Quote Completion at the Quadratic Tail Boundary for Joint Rates Panels

Shaosai Huang

doi:10.2139/ssrn.7435978

Working paper · September 2026

Quote Completion at the Quadratic Tail Boundary for Joint Rates Panels

Shaosai Huang

∗ Working paper. Comments welcome.

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

Abstract

In a joint caplet–swaption market based on overnight risk-free rates (RFRs), the companion paper shows that, under its escape hypotheses, no finite standard-timing book controls the superlinear annuity-weight moment governing the numeraire tail. It proposes a single in-advance convexity-carrier type, quoted at every post-initial date. We determine what this completion buys. At one date an unbudgeted piecewise-linear quote image can omit boundary points from its polyhedral closure; the quoted second moment removes this nonattainment and yields an exact compact projection with one rotated second-order-cone block per cell and attained certificates. On a finite calendar, under the companion paper’s weak-compactness hypotheses, all-date ask-only carrier calibration, recursive support and selection hypotheses, joint moment-Slater, and subcritical objective and non-carrier calibration rows, the maximizing model and optimal semistatic hedge are attained. At exact quadratic growth, under the recursive selection, nonemptiness, all-date ask, and joint relative-interior hypotheses, an optimal hedge still exists although a maximizing law need not. Thus market-data completion and model attainment differ. For exact and one-sided ask panels of at most quadratic growth, an ordered pathwise hull determines whether an inconsistency requires dynamic trading. When the exact and non-carrier ask rows are subcritical and the corresponding completed-market compactness, nonemptiness, all-date ask-only carrier, and recursive-selection hypotheses hold, every inconsistency has an attained strict certificate. Across dates the polyhedral description is sufficient, not characteristic. Finally, on a common raw-history market with the same finite compilation, replacing one universally measurable, pointwise-finite integrability gauge by another does not change feasibility of integrable finite books.

Contents

1Introduction
2The single-date feasibility body
3Quote completion and attained hedges
4A cross-date primitive: perspective-calendar monotonicity
5Multi-date polyhedrality: a sufficient grammar and its boundary
6The ordered certificate boundary after quote completion
7Conclusion
AThe finite-date constant-weight martingale image
References
How to cite

Keywords: joint calibration; martingale optimal transport; semistatic superhedging; second-order-cone representation; attainment; interest-rate volatility.

MSC 2020: 91G30, 91G20, 90C25, 60G42. JEL: G13, C61.

1Introduction

Caps and swaptions are written on the same curve, and they are not always priced consistently with each other: Longstaff et al. (2001) document that cap prices periodically deviate significantly from the no-arbitrage values implied by the swaptions market. A cap is a strip of caplets, and it is caplet volatilities—stripped from cap quotes, or quoted directly where a caplet market exists—together with swaption volatilities that a model must reproduce. Joint calibration is the response, and it is usually posed as a fitting problem: find one arbitrage-free model reproducing the quoted caplet and swaption panels. Behind the fitting problem lies a feasibility problem: which quote vectors admit any consistent model at all? For a finite call book on one underlying the answer is classical and polyhedral (Davis and Hobson, 2007; Cousot, 2007), and the two-stage view of calibration—first decide whether the quotes are consistent, then select one model from the consistent set—is the programme that Guyon (2024) carries out for equity smiles. Mathematically the feasibility question is a moment problem over martingale laws, the territory of moment-constrained price bounds (Bertsimas and Popescu, 2002; Lasserre, 2001) and of martingale optimal transport (Beiglböck et al., 2013, 2017). What is specific to the rates panel is that the pricing class carries an unbounded numeraire direction, so the first question is not which bounds are sharp but whether the moment problem is well posed at all. This paper develops the feasibility stage for the rates panel, in the market completed by the companion paper described next.

The setting inherited from the companion paper. Quotes are observed on a bond curve at dates $t_0<t_1<\cdots <t_N$ . Throughout we work in the anchor units of Huang (2026): for a fixed terminal maturity $T_\star$ ,

\Zbar _t(u):=\frac {P_t(u)}{P_t(T_\star )},\qquad \zeta _t:=\frac {1}{P_t(T_\star )}=\Zbar _t(t),

so that the normalized bond prices are martingales under a pricing law, and $\zeta _t$ , the reciprocal terminal bond, is the coordinate through which the numeraire tail enters every payoff. The admissible pricing laws form the class $\cM$ of martingale laws on the ambient-closed path space with pinned initial curve $r_0$ and the currency-price bound $\Zbar _t(u)\le C_P\zeta _t$ . A quoted instrument is a payoff on that path space; a panel $f=(f_1,\ldots ,f_n)$ with quotes $c$ is consistent if some $Q\in \cM$ has $\E _Qf_i=c_i$ for every exact row and $\E _Qf_i\le c_i$ for every one-sided ask. For a finite book $\cB$ , write $\cM (\cB )$ for the subclass of $\cM$ in which every book payoff is absolutely integrable and all of the book’s exact and one-sided ask constraints are satisfied.

The companion paper shows that on the standard quote set the calibration question is not yet well posed. Caplet and swaption payoffs are piecewise linear in the anchor coordinates, but the quantity the standard book fails to control is a superlinear moment of the annuity weight $\Abar _e=\sum _j\delta _j\Zbar _e(q_j)$ (coupon dates $q_j$ , accruals $\delta _j$ , stopped coordinates for coupons already paid): under its base-date or multi-date surgery hypotheses,

\begin{equation} \sup \bigl \{\E _Q\Abar _e^{\,2}:\ Q\in \cM \text { calibrated to the standard book}\bigr \}=\infty , \label {eq:pc2-intro-escape} \end{equation}

because mass can escape along the tail of $\zeta$ while every standard-timing quote is recalibrated exactly. The tail direction that controls prices is left free by the data, and with it the value of any claim sensitive to that direction. The remedy proposed there is one quoted in-advance convexity carrier per post-initial date, the strip

\begin{equation} s_i:=\frac {\zeta _{t_i}^2-\zeta _{t_i}}{\tau _i} \quad \text {quoted at an ask }\beta _i ,\qquad i=1,\ldots ,N, \label {eq:pc2-intro-strip} \end{equation}

with accrual $\tau _i>0$ . Write $\chi :=1+\sum _{i=1}^N\zeta _{t_i}^2$ for the resulting gauge and $\cM _\chi :=\{Q\in \cM :\E _Q\chi <\infty \}$ for the completed class. Since $\tau _is_i+\zeta _{t_i}=\zeta _{t_i}^2$ and holding each $t_i$ -bond from $t_0$ produces the zero-cost gain $\zeta _{t_i}-\Zbar _0(t_i)$ , the gauge is replicated pathwise by cash, the strips and a buy-and-hold bond position (Proposition 3.2), whence

