and

Q

-covariance characteristics. It does not require literal equality of diffusion factorizations:

\sigma _t

and

\sigma _tO_t

give the same covariance for a predictable orthogonal rotation

O_t

Remark 5.11 (Drift changes and compatibility distance) .Suppose, in addition, that a common Brownian filtration and a density representation have been verified. With one fixed sign convention for the density, Girsanov then gives a drift displacement of the form

\begin{equation} b_t^P-b_t^Q=\sigma _t\theta _t. \label {eq:girsanov-range} \end{equation}

Hence even the drift displacement is restricted to the range of

\sigma _t

; it is not a free coordinate. The stochastic exponential must be a true martingale and satisfy the integrability and likelihood conditions of the declared bridge class. Neither a formal choice of

\theta

nor a local-martingale density establishes an admissible linked pair.

An isolated projection of covariance estimates onto equality is elementary only after both estimates have been aligned on the same state, clock, and filtration. For example, under a declared weighted Hilbert metric, positive weights $w_P,w_Q$ , and aligned covariance characteristics $a^P,a^Q$ , the pointwise diagonal projection is

\begin{equation} \bar a_t =\operatorname *{argmin}_{a=a^\top \succeq 0} \left \{w_P\|a-a_t^P\|_F^2+w_Q\|a-a_t^Q\|_F^2\right \} =\frac {w_Pa_t^P+w_Qa_t^Q}{w_P+w_Q}. \label {eq:weighted-covariance-projection} \end{equation}

The formula projects

a

, not a chosen square root

\sigma

, and changes with the metric and weights. Separately fitted

P

- and

Q

-models may not possess a common state alignment at all, in which case the pointwise formula is undefined. The useful global object is instead a distance, under a declared metric, from the complete target to the compatible data/quote image

\cY _{\rm feas}=\Gamma (\cK )

. Computing that distance must enforce jointly the support, likelihood, raw-filtration, price, bracket, and other structural rows defining

\cK

. No such numerical or empirical projection is claimed here.

Example 5.12 (Why ordinary weak convergence is insufficient) .Let $U$ be symmetric on $\{-1,1\}$ , $Y_0^n=U/n$ , and $Z_1^n=1+U$ . Then $Y_0^n$ reveals $U$ and $\E [Z_1^n\mid Y_0^n]=1+U$ . In the weak limit $Y_0=0$ , its natural sigma-field is trivial and the corresponding conditional mean is $1$ . Marginal weak convergence has lost information. This motivates adapted row closure in Assumption 2.5.

6Inference on the identified set

This section answers the third question: what a nonempty fibre says about a physical quantity. The result is an identified set; without further likelihood-class restrictions, finitely many physical moments need not make it a point (Proposition 6.1). Its projection onto any continuous affine readout is an exact attained interval (Section 6.1); and, whenever the declared band family contains a compatible member, the band is the hinge at which the fibre appears, with an attained minimal width (Section 6.2).

Proposition 6.1 (Finite physical moments do not identify a likelihood) . Let $Q$ be atomless and $f_1,\ldots ,f_m$ bounded. There are distinct bounded positive normalized likelihoods $L^+,L^-$ such that

\begin{equation} \E _Q[L^+f_i]=\E _Q[L^-f_i],\qquad i=1,\ldots ,m. \label {eq:same-moments} \end{equation}

They may be chosen inside any band of Definition 2.1 that has slack at

L\equiv 1

: for some

\eta >0

\varphi \le C_+

and

\psi \le C_-

[1-\eta ,1+\eta ]

. For the almost-sure band this holds whenever

\ell _-<1<\ell _+

; for the moment band it is the strict slack

C_+>1

and

C_->1

, and for the entropy band it is

C_+>0

and

C_->0

Proof.Choose a nonzero bounded function $g$ orthogonal in $L^2(Q)$ to the finite span of $1,f_1,\ldots ,f_m$ . Such a bounded simple function exists by partitioning an atomless space into more positive-mass cells than the number of restrictions. For small $\varepsilon >0$ , put $L^\pm =1\pm \varepsilon g$ . Orthogonality to constants normalizes the two densities, and orthogonality to each $f_i$ gives (43). For $0<\varepsilon <\min \{\eta ,1\}/\|g\|_\infty$ both densities take values in $[1-\eta ,1+\eta ]$ , so they are positive and satisfy $\varphi (L^\pm )\le C_+$ and $\psi (L^\pm )\le C_-$ pointwise, hence the band.∎

The slack condition cannot be dropped. At zero slack a strictly convex row is at its Jensen minimum: a zero entropy cap, or a moment cap $\E _Q[L^p]\le 1$ , forces $L=1$ almost surely, so the likelihood is identified and $P=Q$ . A physical readout then equals its $Q$ counterpart and is identified by the pricing rows only when those rows determine that counterpart. Proposition 6.1 shows that, absent further class restrictions on the likelihood, a band containing an $L^\infty$ neighborhood of $1$ leaves directions invisible to every finite moment panel; a particular declared class $\cK$ can still eliminate those directions.

The proposition separates compatibility, selection, and identification. A nonempty fibre answers the first question. Entropy or another criterion chooses a point for the second. Only sufficient observations plus a valid inverse argument answer the third.

6.1Exact identified intervals

We next give the identified set of a readout, in the sense of partial identification [16], and separate it from the simpler band envelope. Let $G:\cK \to \R$ be any continuous affine readout and, for a feasible target $y$ , put

\begin{equation} \underline G(y):=\min _{R\in \mathfrak F(y)}G(R),\qquad \overline G(y):=\max _{R\in \mathfrak F(y)}G(R). \label {eq:exact-readout-endpoints} \end{equation}

Proposition 6.2 (Exact compact-fibre identified interval) . The identified set of $G$ is exactly the attained compact interval

\begin{equation} \{G(R):R\in \mathfrak F(y)\} =[\underline G(y),\overline G(y)]. \label {eq:identified-range} \end{equation}

For a finite panel

T_J(R)=y_J

, define the augmented support oracle

\begin{equation} h_{J,G}(a,c):=\max _{R\in \cK }\{a^\top T_J(R)+cG(R)\}. \label {eq:augmented-support} \end{equation}

Then its sharp slice endpoints obey

\begin{align} \overline G_J(y_J) &=\inf _{a\in \R ^{d_J}} \{h_{J,G}(a,1)-a^\top y_J\},\nonumber \\ \underline G_J(y_J) &=-\inf _{a\in \R ^{d_J}} \{h_{J,G}(a,-1)-a^\top y_J\}. \label {eq:slice-support-endpoints} \end{align}

The infima in (47) need not be attained.