\E _Q\chi \;\le \;1+\sum _{i=1}^N\bigl (\tau _i\beta _i+\Zbar _0(t_i)\bigr ) \qquad \text {for every pricing law meeting the asks}:

a finite ask on every strip is exactly a uniform budget on the gauge. Under its declared support, price-bound, observation-regularity, all-date quoted-strip and nonempty augmented-class hypotheses, the companion paper proves that this budget closes the weighted-smile image and yields an exact support-function dual. Two growth tiers relative to the gauge organize everything below, for continuous $f$ :

\begin{equation} \cC _\chi ^0:=\bigl \{f:\ \forall \varepsilon >0\ \exists C_\varepsilon ,\ |f|\le C_\varepsilon +\varepsilon \chi \bigr \} \ \subsetneq \ \cC _\chi :=\bigl \{f:\ |f|\le C_f\chi \bigr \}, \label {eq:pc2-intro-tiers} \end{equation}

the subcritical and the critical class. The carrier itself lies in $\cC _\chi \setminus \cC _\chi ^0$ , and the annuity square in (1) is of critical growth $O(\chi )$ (Proposition 3.9). Hypotheses and results imported from the companion paper are named where they are used.

This paper takes the completion as given and asks the question it leaves open: what does one carrier type, quoted across the calendar, actually buy? We answer in three steps, of which the second is the point of the paper.

Main results. 1. The consistent-quote set. At a single date the class collapses to $\cP (r_0)$ , the probability laws with mean $r_0$ on a polyhedral state space $\Omega \subseteq \R ^k_+$ all of whose nonzero recession directions have $\zeta \to \infty$ (Assumption 2.1 and Definition 2.2), and the object is the feasibility body $\cV =\{(\int f_i\,\dd \mu )_{i\le n}:\mu \in \cP (r_0)\}$ . For a piecewise-linear panel with $r_0\in \ri \Omega$ , Theorem 2.7 shows that $\overline \cV$ is a polytope $\widehat \cV$ with $\ri \widehat \cV \subseteq \cV \subseteq \widehat \cV$ , that its support function

h(\lambda )=\min \bigl \{a+b\cdot r_0:\ a+b\cdot x\ge \lambda \cdot f(x)\ \text {on }\Omega \bigr \}

is attained by an explicit cash-plus-curve portfolio, and that a point of $\widehat \cV$ belongs to $\cV$ exactly when it admits a representation with zero horizon defect: some boundary quote vectors can only be approached, by mass sent along the recession directions. This is the single-date geometry of (1). Adjoining the quoted second moment $q(x)=x_\zeta ^2$ at ask $\beta$ removes the defect. Theorem 2.16 describes the exact body $\cV _\beta$ as the compact projection of a system with one scalar rotated second-order-cone block $m_{j,\zeta }^2\le t_js_j$ per linearity cell, with support function

h_\beta (\lambda )=\min _{\nu \ge 0}\Bigl [\nu \beta + \min \bigl \{a+b\cdot r_0:\ a+b\cdot x+\nu q(x)\ge \lambda \cdot f(x) \ \text {on }\Omega \bigr \}\Bigr ],

both minima attained, so that a positive multiplier $\nu ^*$ is a shadow price of the budget and not a sign of curvature. Piecewise-rational panels—constant-maturity swaps (CMS) and cash settlement—remain semialgebraic (Theorem 2.15).

2. What the quote lets you hedge, and where it stops. On the finite calendar, with exact rows $H$ , ask rows $F$ containing every strip of (2), and a claim $X$ , the completed problem is

P=\sup \bigl \{\E _QX:\ Q\in \cM _\chi ,\ \E _QH=c,\ \E _QF\le q\bigr \}.

Under a rectangular-sections hypothesis (Assumption 3.4), the weak-compactness hypotheses of the companion paper, and one joint relative moment-Slater condition $(c,q)\in \ri \cA$ on the attainable moment domain, Theorem 3.6 proves for subcritical $X$ , $H$ and non-strip asks that

P=\min _{\lambda ,\ \eta \ge 0} \bigl \{\lambda \cdot c+\eta \cdot q+h(X-\lambda \cdot H-\eta \cdot F)\bigr \} =\min \{\text {cost of a semistatic superhedge of }X\},

with the maximizing model, the multipliers and the hedge all attained. At the critical tier $X\in \cC _\chi$ —the growth of the carrier itself—Theorem 3.7 keeps the attained multipliers and the attained hedge but makes no model assertion, and the example following Proposition 2.18 shows why: second-order mass can escape to the horizon, so a maximizing law can fail to exist. Applied to the annuity square, Proposition 3.9 turns the infinite value (1) into the explicit finite superhedge $\Abar _r^{\,2}\le K_r\sum _d\gamma _d\zeta _d^{\,2}$ , of cost $K_r\sum _d\gamma _d(\tau _d\beta _d+\Zbar _0(d))$ ; under the critical-tier hypotheses, Corollary 3.10 makes that hedge optimal. Completing the market data and obtaining a model are therefore different achievements, and one carrier type quoted across the calendar delivers the first without the second.

3. What certifies an inconsistency, and whether it needs dynamics. The carrier trades one-sidedly. For an exact/ask quote $z=(c,q)$ two ordered sets decide the question: the model set $\cA =\{(\E _QH,r):Q\in \cM _\chi ,\ \E _QF\le r\}$ and the ordered pathwise hull $\cK ^{\uparrow }_{\rm stat}=\cl \bigl (\conv G(\Omega )+\{0\}^p\times \R ^m_+\bigr )$ , $G=(H,F)$ , which remembers only pathwise support and the right to pay more for an ask. Theorem 6.3 shows that a static certificate exists exactly when $z\notin \cK ^{\uparrow }_{\rm stat}$ ; that on $\cK ^{\uparrow }_{\rm stat}\setminus \cl \cA$ the dynamic leg of the certificate is attained and indispensable; and that $\cl \cA \setminus \cA$ is a zero-gap region with supporting hedges but no strict certificate. With every strip ask present and subcritical rows, $\cA$ is closed (Corollary 6.4), so the last region is empty and every inconsistency carries an attained strict certificate. At one date the test is two second-order-cone feasibility problems that differ by a single mean row (Corollary 6.6).

Two bounding results. Across dates the polyhedral description is sufficient rather than characteristic: for separated panels, each payoff fixing at one date, the body remains a polytope at every finite depth with a complete semistatic certificate grammar (Theorems 5.2 and 5.3), while a path-mixed forward-start panel has the strictly convex support function $h(\theta )=\tfrac 14+\theta ^2/16$ (Proposition 5.5). On a common raw-history structural market, a nodewise martingale cubature shows that any finite panel integrable in the relevant classes has the same exact moment image under any two universally measurable, pointwise-finite gauges, both equal to the finite-support image (Theorem 6.15). Thus, for the same finite quote compilation, changing only the integrability gauge leaves feasibility unchanged—an equality of finite observation images, not of law classes.

Relation to prior work. (i) Finite-panel consistency. Polyhedrality of the arbitrage-free price set of a finite call book is classical: Davis and Hobson (2007) separate model-independent from weak arbitrage, Cousot (2007) treats bid–ask quotes, Gerhold and Gülüm (2020) study spread structure, and Cohen et al. (2020) run the resulting linear programme on market data. The separated results here extend this line to weighted, unnormalized rates payoffs and, across dates, to genuine martingale couplings, where polyhedrality is no longer a statement about marginals alone.

(ii) Conic price sets from moments. Sharp bounds from moment data by convex duality include Bertsimas and Popescu (2002) and the moment–sum-of-squares (SOS) hierarchy of Lasserre (2001). Classical finite-support reductions for moment problems go back to Richter (1957), Rogosinski (1958) and Isii (1962); d’Aspremont and El Ghaoui (2006) obtain closed-form or linear-programming bounds in special basket-option cases and an efficient linear-programming relaxation of the general problem, tight in some of those special cases. Theorem 2.16 differs in describing the exact body rather than bounds on one payoff or a general-case relaxation, in using one scalar rotated second-order-cone block per linearity cell, and in attaining its support certificates, which is what later converts them into hedges.

(iii) Duality and attainment. In martingale optimal transport, classical dual attainment can fail and is restored in a quasi-sure form (Beiglböck et al., 2017); pointwise attainment can fail even for regular costs (Beiglböck et al., 2019); and semistatic outcome sets need not be closed (Acciaio et al., 2017; Nutz et al., 2023). Closest in constraint structure are Kolesnikov et al. (2026), who specify a marginal through finitely many moments and prove strong duality and existence of optimal couplings at interior moment vectors, and Carlier et al. (2025), who study constraint qualification and dual attainment for weak transport. The critical-tier statements here add no new attainment mechanism—loss of uniform integrability at critical growth is familiar in this tradition—but pin the tier to a quoted budget, so that completion of the data and existence of a model separate.

(iv) Pathwise superhedging and the need for dynamics. Burzoni et al. (2017) show that the pathwise superhedging price can strictly exceed the martingale supremum, with the gap vanishing exactly on an analytic set. This result belongs to the classification of arbitrage by significant sets developed by Burzoni et al. (2016) and Burzoni et al. (2019); d’Aspremont (2004) contrasts static arbitrage bounds with model-consistent prices for multivariate options. The ordered boundary of Section 6 is the finite-panel, one-sided analogue: quotes are vectors, the deciding object is the ordered pathwise hull, and the test is conic. Closedness under one-sided quotes is the difficulty identified for options with spreads by Burzoni (2016) and Kühn (2025); it is resolved here by the quote-controlling ask carrier.

(v) Curvature and cubature. The path-mixed mechanism is the tangency phenomenon of robust forward-start bounds (Hobson and Neuberger, 2012; Hobson and Klimmek, 2015) and of martingale optimal transport (Beiglböck et al., 2013), there for one exotic under full marginals, here for the finite-panel body itself. The finite-support reduction at the selection interface is a raw-history extension of martingale Tchakaloff cubature (Bayer and Teichmann, 2006; Beiglböck and Nutz, 2014), retaining the progressively revealed tape and every pathwise production identity.

Structure. Section 2 proves the single-date geometry of step 1; Section 3 the calendar duality and critical-tier boundary of step 2, using the cross-date primitive of Section 4; Section 5 the separated grammar and its path-mixed limit; and Section 6 the ordered certificate boundary of step 3 and the selection interface. The appendix records a full-marginal complement whose compactness comes from prescribed marginals rather than from a quoted budget.

2The single-date feasibility body

This section fixes one observation date and asks what the set of attainable quote vectors looks like. Three answers follow, in increasing generality of the payoff panel and decreasing simplicity of the geometry: the closure of a piecewise-linear panel’s quote image is a polytope, while the image can omit boundary points; piecewise-rational panels give a semialgebraic body; and adjoining the quoted second moment to a piecewise-linear panel gives an exactly described second-order-cone body in which the defect is gone. The single date is not a restriction of interest in itself—the calendar is treated in Section 3—but it is where the geometry is visible in closed form, and the later multi-date geometric arguments return to it fibrewise.

We work at a single date $e$ with deterministic history, where the martingale class reduces to probability measures on the date- $e$ state space with pinned mean. The state space is polyhedral:

Assumption 2.1 (Polyhedral support with price bound) . The date- $e$ state space is a nonempty polyhedron $\Omega \subseteq \R ^k_{+}$ (coordinates $x=(\Zbar _e(u))_{u\in U_e}$ , one of them the terminal coordinate $x_\zeta =\zeta _e$ ), containing the declared curve constraints (positivity, maturity monotonicity) and the currency-price bound $x_u\le C_P\,x_\zeta$ for all $u$ . Consequently every nonzero recession direction $d\in \rec \Omega$ has $d_\zeta >0$ .