Proof.The fibre is nonempty, compact, and convex. Its continuous affine image in $\R$ is therefore an attained compact interval. For the upper formula, weak duality gives $G(R)\le h_{J,G}(a,1)-a^\top y_J$ on the slice. Conversely, strictly separate $(y_J,\overline G_J(y_J)+\varepsilon )$ from the compact convex augmented image $\{(T_J(R),G(R)):R\in \cK \}$ . The coefficient of the final coordinate is positive because $(y_J,\overline G_J(y_J))$ belongs to that image; normalize it to one and let $\varepsilon \downarrow 0$ . Apply the same argument to $-G$ for the lower formula.∎

For a readout $f$ with $0\le f\le 1$ whose $Q$ - and $P$ -readout rows are continuous, take $G_f(R)=\E _R[Lf]=\E _{P_R}f$ and write $\underline q_f=\min _{R\in \mathfrak F(y)}\E _Rf$ and $\overline q_f=\max _{R\in \mathfrak F(y)}\E _Rf$ . The band alone gives only an outer envelope: under the almost-sure band,

\begin{equation} [\underline G_f(y),\overline G_f(y)] \subseteq [\ell _-\underline q_f,\ell _+\overline q_f], \label {eq:band-envelope} \end{equation}

and under the moment band, by the Hölder argument of (15),

\begin{equation} [\underline G_f(y),\overline G_f(y)] \subseteq \bigl [\underline q_f^{\,(r+1)/r}C_-^{-1/r},\ C_+^{1/p}\,\overline q_f^{\,1-1/p}\bigr ]. \label {eq:moment-envelope} \end{equation}

If the quote rows fix $\E _Qf=c_f$ , the first reduces to $[\ell _-c_f,\ell _+c_f]$ , so positive physical means differ by at most $\ell _+/\ell _-$ . These band endpoints need not themselves be attainable; the exact interval is (45) or, for a finite panel, (47), and it is sharp under every band.

Conditional forecasts require a different optimization.

Proposition 6.3 (Sharp conditional cell range) . Let $w\ge 0$ be a bounded $\cF _t^0$ -measurable history weight and let $f$ be a bounded future readout. Suppose

\begin{equation*} n(R):=\E _R[Lwf],\qquad d(R):=\E _R[Lw] \end{equation*}

are continuous affine rows and

\inf _{R\in \mathfrak F(y)}d(R)\ge \delta >0

. Then the exact identified set of the cell forecast

\begin{equation} m_{f,w}(R):=\frac {n(R)}{d(R)} =\frac {\E _{P_R}[wf]}{\E _{P_R}w} \label {eq:conditional-cell} \end{equation}

is an attained interval

[\underline m_{f,w},\overline m_{f,w}]

. Its upper endpoint is the unique zero of

\begin{equation} \phi ^+(c):=\max _{R\in \mathfrak F(y)}\{n(R)-cd(R)\}, \label {eq:fractional-upper} \end{equation}

and its lower endpoint is the unique zero of

\begin{equation} \phi ^-(c):=\min _{R\in \mathfrak F(y)}\{n(R)-cd(R)\}. \label {eq:fractional-lower} \end{equation}

Thus each trial value of

c

requires an affine support-oracle evaluation. The zero characterization is the parametric fractional-programming device of Dinkelbach [23]; one-dimensional bisection is one valid root solver. For a finite-panel fibre, each oracle evaluation has the dual representation (47) with

G=n-cd

Proof.The denominator bound makes $m_{f,w}$ continuous on the compact fibre, so its extrema are attained; its image is an interval because a ratio of affine functions with positive denominator maps every line segment continuously and monotonically between its endpoint ratios. For any $R$ , $n(R)-cd(R)=d(R)(m_{f,w}(R)-c)$ . At the maximal ratio the maximum in (51) is zero; it is positive below and, using $d\ge \delta$ , strictly negative above. The lower formula is identical with minimum in place of maximum. The objectives $n-cd$ are affine, so the last claim follows from Proposition 6.2.∎

For $w=\1_C$ and a history cell $C$ whose $Q$ -probability is fixed at $q_C>0$ , (50) is $\E _{P_R}[f\mid C]$ , and the positive-denominator hypothesis holds under every band of Definition 2.1. The almost-sure band supplies $d(R)\ge \ell _-q_C$ and the moment band supplies $d(R)\ge q_C^{(r+1)/r}C_-^{-1/r}$ by (15). For a general band put $\psi _*(\varepsilon ):=\inf _{0\le x<\varepsilon }\psi (x)$ , which tends to infinity as $\varepsilon \downarrow 0$ by (4); Markov’s inequality gives $Q(L<\varepsilon )\le C_-/\psi _*(\varepsilon )$ , hence

\begin{equation} d(R)\ \ge \ \sup _{\varepsilon >0:\,\psi _*(\varepsilon )>0}\ \varepsilon \Bigl (q_C-\frac {C_-}{\psi _*(\varepsilon )}\Bigr )\ >\ 0. \label {eq:general-denominator} \end{equation}

A point-history forecast $\E _{P_R}[f\mid \cF _t^0](\omega )$ additionally requires a common version and continuity of evaluation at that history; a band on the terminal likelihood alone does not provide either condition. Proper scoring, selector choice, and prequential evaluation are separate statistical questions.

Remark 6.4 (Relation to sensitivity analysis under a bounded density ratio) . The device of this section is a new application, to a martingale-constrained law class, of sensitivity analysis under a bounded density ratio. A two-sided bound on a density ratio between a target law and a reference law, used to convert a point estimand into an identified interval, is analogous to the marginal sensitivity model of Tan [52], an extension of the propensity-odds sensitivity model of Rosenbaum [45]. The causal model bounds a conditional propensity-odds ratio, whereas our global, generally asymmetric band bounds $dP/dQ$ itself. If the latter is parameterized symmetrically as $L\in [\Lambda ^{-1},\Lambda ]$ , its gain–loss endpoint ratio is $\Lambda ^2$ , not $\Lambda$ . In that literature the interval for a mean is obtained by linear-fractional programming [54], and the sharp version of the bound is due to Dorn and Guo [24]. The one-dimensional root-finding step in Proposition 6.3 uses the parametric zero characterization of Dinkelbach [23]. For a finite-dimensional linear-fractional program, the Charnes–Cooper substitution is a distinct one-shot LP reformulation [13]. Readers who know that literature will recognize Propositions 6.2 and 6.3 as its instance on the compatible fibre.

What is new is the constraint set. In sensitivity analysis the reweighting is constrained only by covariate-wise normalization, so each bound is a sort or an explicit quantile-balancing formula. Here the reweighted law must simultaneously remain in $\cK$ : one raw filtration, a single additive forward-variance budget, true-martingale spot and surface rows, settlement identities, and full-history physical conditional rows. Every evaluation of the support oracle in (46) is therefore a martingale-constrained optimization over an adapted-compact law class rather than a pointwise reweighting, and the resulting interval is a statement about the compatible fibre, not about a reweighting of a fixed sample. The transplant is what makes the sensitivity parameter interpretable as a gain–loss cap or a maximal Sharpe ratio rather than as unmeasured confounding. The divergence bands of Definition 2.1 likewise have a sensitivity-analysis counterpart: the $f$ -sensitivity model of Jin et al. [38] constrains an $f$ -divergence generated by the relevant likelihood ratio, and its counterfactual-mean bounds admit distributionally robust dual formulations. The global moment and entropy rows here are finance-side analogues, not the same causal restriction.