The last sentence is the geometric form of the companion paper’s currency-price bound and is used in the horizon arguments and in the gauge control behind Theorem 2.16 and Proposition 3.2.

Definition 2.2 (Class, panel, feasibility body) . Let $r_0\in \ri \Omega$ (quote-Slater: the initial curve is non-degenerate in the declared support). The single-date class is

\cP (r_0) :=\Bigl \{\mu \ \text {Borel probability on }\Omega :\ \textstyle \int |x|_1\,\dd \mu <\infty ,\ \int x\,\dd \mu =r_0\Bigr \},

the collapse of the martingale rows at one date. A panel is a finite family

f=(f_1,\ldots ,f_n)

of Borel payoffs

f_i:\Omega \to \R

with

|f_i(x)|\le \kappa (1+|x|_1)

. The feasibility body is

\cV :=\Bigl \{\Bigl (\textstyle \int f_1\,\dd \mu ,\ldots , \int f_n\,\dd \mu \Bigr ):\mu \in \cP (r_0)\Bigr \}\subseteq \R ^n ,

a convex set; a bid–ask panel

[\ell ,u]

is feasible iff

[\ell ,u]\cap \cV \ne \emptyset

2.1Piecewise-linear panels: the polytope theorem

Caplets and physical swaptions are piecewise linear in the anchor coordinates, so this is the case to understand first. After Definition 2.3 fixes the class of panels, Lemma 2.4 describes the affine functions that are nonnegative on a polyhedral cell, and Lemma 2.5 turns this description into a finite polyhedral support function. The horizon cone of Definition 2.6 records the mass that can escape along recession directions. Theorem 2.7 combines these pieces: the closure of the body is a polytope, every relative facet carries an explicit static certificate, and a quote in the closed body is attained exactly when it admits a representation without escaping mass. Corollary 2.8 and Remark 2.10 make the resulting inequality system concrete.

Definition 2.3 (Piecewise-linear panel) . The panel $f$ is piecewise linear (PL) if there are finitely many polyhedral cells $C_1,\ldots ,C_m$ with $\Omega =\bigcup _jC_j$ and, on each $C_j$ , $f(x)=\alpha _j+A_jx$ for a vector $\alpha _j\in \R ^n$ and a matrix $A_j\in \R ^{n\times k}$ .

Caplets, physical payer and receiver swaptions, forward rate agreements (FRAs), and bond options are PL in the anchor coordinates: for example the natural-timing caplet anchor payoff is $\delta \Zbar (b)\,(L-K)^+=(x_a-(1+\delta K)x_b)^+$ , and the payer swaption is $(\,x_{a_0}-x_{q_m}-K\sum _j\delta _jx_{q_j})^+$ . CMS and cash-settled payoffs are piecewise rational and are treated in Theorem 2.15.

Lemma 2.4 (Affine domination cone over a polyhedron) . Let $C=\{x:Ex\le e\}$ be nonempty. Then

K(C):=\bigl \{(c_0,c)\in \R \times \R ^k:\ c_0+c\cdot x\ge 0\ \forall x\in C\bigr \} =\bigl \{(c_0,c):\ \exists \,y\ge 0,\ c=-E^{\mathsf T}y,\ c_0\ge e\cdot y\bigr \},

a polyhedral convex cone.

Proof.If $c=-E^{\mathsf T}y$ with $y\ge 0$ and $c_0\ge e\cdot y$ , then for $x\in C$ : $c\cdot x=-y\cdot Ex\ge -y\cdot e$ , so $c_0+c\cdot x\ge c_0-e\cdot y\ge 0$ . Conversely, if $\min _{x\in C}(c_0+c\cdot x)\ge 0$ , the minimum is finite, and linear-programming duality for $\min \{c\cdot x:Ex\le e\}$ supplies $y\ge 0$ with $c=-E^{\mathsf T}y$ and optimal value $-e\cdot y$ , whence $c_0\ge e\cdot y$ . The set is the image of the polyhedron $\{(y,c_0):y\ge 0,\ c_0\ge e\cdot y\}$ under a linear map, hence polyhedral (Rockafellar, 1970, Section 19).∎

Applied cell by cell, the lemma yields the support function of a piecewise-linear panel.

Lemma 2.5 (Polyhedral support function) . For a PL panel define

h(\lambda ):=\inf \bigl \{a+b\cdot r_0:\ a+b\cdot x\ge \lambda \cdot f(x)\ \forall x\in \Omega \bigr \}, \qquad \lambda \in \R ^n .

Then

h

is a polyhedral, positively homogeneous, convex function, finite everywhere, and for every

\lambda

the infimum is attained.

Proof.The constraint set $F:=\{(\lambda ,a,b):a+b\cdot x\ge \lambda \cdot f(x)\ \text {on }\Omega \}$ decomposes over cells as $F=\bigcap _j\{(\lambda ,a,b):(a-\lambda \cdot \alpha _j,\ b-A_j^{\mathsf T}\lambda )\in K(C_j)\}$ , an intersection of preimages of the polyhedral cones of Lemma 2.4 under linear maps: $F$ is a polyhedral cone. The epigraph of $h$ is the image of $F$ under $(\lambda ,a,b)\mapsto (\lambda ,\,a+b\cdot r_0)$ together with the vertical ray; linear images of polyhedra are polyhedra, so $h$ is polyhedral convex, and positive homogeneity is the homogeneity of $F$ . Finiteness: with $|f_i|\le \kappa (1+|x|_1)$ , the affine function $a=n\kappa |\lambda |_\infty$ , $b=n\kappa |\lambda |_\infty \one$ dominates $\lambda \cdot f$ on the positive orthant, so $h(\lambda )<\infty$ ; and $h(\lambda )\ge$ the value at any single $\mu \in \cP (r_0)$ , hence $h>-\infty$ . Attainment: the infimum of a linear function over the nonempty polyhedral fiber $\{(a,b):(\lambda ,a,b)\in F\}$ is attained whenever finite.∎

Definition 2.6 (Horizon cone) . The horizon cone of the PL panel is the polyhedral cone

\cH :=\sum _{j=1}^{m}\bigl \{(d,\,A_jd):\ d\in \rec C_j\bigr \} \subseteq \R ^k\times \R ^n ,

a finite Minkowski sum of finitely generated cones (Minkowski–Weyl), hence finitely generated and closed. The relaxed body is

\widehat \cV :=\Bigl \{\textstyle \int f\,\dd \mu +w:\ \mu \ \text {prob on }\Omega ,\ \int |x|_1\dd \mu <\infty ,\ \int x\,\dd \mu +z=r_0,\ (z,w)\in \cH \Bigr \}.

The pair $(z,w)$ is the defect: first-moment mass $z$ escaped to infinity along cell directions, carrying payoff value $w$ at the cells’ asymptotic slopes. This is the single-date form of the escape phenomenon of Huang (2026).

Theorem 2.7 (Polytope theorem) . Let the panel be PL under Assumption 2.1 with $r_0\in \ri \Omega$ . Then:

1.: $\widehat \cV$ is a polytope whose support function is $h$ ;
2.: $\overline {\cV }=\widehat \cV$ , and $\ri \widehat \cV \subseteq \cV \subseteq \widehat \cV$ ;
3.: every relative-facet inequality $\lambda ^*\!\cdot v\le h(\lambda ^*)$ in $\operatorname {aff}\widehat \cV$ is certified by an explicit static portfolio: the attained dominator $(a^*,b^*)$ of Lemma 2.5 satisfies $a^*+b^*\cdot x-\lambda ^*\!\cdot f(x)\ge 0$ pathwise, i.e. cash $a^*$ plus the curve position $b^*$ superreplicates the panel combination $\lambda ^*\!\cdot f$ at cost $h(\lambda ^*)$ ;
4.: every $v\in \widehat \cV$ (in particular, every point of its relative boundary) belongs to $\cV$ if and only if it admits a representation with defect $z=0$ .

Proof.(a) The generator cone. Let $G_0:=\cone \{(1,x,f(x)):x\in \Omega \}\subseteq \R ^{1+k+n}$ . Over each cell, $\{(1,x,\alpha _j+A_jx):x\in C_j\}$ is a polyhedron; by the Motzkin decomposition $C_j=Q_j+\rec C_j$ with $Q_j$ a polytope, the closed conic hull of a polyhedron $P=Q+R$ is $\overline {\cone (\{1\}\times P)} =\cone (\{1\}\times Q)+\{0\}\times R$ . Applying this cellwise and taking convex hulls of finitely many polyhedral cones,

G:=\overline {G_0} =G_0+\{0\}\times \cH ' , \qquad \cH ':=\{(d,A_jd):d\in \rec C_j,\ j\le m\}\text {-cone},

is a polyhedral cone, and

\cH '=\cH

. (All hulls here are of finitely many polyhedra, and closures of such hulls are polyhedral (Rockafellar, 1970, Section 19).)

(b) $\widehat \cV$ as a slice. Every $\mu \in \cP (r_0)$ has barycenter $\int (1,x,f(x))\,\dd \mu \in G$ : the integrand is supported in the closed convex $G$ with integrable norm, so the barycenter lies in $G$ . Conversely, points of $G$ with first coordinate $1$ are, by Carathéodory for the polyhedral cone, finite convex combinations of generators: atoms in $\Omega$ plus horizon generators. Hence

\widehat \cV =\bigl \{v:\ (1,r_0,v)\in G\bigr \},

a slice of a polyhedral cone: a polyhedron; bounded by the growth bound (the same estimate as after Definition 2.2, since horizon values obey

|w|\le \kappa '|z|_1\le \kappa '|r_0|_1

): a polytope. Its support function:

\sup \{\lambda \cdot v:(1,r_0,v)\in G\}

equals

h(\lambda )

by polyhedral linear-programming duality. The dual constraints of the slice program say exactly that the dual variables

(a,b)

—the multipliers of the mass and moment rows—belong to the domination cone

F_\lambda

, including the horizon rows

b\cdot d\ge \lambda \cdot A_jd

, which are implied by domination on the unbounded cells (divide

a+b\cdot (x_j+td)\ge \lambda f(x_j+td)

t\to \infty

). Polyhedral programs have no duality gap and attain (Rockafellar, 1970, Section 28).

(c) Sandwich. $\cV \subseteq \widehat \cV$ is the barycenter statement with $z=0$ . Density and the relative-interior inclusion: let $v\in \ri \widehat \cV$ with representation $(\mu ,z,w)$ , $(z,w)=\sum _l s_l(d_l,A_{j_l}d_l)$ . For $t>0$ replace each horizon generator by a far atom: weight $w_l(t):=s_l/t$ at the point $x_{j_l}+t\,d_l\in C_{j_l}$ , with $x_{j_l}\in C_{j_l}$ fixed; then $w_l(t)\to 0$ while $w_l(t)\,(x_{j_l}+td_l)\to s_ld_l$ and $w_l(t)\,f(x_{j_l}+td_l)\to s_lA_{j_l}d_l$ . Renormalizing the $\mu$ -part accordingly, the resulting probability measures $\mu _t\in \cP (r_t)$ have moments $r_t\to r_0$ and values $v_t\to v$ .

Moment correction. The corrected measures are constructed explicitly. Since $r_0\in \ri \Omega$ , choose points $x_0,\ldots ,x_p\in \Omega$ whose affine hull is $\operatorname {aff}\Omega$ and weights $\eta _i>0$ , $\sum _i\eta _i=1$ , with $r_0=\sum _i\eta _ix_i$ (Rockafellar, 1970, Theorem 6.4). For $\varepsilon \in (0,1)$ and $m\in \Omega$ consider the affine system

(1-\varepsilon )\,m+\sum _i\gamma _i x_i=r_0, \qquad \sum _i\gamma _i=\varepsilon ,\qquad \gamma _i\ge 0 .