6.2The minimal compatible divergence

Band sensitivity is meaningful only if the nonband ingredients are held fixed. Accordingly, let $\cK ^0$ be one compact parent class satisfying the structural, topology, envelope, and row-continuity requirements above with the band rows omitted, and let $\{(\varphi _\theta ,\psi _\theta ,C_+(\theta ),C_-(\theta )):\theta \in \Theta \}$ be a family of bands indexed by a compact parameter set $\Theta \subset \R ^d$ , with $(\theta ,x)\mapsto \varphi _\theta (x)$ and $(\theta ,x)\mapsto \psi _\theta (x)$ jointly lower semicontinuous, with levels $C_\pm :\Theta \to [0,\infty )$ continuous, and nested in a specified partial order: $\theta \preceq \theta '$ implies that every law satisfying the band at $\theta$ satisfies it at $\theta '$ . For the almost-sure band, $\theta =(a,b)$ ranges over $[\underline \ell ,1]\times [1,\overline \ell ]$ , where $0<\underline \ell \le 1\le \overline \ell <\infty$ , and $(a,b)\preceq (a',b')$ means $a'\le a$ and $b\le b'$ . For the moment band, $\theta =(C_+,C_-)$ ranges over a compact box contained in $[1,\infty )^2$ , with the usual product order; for the entropy band it ranges over a compact box contained in $[0,\infty )^2$ , again with the usual product order. Define

\begin{equation} \mathfrak F^0_{\theta }(y) :=\{R\in \cK ^0:\Gamma (R)=y,\ \E _R[\varphi _\theta (L)]\le C_+(\theta ),\ \E _R[\psi _\theta (L)]\le C_-(\theta )\}, \label {eq:band-fibre} \end{equation}

so that

\begin{equation} \theta \preceq \theta '\quad \Longrightarrow \quad \mathfrak F^0_{\theta }(y)\subseteq \mathfrak F^0_{\theta '}(y). \label {eq:nested} \end{equation}

Fix a continuous width $\rho :\Theta \to [0,\infty )$ , nondecreasing in the specified order. For the almost-sure band, $\rho (a,b)=b/a$ . For the moment band with $(p,r)=(2,1)$ and the $\varphi$ -level held fixed, take a compact slice $C_-\in [1,\overline C_-]$ and the natural Sharpe width $\rho (C_-)=\sqrt {C_--1}$ . The domain starts at one because every feasible equivalent pair satisfies $\E _Q[L^{-1}]\ge 1$ by Jensen’s inequality. For the entropy band, vary the selected level from its Jensen lower bound, either $C_-\in [0,\overline C_-]$ or $C_+\in [0,\overline C_+]$ , while holding the opposite level fixed.

Proposition 6.5 (Minimal compatible divergence) . For a fixed target $y$ , define

\begin{equation}\label {eq:minimal-band} \kappa _*(y):=\inf \left \{\rho (\theta ): \theta \in \Theta ,\quad \mathfrak F^0_{\theta }(y)\ne \varnothing \right \}, \end{equation}

with the convention

\inf \varnothing =+\infty

. Whenever

\kappa _*(y)<+\infty

, this infimum is a minimum and is attained. For a compact uncertainty set

\mathcal B

of target vectors, define

\kappa _*(\mathcal B)

analogously after replacing

\Gamma (R)=y

\Gamma (R)\in \mathcal B

; whenever finite, its infimum is likewise attained.

Proof.The likelihood coordinate is continuous on the canonical carrier and $(\theta ,x)\mapsto \varphi _\theta (x)$ is nonnegative and lower semicontinuous, so $(\theta ,R)\mapsto \E _R[\varphi _\theta (L)]$ is lower semicontinuous on $\Theta \times \cK ^0$ by the Portmanteau theorem applied to the joint laws $\delta _{\theta _n}\otimes R_n$ ; likewise for $\psi _\theta$ . Since $C_\pm$ are continuous, $(\theta ,R)\mapsto \E _R[\varphi _\theta (L)]-C_+(\theta )$ and its $\psi$ counterpart are lower semicontinuous, so the set of $(\theta ,R)$ satisfying both band rows is closed. Upper semicontinuity of $C_\pm$ would suffice here; continuity is what the instances provide. Intersecting with the closed row constraints gives a compact feasible graph inside $\Theta \times \cK ^0$ , whose projection onto $\Theta$ is compact, possibly empty. If it is nonempty, the continuous function $\rho$ attains its infimum there; if it is empty, the stated convention gives $+\infty$ . Adding a compact target uncertainty set preserves the same argument.∎

Remark 6.6 (Three readings of $\kappa _*$ ) . Under the almost-sure band, when finite, $\kappa _*$ is the minimal gain–loss ratio that restores compatibility; in Example 5.5, if the widest envelope contains the unique density, $\kappa _*=1.8/0.2=9$ . Under the moment band with $(p,r)=(2,1)$ , fix the opposite cap $\overline C_+$ and write

\begin{equation} \mathfrak F^0_\varphi (y) :=\{R\in \cK ^0:\Gamma (R)=y,\ \E _R[L^2]\le \overline C_+\}. \label {eq:phi-fibre} \end{equation}

Extend

x\mapsto x^{-1}

0^{-1}=+\infty

. For the compact slice

C_-\in [1,\overline C_-]

described above, whenever

\kappa _*(y)

is finite,

\begin{equation} \kappa _*(y) =\min _{R\in \mathfrak F^0_\varphi (y)} \sqrt {\E _R[L^{-1}]-1}. \label {eq:kappa-sharpe} \end{equation}

Indeed, the objective in (58) is extended lower semicontinuous, and (57) is compact, so the displayed minimum is attained. Any feasible point of the compact

C_-

slice shows that the minimizing inverse moment is at most

\overline C_-

; choosing

C_-=\E _R[L^{-1}]

at a minimizing law then proves the equality with the band-width problem. Finiteness forces

L>0

R

-a.s.; the minimizing law therefore defines an equivalent pair, and Lemma 2.7 identifies its objective with

\sup _{\substack {X\ \mathrm {bounded}\\\sigma _{P_R}(X)>0}} \frac {\E _{P_R}[X]-\E _{Q_R}[X]}{\sigma _{P_R}(X)},

with the empty-class convention of that lemma. Thus (58) is the smallest maximal physical Sharpe ratio that any compatible pricing kernel satisfying the fixed opposite cap must admit. This is the good-deal bound of Cochrane and Saá-Requejo [17] read in reverse, with the option surface and the physical rows as inputs and the Sharpe cap as output. In Example 5.5 it equals

4/3

, because conditional randomization of

L

given the state cannot lower

\E _Q[L^{-1}]

, by Jensen’s inequality. Under the entropy slices anchored at zero above, with

\rho =C_-

, whenever finite,

\kappa _*

is the minimal

H(Q_R\,|\,P_R)

over the compatible fibre, the orientation of entropic calibration; with

\rho =C_+

, whenever finite, it is the minimal

H(P_R\,|\,Q_R)

. Either is a Schrödinger-type value for the linked pair. The three thresholds are different numbers selecting different fibre points, as Example 6.9 shows.

Remark 6.7 (The bridge literature as a special case) . Under the entropy slice $C_-\in [0,\overline C_-]$ with $\rho =C_-$ , an exact reduction to a standard entropic calibration requires more than point identification of the physical rows. Let $\Omega ^{\rm mkt}$ denote the entire common declared market-history carrier, before the pair-dependent coordinates $L$ and $D$ are adjoined. Assume that the physical restrictions fix one full law $P$ on $\Omega ^{\rm mkt}$ , not merely finitely many of its moments, and let $\mathcal Q$ be the intended pricing-feasible class on that same carrier. For each candidate $Q$ , the relevant object is its canonical linked lift, which appends $L=dP/dQ$ and its density process. Suppose the $Q$ -projection of every relevant lift in $\cK ^0$ belongs to $\mathcal Q$ , and an ordinary minimizer

Q^\dagger \in \arg \min _{Q\in \mathcal Q}H(Q\,|\,P)

exists, is equivalent to

P

, admits a linked lift in

\cK ^0

satisfying all the nonband rows, and obeys the fixed opposite cap

H(P\,|\,Q^\dagger )<\overline C_+

and the slice bound

H(Q^\dagger \,|\,P)\le \overline C_-

. If, conversely, every

Q\in \mathcal Q

has a canonical lift in

\cK ^0

satisfying all nonband rows, the fixed opposite cap, and

H(Q\,|\,P)\le \overline C_-

, the two feasible classes agree exactly; for equality of the minimum it is enough that the displayed minimizer has such a lift. Under these conditions the opposite cap is nonbinding at the standard solution and

\kappa _*=\min _{Q\in \mathcal Q}H(Q\,|\,P).

This is entropic calibration to the option surface with the historical law as prior, the martingale Schrödinger-bridge orientation [2, 30, 42]. The present framework includes that corner under the stated lifting conditions, while also allowing

P

to be set-valued, a moment rather than entropy band, and existence or range questions rather than selection. Existing entropic calibration machinery computes the present

\kappa _*

only when it enforces the declared parent class, every nonband row, and the fixed opposite cap.

The threshold is class-relative but not ambiguous: all nonband conditions come from the single parent $\cK ^0$ . Enlarging the band increases every support value and weakly decreases every fixed-direction separator gap. A reported empirical threshold must also disclose $\cK ^0$ , the band family and width $\rho$ , the target uncertainty set, and the normalization of $L$ .

Remark 6.8 (What $\kappa _*$ would measure on a reported residual panel) . The intended use of $\kappa _*$ is not to adjudicate whether a joint fit succeeded, but to put a scalar on the gap that a successful-looking fit leaves behind. A concrete instance is available in the literature. Guyon and Parent [31] estimate a path-dependent volatility model under $P$ from index returns alone and compare its implied pricing quantities with a market panel across many dates. Their reported disagreement is small in aggregate but signed and structured: model implied volatilities below the market for short- to medium-dated out-of-the-money index options and above it at the longest maturity, and below the market for large-strike VIX options. They suggest this pattern may reflect a volatility tail risk premium.

Such a panel of signed residuals, together with bid–ask and sampling tolerances, can be encoded as a compact target uncertainty set $\mathcal B$ in the sense of Proposition 6.5 only after a common linked $P/Q$ carrier, its row map, and a parent class $\cK ^0$ have been specified. The empirical comparison in Guyon and Parent [31] does not itself construct that linked pair. Once those ingredients are supplied, and whenever $\kappa _*(\mathcal B)<+\infty$ , the proposition gives the attained smallest divergence level consistent with the panel inside the declared $\cK ^0$ ; under the moment band, by (58), it is a maximal Sharpe ratio. Reporting that number would replace a qualitative statement that two measures are close, or that a residual looks like a premium, with a disclosed quantity on a disclosed class, in units the good-deal literature already uses. We emphasize what this would and would not settle. It would not show that the residual is a risk premium rather than misspecification, and by Proposition 7.1 a threshold computed against one architecture is architecture-relative. It would bound how large a kernel distortion, in gain–loss, Sharpe-ratio, or entropy units, any compatible bridge must carry to reproduce the observed panel. Lemma 5.8 verifies the required moment integrability only for the deliberately linked fixed-parameter drift-shift construction described in Remark 5.9; it does not place the separately estimated empirical comparison inside a band. No such empirical computation is performed here. Besides Assumption 2.5 on a continuous carrier, it first requires the linked construction and every pricing and physical row. A fully specified finite-grid carrier, where compactness is direct, is one tractable route; Section 6.3 supplies only a synthetic proxy calculation.

Example 6.9 (An analytic bridge fibre with a strict band envelope) . On a one-date carrier let $Z\in \{-1,0,1\}$ have fixed pricing probabilities $Q\{Z=z\}=1/3$ . Hold all other structural coordinates fixed, impose $1/2\le L\le 3/2$ , and require the physical row $\E _PZ=\E _Q[LZ]=1/6$ . For any compatible lifted law define the conditional likelihood means

\lambda _z:=\E _Q[L\mid Z=z].

Normalization and the physical row give, for some scalar

s

\begin{equation} \lambda _{-1}=s,\qquad \lambda _0=\frac 52-2s,\qquad \lambda _1=s+\frac 12. \label {eq:three-state-likelihoods} \end{equation}

The band forces

s\in [1/2,1]

, and conversely every such point is attained by taking

L=\lambda _Z

. Hence the exact identified interval for the physical success probability is

\begin{equation} P\{Z=1\}=\frac {s+1/2}{3}\in \left [\frac 13,\frac 12\right ], \label {eq:three-state-range} \end{equation}

whereas the generic band envelope based only on

Q\{Z=1\}=1/3

is the strictly wider

[1/6,1/2]

. If the band is allowed to vary while the pricing and physical rows remain fixed, minimizing the largest conditional likelihood mean divided by the smallest in (59) gives

\begin{equation} \kappa _*=\frac 85, \qquad s=\frac 56, \qquad (\lambda _{-1},\lambda _0,\lambda _1) =\left (\frac 56,\frac 56,\frac 43\right ). \label {eq:three-state-minimal-band} \end{equation}

Randomizing

L

conditionally on

Z

cannot improve this ratio, because every admissible lower bound is no larger than the smallest conditional mean and every admissible upper bound is no smaller than the largest conditional mean.