m=r_0

it has the strictly positive solution

\gamma _i=\varepsilon \eta _i

; the solution of the linear system depends continuously on

m

within

\operatorname {aff}\Omega

, so there are

\varepsilon _t\downarrow 0

and, for all large

t

, nonnegative solutions

\gamma (t)

with

m=r_t

. The corrected laws

\tilde \mu _t:=(1-\varepsilon _t)\mu _t+\sum _i\gamma _i(t)\, \delta _{x_i}

lie in

\cP (r_0)

and their values satisfy

\int f\,\dd \tilde \mu _t=(1-\varepsilon _t)v_t +\sum _i\gamma _i(t)f(x_i)\to v

. Hence every

v\in \ri \widehat \cV

lies in

\overline {\cV }

, so

\widehat \cV =\overline {\ri \widehat \cV }\subseteq \overline {\cV } \subseteq \widehat \cV

(the last inclusion because

\widehat \cV

is closed and contains

\cV

by the barycenter statement), giving

\overline {\cV }=\widehat \cV

Exactness on the relative interior. $\cV$ is convex, so $\ri \widehat \cV =\ri \,\overline {\cV }=\ri \cV \subseteq \cV$ (Rockafellar, 1970, Theorem 6.3). No correction with exact value is required: exactness on the relative interior is a consequence of convexity once the closure identity is proved.

(d) If $v\in \cV$ then $z=0$ by definition. Conversely a zero-defect representation is a measure in $\cP (r_0)$ attaining $v$ . Item 3 of the statement is Lemma 2.5 attainment plus weak duality.∎

Corollary 2.8 (Finite closed-body certificate system) . Let $A=\operatorname {aff}\widehat \cV$ . There are affine equations $Hv=d$ defining $A$ and finitely many relative-facet inequalities $\lambda _l\cdot v\le h_l$ such that

\widehat \cV =\{v:Hv=d,\ \lambda _l\cdot v\le h_l\ \text {for every }l\}.

Each inequality has the explicit cash-plus-curve certificate of Theorem 2.7. Thus a bid–ask box

[\ell ,u]

is feasible to arbitrarily small calibration error exactly when its intersection with this system is nonempty. Exact feasibility additionally requires a zero-defect representation. Meeting

\ri _A\widehat \cV

is sufficient; relative-facet inequalities alone cannot decide attainability on boundary faces.

Remark 2.9 .The classical single-maturity call-panel conditions (Davis and Hobson, 2007) are the case $k=1$ , cells given by the strikes, with mass and mean pinned; Theorem 2.7 is their weighted, multi-coordinate closure with the attainability defect made explicit.

Remark 2.10 (Static inequalities of a caplet–swaption panel) . Under Assumption 2.1 the relative facets of Corollary 2.8 are certified by static positions, and for the standard rates panel the most familiar static inequalities can be written down together with their certificates. For an accrual $[a,b]$ with accrual factor $\delta$ and strike $K$ let $c_K(x):=(x_a-(1+\delta K)x_b)^+$ be the natural-timing caplet, and for a swap on $a_0=q_0<q_1<\cdots <q_m$ with accrual factors $\delta _j$ on $[q_{j-1},q_j]$ let $p_K(x):=(x_{q_0}-x_{q_m}-K\sum _j\delta _jx_{q_j})^+$ be the physical payer swaption. Each payoff inequality below holds at every point of $\Omega$ . Integrating it against any $\mu \in \cP (r_0)$ gives the same inequality between quotes, with each curve coordinate $x_u$ replaced by $r_{0,u}$ . A curve position on the right-hand side is the static certificate; where none appears the certificate is zero, that is, the panel combination is nonpositive pathwise.

(i): Bounds. For $K\ge 0$ , maturity monotonicity gives $0\le c_K\le x_a-x_b$ . The upper certificate is the curve position $x_a-x_b$ , of cost $r_{0,a}-r_{0,b}$ .
(ii): Strike spreads. For $K_1<K_2$ , $0\le c_{K_1}-c_{K_2}\le \delta (K_2-K_1)\,x_b$ . The upper certificate is the curve position $\delta (K_2-K_1)\,x_b$ , of cost $\delta (K_2-K_1)\,r_{0,b}$ .
(iii): Butterflies. For $K_1<K_2<K_3$ and $\theta :=(K_3-K_2)/(K_3-K_1)$ , $c_{K_2}\le \theta \,c_{K_1}+(1-\theta )\,c_{K_3}$ , because $K\mapsto c_K(x)$ is convex.
(iv): Support wings. If $\Omega$ carries the declared rate cap $x_a\le (1+\delta L_{\max })\,x_b$ and $K_1<K_2<L_{\max }$ , then
$c_{K_2}\le \frac {L_{\max }-K_2}{L_{\max }-K_1}\,c_{K_1} \qquad \text {on }\Omega .$
Indeed, for fixed $x_b>0$ the left side minus the right side vanishes at the cap, is affine in $x_a$ with positive slope on $[(1+\delta K_2)x_b,(1+\delta L_{\max })x_b]$ , and is nonpositive below that interval, while $x_b=0$ forces $x_a=0$ . This inequality uses the declared support and fails without the cap.
(v): Swaptions against caplets. The telescoping identity
$x_{q_0}-x_{q_m}-K\sum _{j=1}^m\delta _jx_{q_j} =\sum _{j=1}^m\bigl (x_{q_{j-1}}-(1+\delta _jK)\,x_{q_j}\bigr )$
and subadditivity of the positive part give $p_K\le \sum _{j=1}^mc^{(j)}_K$ , where $c^{(j)}_K:=(x_{q_{j-1}}-(1+\delta _jK)\,x_{q_j})^+$ is the natural-timing caplet on $[q_{j-1},q_j]$ fixed at $e$ . The familiar comparison of a payer swaption with the cap at the same strike, whose caplets fix at their own dates $q_{j-1}\ge e$ , follows by combining this static inequality with Lemma 4.1 under that lemma’s hypotheses. That second step holds in expectation under the class, not pathwise.

These are valid inequalities, not a facet list: which of them are relative facets of $\widehat \cV$ depends on $\Omega$ , on $r_0$ and on the panel, and Corollary 2.8 guarantees only that every relative facet carries a static certificate of this kind.

2.2A full-marginal complement: constant-weight baskets

The body of Definition 2.2 is determined by finitely many expectations of a fixed panel. It is worth recording, alongside it, the sharper question in which the whole marginal of each component is prescribed, because the two are answered by different mechanisms and the contrast is informative for the attainment discussion of Section 3. The results of this subsection are self-contained, are used nowhere else in this paper, and may be skipped.

Remark 2.11 (This is a different problem, not a reformulation) . Two differences from Definition 2.2 are essential and must not be elided. (i) Data: here each component law $\mu _i$ is prescribed in full, i.e. all-strike information, and the criterion quantifies over all continuous tests of at most linear growth; finitely many expectations of a panel do not determine a marginal, so the results below do not transfer to a finite quote vector without a separate reduction. (ii) Support: the support is a Cartesian product $K=\prod _{i\le d}K_i$ of compact sets, with couplings ranging over $\Pi (\mu _1,\ldots ,\mu _d)$ , whereas the $\Omega$ of Assumption 2.1 carries cross-coordinate curve constraints and is not a product. The statements below are therefore a parallel branch on a product support, not a specialization of the body of this paper. Re-proving them for couplings constrained to a non-product $\Omega$ is possible in principle but changes the dual class and is not attempted here.

Theorem 2.12 (Exact constant-weight component–basket criterion) . Let $K=\prod _{i\le d}K_i$ with each $K_i\subset \R$ compact, let $w_i>0$ with $\sum _iw_i=1$ , let $B(x)=\sum _i w_ix_i$ , and put $K_B=B(K)$ . Fix $\mu _i\in \cP (K_i)$ and a proposed basket law $\nu \in \cP (K_B)$ . The following are equivalent.

1.: There is a coupling $\pi \in \Pi (\mu _1,\ldots ,\mu _d)$ with $B_\#\pi =\nu$ .
2.: For all $\varphi _i\in C(K_i)$ and $\psi \in C(K_B)$ with $\sum _i\varphi _i(x_i)+\psi (B(x))\ge 0$ on $K$ , one has $\sum _i\int \varphi _i\,\dd \mu _i+\int \psi \,\dd \nu \ge 0$ .
3.: $\int \psi \,\dd \nu \le \sup _{\pi \in \Pi (\mu _1,\ldots ,\mu _d)} \int \psi (B(x))\,\dd \pi$ for every $\psi \in C(K_B)$ .

When they fail, the violating $(\varphi _i,\psi )$ is a pointwise nonnegative, strictly negatively priced continuous observable-test portfolio. If cash, forwards, and all component and basket calls are executable, it may be chosen as a finite portfolio of those vanilla claims.

Proof. $\Pi (\mu _1,\ldots ,\mu _d)$ is weakly compact and $\pi \mapsto B_\#\pi$ is weakly continuous into $\cP (K_B)$ , so the attainable set of basket laws is compact and convex; membership of $\nu$ is therefore equivalent to the support-function inequality in (3), which is (2) after moving $\psi$ to the other side. The equivalence of (2) with (1) is the compact continuous-cost multimarginal Kantorovich duality with the linear constraint $B_\#\pi =\nu$ adjoined; see Villani (2009) for the duality and Zaev (2015) for the linear-constraint extension used here. For the vanilla statement, extend each continuous potential from its compact support to the compact convex-hull interval by linear interpolation across the complementary intervals, and approximate the extension uniformly by a piecewise-affine function. Add a cash buffer equal to the sum of the uniform errors, so pathwise nonnegativity is preserved; choosing the errors below the strict price gap preserves strict negativity. Every continuous piecewise-affine function on a compact interval is a finite linear combination of cash, the underlying forward, and call hinges.∎

Corollary 2.13 (Jensen bounds are necessary) . If $\nu$ is attainable then $\int x\,\dd \nu =\sum _iw_i\int x\,\dd \mu _i$ and, for every convex $\phi$ , $\int \phi \,\dd \nu \le \sum _iw_i\int \phi \,\dd \mu _i$ whenever the right side is defined.

Proof.Linearity of $B$ gives the mean identity. For convex $\phi$ , $\phi (B(x))\le \sum _iw_i\phi (x_i)$ pointwise when $\sum _iw_i=1$ and $w_i\ge 0$ ; integrate against a coupling with the prescribed marginals.∎

Example 2.14 (Convex order is not sufficient) . Let $d=2$ , $w_1=w_2=1/2$ , and let both components put mass $1/2$ on each of $-1$ and $1$ (an abstract instance: no positivity is claimed, and Assumption 2.1 is not in force in this subsection). Every attainable basket law is supported on $\{-1,0,1\}$ . The uniform law on $[-1,1]$ has mean zero and is dominated in convex order by the symmetric Bernoulli law, so it satisfies every bound of Corollary 2.13; it is nevertheless unattainable, its support being uncountable. Hence the full family of Theorem 2.12(2), not a scalar list of convex-order inequalities, is what decides the question.

2.3Piecewise-rational panels: semialgebraic bodies

We return to the finite-expectation body of Definition 2.2. CMS caplets and coupons and cash-settled swaptions are not piecewise linear in the anchor coordinates: their payoffs are piecewise rational, with denominators positive on $\Omega$ . Polyhedrality can then no longer be expected, but the body keeps a finite algebraic description.

Theorem 2.15 (Semialgebraic quote body) . Let the panel consist of semialgebraic payoffs of linear growth (this includes CMS caplets and coupons and cash-settled swaptions, whose payoffs are piecewise rational with denominators positive on $\Omega$ ). Then $\cV$ is semialgebraic; consequently $\overline {\cV }$ is a compact semialgebraic convex set. In particular it admits a finite Boolean description by polynomial equalities and strict or weak inequalities; no facet description is asserted.

Proof.By the Richter–Rogosinski reduction (Richter, 1957; Rogosinski, 1958), any $\mu \in \cP (r_0)$ admits an atomic $\nu$ with at most $k+n+2$ atoms in $\Omega$ matching the $k+n+1$ integrals $(1,x,f)$ . Hence ∎

\cV =\Bigl \{\textstyle \sum _lp_lf(x^{(l)}):\ x^{(l)}\in \Omega ,\ p\in \Delta _{k+n+1},\ \sum _lp_lx^{(l)}=r_0\Bigr \},

the image of a semialgebraic set under a semialgebraic map (the graph of

f

is semialgebraic), which is semialgebraic by Tarski–Seidenberg (Bochnak et al., 1998). Boundedness is the growth estimate; finite presentability is the finiteness of semialgebraic descriptions.