The central moment and entropy orientations also select different points. For the Sharpe width of Remark 6.6,

\begin{equation} C_-(s)-1 =\E _Q[L^{-1}]-1 =\frac 13\left (\frac 1s+\frac 1{\frac 52-2s} +\frac 1{s+\frac 12}\right )-1. \label {eq:three-state-inverse-moment} \end{equation}

This strictly convex function has its unique minimum on

[1/2,1]

s=0.779360176199\ldots

, the unique root in that interval of

\begin{equation} 96s^4-288s^3+48s^2+60s+25=0, \label {eq:three-state-inverse-root} \end{equation}

where

\begin{equation} \min _s\{C_-(s)-1\}=0.042376014556\ldots , \qquad \min _s\sqrt {C_-(s)-1}=0.205854352774\ldots . \label {eq:three-state-sharpe} \end{equation}

At this point

C_+(s)=\E _Q[L^2]=1.043390706560\ldots

, so any fixed opposite moment cap strictly above that value is nonbinding. The mirror moment criterion

\begin{equation} C_+(s)-1=\operatorname {Var}_Q(L) =\tfrac 13\bigl (s^2+(\tfrac 52-2s)^2+(s+\tfrac 12)^2\bigr )-1 \label {eq:three-state-variance} \end{equation}

is instead minimized at

s=3/4

, where its value is

1/24

; at the gain–loss optimum

s=5/6

its value is

1/18

For the entropic-calibration orientation,

\begin{equation} H(Q\,|\,P)=-\frac 13\sum _z\log \lambda _z \label {eq:three-state-forward-entropy} \end{equation}

has its unique minimum at

\begin{equation} s=\frac 14+\frac {\sqrt {39}}{12} =0.770416499867\ldots , \qquad \min _s H(Q\,|\,P)=0.021056427117\ldots . \label {eq:three-state-forward-entropy-min} \end{equation}

There

H(P\,|\,Q)=\frac 13\sum _z\lambda _z\log \lambda _z =0.021045573240\ldots

, so a fixed reverse-entropy cap strictly above this value is nonbinding. The reverse-KL mirror

H(P\,|\,Q)

is itself minimized at the root

s=(21-\sqrt {141})/12=0.760471492747\ldots

s(s+\tfrac 12)=(\tfrac 52-2s)^2

. Conditional randomization of

L

given

Z

cannot lower any of these four criteria, by conditional Jensen applied to

x^2

x^{-1}

x\log x

, and

-\log x

. Thus the compact fibre, its exact readout interval, the loose band envelope, and the three central minimal widths—gain–loss, physical Sharpe, and

H(Q\,|\,P)

—are distinct objects even in three states; their opposite-orientation mirrors are distinct as well.

6.3A finite-grid illustration

The exact finite-carrier setting of Section 5.2 is computable as stated. The synthetic calculation below implements the lifted convex geometry and an exact discrete counterpart of all three structural martingale identities in Definition 2.2. It is a finite quoted panel rather than a complete continuous option surface, and it is not evidence about any market.

Setting. On a finite path set the substitution $p_i=r_iL_i$ represents the lift of Section 2: one optimizes jointly over two probability vectors on the same paths, $r$ the pricing weights and $p$ the physical weights. Every declared pilot pricing row is linear in $r$ , every physical row is linear in $p$ , and the prefix ratios define the discrete density process. The three bands become linear inequalities, rotated second-order cones, and exponential cones, respectively. The moment and entropy widths each require one conic program. Every finite conic value is recomputed from a residual-verified primal witness; its global optimality is based on solver status, without a separately checked dual certificate. The displayed recovery endpoints are likewise raw-primal-verified solver-status optima; no independently checked endpoint dual certificate is claimed. The gain–loss width is a bisection on $\kappa$ over linear programs after the substitution $\tilde r=ar$ . Numerically it uses the exact complete pricing-fibre reduction $N\tilde r=a(Nq^{\rm true})+B\gamma$ and $P=Np$ , then checks the reconstructed probabilities against every original row. Every finite reported gain–loss value is a verified-feasible upper bound on $\kappa _*$ , and the largest solver-reported-infeasible trial lies within $\pilotGLbracketMax$ of it.

The carrier is a non-recombining hybrid tree over six trading days with $m=6$ steps ( $\pilotN$ paths, $\pilotNodes$ nodes), on which volatility follows the four-factor path-dependent recursion of Guyon and Parent [31] at their published parameters. Its finite innovations are constructed for the present structural identities and are not their Gaussian mixture. At a node put $a=\sigma ^2\,dt$ , $\Delta A=a$ , $Y=\Delta N$ , and $\Delta X=Y-a/2$ . An ordinary node has $Y\in \{-h,0,h\}$ , where $h\in (\sqrt a,2\sqrt a)$ solves

\begin{equation} 1+\frac {a}{h^2}(\cosh h-1)=e^{a/2}, \label {eq:pilot-structural-rule} \end{equation}

with probabilities $a/(2h^2),1-a/h^2,a/(2h^2)$ . Consequently $\E _Q[Y]=0$ , $\E _Q[Y^2-a]=0$ , and $\E _Q[e^{Y-a/2}]=1$ exactly. One node at the final step instead has $Y=\pm c_k\sqrt a$ , $c=(0.2,1,2)$ , with positive symmetric pair masses $w_k$ solving $\sum _kw_k=\sum _kw_kc_k^2=1$ and $\sum _kw_k\cosh (c_k\sqrt a)=e^{a/2}$ . This node is chosen so the terminal call rows add at most one local rank. At every ordinary node, normalization and the local structural rows have rank three, so root-to-leaf induction pins all ordinary transitions. Once normalization and all structural and quoted rows are imposed, the remaining complete affine pricing fibre is $\{q^{\rm true}+B\theta :q^{\rm true}+B\theta \geq 0\}$ , where $B$ is supported on the special node and $\dim B=\pilotQNullity$ ; the archive contains an explicit distinct positive pricing law in this fibre. Thus $A$ is predictable and $S$ , $N$ , and $N^2-A$ are exact discrete martingales, while $Q$ is genuinely set-valued.

The pilot imposes $\pilotStructuralRows$ sparse structural rows, SPX calls at the horizon and at its midpoint, the date-zero budget row, budget rows $B(t;t,t+\Delta )$ at each of the $\pilotBudgetRows$ midpoint nodes with tenor $\Delta =\pilotTenorDays$ days, so that $V^\Delta =\sqrt {B/\Delta }$ is a node coordinate, a VIX future and $\pilotVixCalls$ VIX calls written on it: $\pilotPricingRows$ quoted and budget rows in all, with $V_0^\Delta =\pilotVixZero$ . The physical rows are a conditional-mean row at every node, which fixes one of the two free conditional probabilities at an ordinary node. A variant adds standardized conditional powers through one less than the local child count; together with normalization these rows have full Vandermonde rank and pin every physical transition, including the six-way node. The pricing weights, and with them $L=p/r$ , remain set-valued. The remaining rows are a crash probability $P(S_H<0.95)$ , and a terminal-return second moment, all taken from a fixed physical law calibrated at every node to $\E _P[e^{\Delta X}\mid \mathcal F_j]=e^{0.06\,dt}$ and given a tail tilt. In the path-level crash stress displayed in the left panel below, the conditional rows are dropped and the second-moment row is replaced by the mean terminal gross return, so the other path target remains fixed. The code reproduces the analytic calculations in Examples 5.5 and 6.9 to a maximum relative deviation of $\pilotValidationDev$ , commensurate with the bisection tolerance of $10^{-4}$ .¹ Sharpe widths are in six-day units throughout; the annualization factor is about $6.5$ .

Figures 1 and 2 numerically instantiate the same interval and minimal-width optimizations as Propositions 6.2 and 6.5 on this explicitly declared exact finite structural fibre.

Figure 1: Set-valued recovery from option prices alone. Identified intervals for two physical readouts under three bands, with no physical rows declared. The true law’s two readout values lie inside every interval, including under the cap $0.20$ , which excludes the true law itself (its kernel has Sharpe width $\pilotTrueSharpe $ ): each value is attained by some feasible law with a smaller kernel. Once the physical rows are added the minimal distortions are gain–loss $\pilotKappaGL $ on $[\pilotBandLo ,\pilotBandHi ]$ (a verified upper bound; see Table 1 ), Sharpe width $\pilotKappaSharpe $ , and $H(Q\,|\,P)_*=\pilotKappaEnt $ , each below the true kernel’s $\pilotTrueLratio $ , $\pilotTrueSharpe $ , and $\pilotTrueEntropy $ , because the rows do not pin the kernel. With the full local-moment rows added, the physical transition weights and hence the physical market-path projection are pinned. The enhanced-carrier law $P_R$ is not point-identified, because its $L$ and $D$ coordinates vary with the still set-valued pricing weights; the Sharpe width rises to $\pilotFullKappaSharpe $ . — Figure 1. Set-valued recovery from option prices alone. Identified intervals for two physical readouts under three bands, with no physical rows declared. The true law’s two readout values lie inside every interval, including under the cap $0.20$ , which excludes the true law itself (its kernel has Sharpe width $\pilotTrueSharpe$ ): each value is attained by some feasible law with a smaller kernel. Once the physical rows are added the minimal distortions are gain–loss $\pilotKappaGL$ on $[\pilotBandLo ,\pilotBandHi ]$ (a verified upper bound; see Table 1), Sharpe width $\pilotKappaSharpe$ , and $H(Q\,|\,P)_*=\pilotKappaEnt$ , each below the true kernel’s $\pilotTrueLratio$ , $\pilotTrueSharpe$ , and $\pilotTrueEntropy$ , because the rows do not pin the kernel. With the full local-moment rows added, the physical transition weights and hence the physical market-path projection are pinned. The enhanced-carrier law $P_R$ is not point-identified, because its $L$ and $D$ coordinates vary with the still set-valued pricing weights; the Sharpe width rises to $\pilotFullKappaSharpe$ .

$Figure 2: The band as hinge. Minimal kernel distortion as the physical target moves away from the pricing data: Sharpe width, $\sqrt {2H(Q\,|\,P)_*}$ , and $\log \kappa _*$ for the gain–loss band. Every finite gain–loss point is a verified-feasible upper bound; open triangles mark the search ceiling $60$ , hence censoring rather than infeasibility. Grey bars mark targets at which the fixed-band LP reports $[0.5,2]$ infeasible, a solver-status rather than certificate-checked verdict. Left: the crash target is raised with only the path-level crash and mean terminal-return rows; right: the conditional mean log return at every node is shifted by a constant annualized amount. For the left panel, a direct LP with an archived primal–dual certificate places the fixed-band wall at $\pilotBCertifiedBandWall $ times the true crash probability. For the right panel, the transition lies in $\pilotCthrBracket $ per year of added conditional mean log return: the lower endpoint has a verified feasible witness and the upper endpoint is solver-reported infeasible without a separately checked Farkas certificate.$

Figure 2. The band as hinge. Minimal kernel distortion as the physical target moves away from the pricing data: Sharpe width,

\sqrt {2H(Q\,|\,P)_*}

, and

\log \kappa _*

for the gain–loss band. Every finite gain–loss point is a verified-feasible upper bound; open triangles mark the search ceiling

60

, hence censoring rather than infeasibility. Grey bars mark targets at which the fixed-band LP reports

[0.5,2]

infeasible, a solver-status rather than certificate-checked verdict. Left: the crash target is raised with only the path-level crash and mean terminal-return rows; right: the conditional mean log return at every node is shifted by a constant annualized amount. For the left panel, a direct LP with an archived primal–dual certificate places the fixed-band wall at

\pilotBCertifiedBandWall

times the true crash probability. For the right panel, the transition lies in

\pilotCthrBracket

per year of added conditional mean log return: the lower endpoint has a verified feasible witness and the upper endpoint is solver-reported infeasible without a separately checked Farkas certificate.

Discretization and reproducibility. Table 1 refines the same six-day horizon with $m=\pilotRefMs$ steps ( $\pilotRefNs$ paths; $\pilotRefSecs$ seconds for the base block) and records the solver settings. Trees at different step sizes are not nested, so this is a sensitivity test rather than convergence along a refining sequence. The true kernel’s width and the displayed expected-return recovery interval are stable across the three grids, whereas the crash probability and its recovery intervals are not. The Sharpe and entropy minima decrease across the three displayed grids, whereas the gain–loss minimum is stable but nonmonotone. Because the grids are not nested, these patterns are reported as observed sensitivity and are not explained by any inclusion of fibres; the spreads are shown rather than summarized. Tree, pricing law, and physical law are synthetic, and the computation on a reported residual panel remains the open step named in Remark 6.8. What the illustration establishes is that the declared finite structural fibre is computationally workable, that the Sharpe width and $\sqrt {2H(Q\,|\,P)_*}$ agree to second order at small distortions and differ by up to $\pilotSharpeEntDev$ at the largest stresses (the gain–loss width is a different functional and is not compared), and that the two emptiness mechanisms of this paper are distinguishable in this synthetic experiment.

In the unplotted crash stress that retains the conditional-mean rows, all three bands fail beyond the direct-LP support wall $\pilotACertifiedSupportWall$ times the true target because the conditional rows, retained terminal-return second-moment row, and finite branch support jointly bound the crash probability: a support obstruction rather than a band obstruction. The wall has an archived primal–dual certificate.

Table 1. Discretization sensitivity at a fixed six-day horizon, and solver settings.



item	value


steps $m$ (paths)	$\pilotRefMs$ ( $\pilotRefNs$ )
exact structural carrier	three martingale rows at every node; $\pilotNstructuralChecks$ independent mean/full checks on the three grids, with largest conditional residual $\pilotMaxStructuralConditionalResidual$
true kernel Sharpe width	$\pilotRefTrueSharpeVals$ (spread $\pilotRefTrueSharpeSpread$ )
true kernel $H(Q\,\|\,P)$	$\pilotRefTrueEntVals$ (spread $\pilotRefTrueEntSpread$ )
true crash probability	$\pilotRefTrueCrashVals$ (spread $\pilotRefTrueCrashSpread$ )
Sharpe width $\kappa _*$ , physical rows added	$\pilotRefKSharpeVals$ (spread $\pilotRefKSharpeSpread$ )
entropy $H(Q\,\|\,P)_*$ , physical rows added	$\pilotRefKEntVals$ (spread $\pilotRefKEntSpread$ )
gain–loss $\kappa _*$ , physical rows added	$\pilotRefKGLVals$ (spread $\pilotRefKGLSpread$ )
recovery of $\E _P[\text {return}]$ , band $[0.5,2]$	lower $\pilotRefRetLo$ ; upper $\pilotRefRetHi$

solvers	Python 3.14.0, cvxpy 1.8.2, CLARABEL 0.11.1, SCS 3.2.11, HiGHS through scipy 1.17.0. Gain–loss ratio LPs: HiGHS through the SciPy interface, three 30 s attempts (automatic method selection, dual simplex, then interior point), then CLARABEL, then SCS; other LPs CLARABEL then HiGHS; cone programs CLARABEL with default settings, then tightened tolerances, then equilibration disabled, then SCS, re-solving until a point verifies
verification	gain–loss ratio LPs: a feasible verdict requires a primal residual of at most $10^{-7}$ on every row and band inequality, whatever the solver reports; fixed-band feasibility LPs: $10^{-6}$ ; Sharpe and entropy programs: affine equalities are row-normalized for conditioning, then the raw returned vectors are checked against the original rows for nonnegativity, linear rows, and simplex within $10^{-6}$ , closed-perspective zero pairs within $10^{-8}$ , the opposite-side cap within a relative $10^{-6}$ , and scaled objective error $\|v_{\rm raw}-v_{\rm solver}\|/\max \{1,\|v_{\rm solver}\|\}\le 10^{-5}$ ; cold-start re-solves continue until a point verifies; the finite conic objectives are primal-verified solver-status optima, not independently dual-certified; each JSON record references a unique archive containing the raw witnesses and diagnostics for every finite reported solution
known-witness guard	on all $\pilotNKnownTrueGLChecks$ base and refinement grids, the independently reconstructed true laws satisfy every active row with residual at most $\pilotMaxKnownTrueGLResidual$ ; each gain–loss search must be uncensored and its verified upper trial cannot exceed the true likelihood-ratio width by more than the bisection tolerance
tolerances	solver defaults on first attempts and the fallback settings just described; gain–loss bisection $10^{-4}$ relative with ceiling $60$
caps, nonnegativity, and closed perspectives	opposite-side cap $50$ in the Sharpe and entropy programs; no strictly positive mass floor is imposed, and the raw returned weight vectors are checked directly for nonnegativity within the stated primal-residual tolerance; moment and entropy terms use their exact closed perspectives; the entropy epigraph uses the exact substitution $T=Nt$ , $\mathrm {ExpCone}(-T,Nx,Ny)$ , and objective or cap $\sum _iT_i/N$ ; a numerical zero denominator is accepted only when its paired numerator is at most $10^{-8}$ ; no logarithmic smoothing is used
statuses	over the $\pilotNrecords$ archived base, full-physical-row, and stress A/B/C records (recovery intervals, refinement, and validation excluded): $\pilotNsolves$ logical optimization calls ( $\pilotNattempts$ solver attempts), $\pilotNfallbackInvoked$ invoked a fallback solver, $\pilotNfallbackVerdict$ obtained their verdict on a fallback, $\pilotNunresolved$ exhausted the chain unresolved and were treated as infeasible, $\pilotNinaccurate$ ended `optimal_inaccurate`, and $\pilotNviol$ cone-program solutions were flagged by the recomputation check and are reported with that flag; the affected gain–loss widths are therefore verified-feasible upper bounds; the largest solver-reported-infeasible trial lies within $\pilotGLbracketMax$ of each, infeasibility verdicts being taken from solver status without a certificate check
infeasibility residuals	smallest phase-1 gap at a reported infeasibility $\pilotMinGap$ (unscaled; reported without further inference)

7Scope: architectures and charts

This section answers the fourth question: what the fibre is a set of. It is a set of laws in $\cK$ . A fixed architecture inside $\cK$ sees only its closed convex quote hull (Section 7.1), and identification of a selected finite model is a separate, chart-level tier (Section 7.2).

7.1Architecture boundary: the closed convex quote hull

Proposition 5.3 is exact because the declared law class $\cK$ has a compact convex quote image. A fixed parametric or structural architecture need not share that geometry. Let $\mathfrak M\subseteq \cK$ be any nonempty declared architecture and put

\begin{equation} \cY _J(\mathfrak M):=T_J(\mathfrak M)\subset \R ^{d_J}, \qquad h_J^{\mathfrak M}(a):=\sup _{z\in \cY _J(\mathfrak M)}a^\top z, \label {eq:architecture-support} \end{equation}

where $d_J$ is the number of pricing and physical rows in the finite panel.

Proposition 7.1 (Linear tests see the closed convex quote hull) . For every finite target $y_J\in \R ^{d_J}$ ,

\begin{equation} a^\top y_J\le h_J^{\mathfrak M}(a) \quad \text {for every }a\in \R ^{d_J} \quad \Longleftrightarrow \quad y_J\in \overline {\operatorname {conv}}\cY _J(\mathfrak M). \label {eq:architecture-hull-test} \end{equation}

Consequently, a strict violation of one support inequality validly proves

y_J\notin \cY _J(\mathfrak M)

, whereas satisfaction of every inequality proves only closed-convex-hull membership. The support test characterizes exact membership in the architecture for every target if and only if

\cY _J(\mathfrak M)

is closed and convex.

Proof.A continuous linear functional has the same supremum on a set and on its closed convex hull. Finite-dimensional strict separation characterizes the complement of that closed convex set, proving (70). Since $\cY _J(\mathfrak M)\subseteq \overline {\operatorname {conv}}\cY _J(\mathfrak M)$ , a strict violation also excludes the original architecture. Exact equivalence for every target holds precisely when the two sets coincide.∎

Example 7.2 (False reassurance from nonconvexity or nonclosure) . If an architecture has scalar quote image $\{-1,1\}$ , the target zero is not attainable, but it satisfies every support inequality because it lies in the convex hull. This is a failure of completeness, not a false rejection: any strict separator would still exclude both the hull and the architecture. Closure is a separate issue. For quote image $(0,1]$ , the target zero again satisfies every support inequality and has distance zero from the image, yet no model attains it. Thus convexity without closedness is insufficient, and a residual alone does not certify exact membership.

Remark 7.3 (A scoped convexification diagnostic) . Fix a disclosed weighted norm $\|\cdot \|_W$ on the finite panel and a market uncertainty set $\mathcal B_J$ containing, for example, bid–ask and sampling bands. Define

\begin{align} r_{\mathfrak M}(\mathcal B_J) &:=\inf _{b\in \mathcal B_J,\,z\in \cY _J(\mathfrak M)} \|b-z\|_W,\nonumber \\ r_{\rm hull}(\mathcal B_J) &:=\inf _{b\in \mathcal B_J,\, z\in \overline {\operatorname {conv}}\cY _J(\mathfrak M)} \|b-z\|_W,\qquad g_{\rm conv}:=r_{\mathfrak M}-r_{\rm hull}\ge 0. \label {eq:convexification-gap} \end{align}

For a singleton target $\mathcal B_J=\{y_J\}$ and dual norm $\|\cdot \|_{W,*}$ , the hull residual is itself the maximal normalized support violation,

\begin{equation} r_{\rm hull}(y_J) =\max _{\|a\|_{W,*}\le 1} \{a^\top y_J-h_J^{\mathfrak M}(a)\}. \label {eq:hull-distance-dual} \end{equation}

The gap measures relaxation by convexification only relative to the declared panel, norm, uncertainty set, and architecture. Computing

r_{\mathfrak M}

is the original global nonconvex problem. The hull problem is an LP or convex program only when its finite representation or a globally valid support oracle is available; an LP over a sampled parameter grid sees only the sampled hull and cannot exclude unsampled models. If the quote image is nonclosed,

r_{\mathfrak M}=0

may still coexist with exact nonattainment.

For nested classes $\mathfrak M_{\le k}$ , a component-order diagnostic must report the whole residual curve and, at tolerance $\varepsilon$ , may define

k_\varepsilon :=\min \{k: r_{\mathfrak M_{\le k}}(\mathcal B_J)\le \varepsilon \}.

This panel-relative order is not an intrinsic factor or predictive-state dimension. Convexifying

\mathfrak M_{\le k}

mixes whole models and can add regimes or internal components, so the single number

g_{\rm conv}

does not answer how many components the market “requires.”

One possible response is to replace $\mathfrak M$ by a separately verified tagged or randomized enlargement. That converts hull points into admissible models only when the enlarged class genuinely preserves the common root, all affine pricing and physical rows, the likelihood restrictions, and the declared filtration after the tag becomes observable. It changes the architecture and its complexity; it does not prove membership in the original $\mathfrak M$ . Otherwise exact architecture feasibility requires a separate global nonlinear membership method. No architecture gap is computed in this paper.

7.2Selected-chart identification is a separate tier

Let $\Theta \subset \R ^d$ and let $\psi :\Theta \to \cK$ map a gauge-fixed finite model parameter to its enhanced canonical law. Gauge fixing removes local label permutations, unreachable states, and other law-preserving symmetries. For a finite panel $J$ , set

\begin{equation} \mathcal O_J(\theta )=(\Gamma _k(\psi (\theta )))_{k\in J}. \label {eq:obs-map} \end{equation}

Assumption 7.4 (Law/chart interface) . On a neighborhood $U$ of $\theta _*$ :

(a): $\psi (\theta )$ lies in the same $\cK$ , including its raw filtration, common budget, density rows, and band;
(b): $\psi$ is one-to-one modulo equality of the induced linked pair;
(c): every coordinate of $\mathcal O_J$ is a canonical row, not a surrogate observation from a different model;
(d): $\mathcal O_J$ is $C^1$ , its derivative is $M$ -Lipschitz, and
$\begin{equation} \sigma _{\min }(D\mathcal O_J(\theta _*))\ge \kappa >0. \label {eq:rank} \end{equation}$

This is an interface assumption, not an imported realization theorem. A model-specific application must prove that its chart maps into $\cK$ , that its observations are the canonical rows, and that its derivative has the stated regularity and rank. Generic rank theory is classical [47].

Proposition 7.5 (Chart-relative local inverse) . If $B_r(\theta _*)\subset U$ and

\begin{equation} 0<r\le \kappa /(2M), \label {eq:radius} \end{equation}

with

\kappa /(2M)=\infty

when

M=0

, then for

\theta _1,\theta _2\in B_r(\theta _*)

\begin{equation} \|\theta _1-\theta _2\| \le \frac {2}{\kappa } \|\mathcal O_J(\theta _1)-\mathcal O_J(\theta _2)\|. \label {eq:inverse} \end{equation}

Thus the selected chart is locally identified by

J

. The conclusion does not imply that the full law-level fibre is a singleton.

Rank failure produces a first-order null direction and invalidates this inverse modulus; it does not automatically prove exact nonidentification. The fallback is an identified set, a null-direction analysis, or more observations. A regularized optimizer is a selector, not evidence that the data identify its output. Conversely, local injectivity of the selected chart does not make global linear support tests exact for the generally nonconvex image $\mathcal O_J(\Theta )$ ; that question remains subject to Proposition 7.1.

8Conclusion

Within one fixed compact convex adapted-law class with a divergence band on the likelihood, the linked laws consistent with a complete SPX/VIX pricing market form a compact convex fibre, nonempty exactly when no finite linked separator exists, on which every continuous affine physical readout has an exact attained interval. Every statement is relative to that class. On a finite grid its compactness is proved (Section 5.2); in continuous time it is Assumption 2.5, and constructing a nonempty adapted-compact continuous SPX/VIX carrier is the one proof obligation this paper leaves open.

The practical message is compatibility before calibration. A failed joint fit can reflect an optimizer, a chosen architecture, noisy targets, or a genuine conflict between physical and pricing requirements; the fibre and its finite separators isolate the last, and architecture-level claims require the hull diagnostic of Section 7.1. When compatibility holds, what remains are decisions the market does not make: how the class is justified, which point is selected, and what further model and data assumptions identification needs.

The symmetric message applies to an apparent success. Current joint $P/Q$ work reports aggregate agreement with signed, maturity- and strike-dependent residuals read informally as risk premia [31], and Remark 5.9 notes that the invariance licensing those comparisons is the one constraining them. A structured residual of that kind is a statement about the likelihood, and the likelihood admits a disclosed scalar summary on a disclosed class. Computing $\kappa _*(\mathcal B)$ on a reported residual panel is the natural next step and the point at which the framework becomes empirically informative. It first requires an equivalent linked pair, verification of the chosen moment or entropy bounds, and a parent carrier carrying every pricing and physical row. The exact structural finite-grid pipeline of Section 6.3 is an engineering template for that exercise, not yet the empirical application.

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.

Notes

Code, scope, raw output, and the generated numbers used here are in the pilot directory of the source archive. ↩

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How to cite

Shaosai Huang (2026). Linked Physical and Risk-Neutral SPX/VIX Laws: The Identified Set under a Divergence-Bounded Pricing Kernel. Working paper, version of September 2026. Kspectra Research. SSRN 7404180 (doi:10.2139/ssrn.7404180). https://kspectra.ai/papers/linked-spx-vix-laws-identified-set/

@misc{huang2026linked,
  author = {Huang, Shaosai},
  title  = {{Linked Physical and Risk-Neutral SPX/VIX Laws: The Identified Set under a Divergence-Bounded Pricing Kernel}},
  year   = {2026},
  month  = sep,
  note   = {Working paper, version of September 2026},
  doi    = {10.2139/ssrn.7404180},
  url    = {https://kspectra.ai/papers/linked-spx-vix-laws-identified-set/}
}

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