2.4The strip: exact conic geometry and the attainment boundary

The polytope theorem leaves one defect: quotes that can be approached only by sending mass along recession directions, along each of which $\zeta$ increases by Assumption 2.1. A quoted budget on the second moment of $\zeta$ rules out exactly this escape, because an escaping atom that carries a bounded defect has diverging quadratic cost (Remark 2.20). This subsection shows what the budget buys at one date. Theorem 2.16 describes the budgeted body exactly: it is second-order-cone (SOC) representable, with one scalar cone block per linearity cell, and its support certificates are attained. Remark 2.17 shows that a positive budget multiplier is a shadow price rather than a sign of curvature, and Proposition 2.18 and Remark 2.19 identify the growth tier at which the optimal hedge is still attained but a maximizing law may fail to exist.

For the next result take full polyhedral descriptions $C_j=\{x:E_jx\le e_j\}$ , including the constraints inherited from $\Omega$ . In particular the positivity and price-bound inequalities are present in every cell description.

Theorem 2.16 (Exact strip-augmented SOC body) . Let the panel be PL, let $q(x):=x_\zeta ^2$ , and let $\beta$ have Slater room: some $\mu \in \cP (r_0)$ satisfies $\int q\,\dd \mu <\beta$ . Then the exact body $\cV _\beta :=\{\int f\dd \mu :\mu \in \cP (r_0),\ \int q\dd \mu \le \beta \}$ is the compact finite-dimensional projection

\begin{equation} \begin {split} \cV _\beta =\bigl \{v:\ \exists (t_j,m_j,s_j)_{j\le m}\quad & t_j,s_j\ge 0,\quad \sum _jt_j=1,\quad \sum _jm_j=r_0,\\ &E_jm_j\le e_jt_j,\quad m_{j,\zeta }^{2}\le t_js_j, \quad \sum _js_j\le \beta ,\\ &v=\sum _j(\alpha _jt_j+A_jm_j)\bigr \}. \end {split} \label {eq:pc2-primal-soc} \end{equation}

Thus one scalar rotated-SOC block

m_{j,\zeta }^2\le t_js_j

suffices per cell. Its support function is

\begin{equation} h_\beta (\lambda ) =\min _{\nu \ge 0}\Bigl [\nu \beta + \min \{a+b\cdot r_0: a+b\cdot x+\nu q(x)\ge \lambda \cdot f(x)\ \text {on }\Omega \} \Bigr ], \label {eq:pc2-hbeta} \end{equation}

and both minima are attained. Cellwise, domination is equivalent to the existence of

y_j\ge 0

such that, with

\ell _j=b-A_j^{\mathsf T}\lambda +E_j^{\mathsf T}y_j, \qquad c_j=a-\lambda \cdot \alpha _j-e_j\cdot y_j,

\begin{equation} (\ell _j)_u=0\ (u\ne \zeta ),\qquad c_j\ge 0,\qquad 4\nu c_j\ge (\ell _j)_\zeta ^2. \label {eq:pc2-soc} \end{equation}

\nu ^*=0

, an unbudgeted PL affine dominator already certifies that support direction. If an optimal multiplier satisfies

\nu ^*>0

, every maximizing law exhausts the budget and

h_{\beta '}(\lambda ) \le h_\beta (\lambda )+\nu ^*(\beta '-\beta ).

Hence

\nu ^*

is a shadow price. Its sign does not determine the dimension or curvature of the exposed face.

Proof.Choose a Borel partition $D_j\subseteq C_j$ of $\Omega$ . Given a feasible law, put

t_j=\mu (D_j),\qquad m_j=\int _{D_j}x\,\dd \mu , \qquad s_j=\int _{D_j}x_\zeta ^2\,\dd \mu .

Integration of the cell inequalities gives

E_jm_j\le e_jt_j

, and Cauchy–Schwarz gives

m_{j,\zeta }^2\le t_js_j

; all the other rows of (4) are immediate. Conversely, if

t_j>0

, then

x_j=m_j/t_j\in C_j

. If

t_j=0

, the cone constraint gives

m_{j,\zeta }=0

and

E_jm_j\le 0

puts

m_j

\rec C_j

; the price-bound recession property forces

m_j=0

. Therefore

\mu =\sum _{t_j>0}t_j\delta _{x_j}

has the required mean and panel value, while

\int q\,\dd \mu =\sum _{t_j>0}\frac {m_{j,\zeta }^2}{t_j} \le \sum _js_j\le \beta .

This proves exactness. The lift is compact:

t

and

s

are bounded, and positivity together with

\sum _jm_j=r_0

bounds every component of every

m_j

. Its projection

\cV _\beta

is therefore compact.

For the support formula, the budget value as a function of its right-hand side is finite, concave and nondecreasing. Slater room puts $\beta$ in the relative interior of its domain, so it has a supergradient $\nu ^*\ge 0$ , which gives the outer equality in (5). For fixed $\nu >0$ , write $g=\lambda \cdot f-\nu q$ . Any maximizing sequence whose objective is bounded below has uniformly bounded second moment: linear growth, the pinned first moment and the term $-\nu q$ give $\int g\le C_0+C_1|r_0|_1-\nu \int q$ . The price bound then gives uniform integrability of every state coordinate, so a weak limit preserves the mean. Tightness and the upper-semicontinuity bound give an optimizer. The resulting finite concave mean-value function has a supergradient at $r_0\in \ri \Omega$ ; testing point masses gives an attained affine dominator. When $\nu =0$ this is Lemma 2.5. Finally, minimizing the convex quadratic over each polyhedral cell and applying polyhedral quadratic-programming duality gives (6). Complementary slackness and the supergradient inequality give the last assertions.∎

Remark 2.17 (Positive shadow price can expose a flat facet) . Let

\Omega =\{(\zeta ,y):\zeta \ge \tfrac 12, \ \tfrac 14\le y\le \zeta -\tfrac 14\},\quad r_0=(1,\tfrac 12),\quad \beta =\tfrac 54,

and

f=(|\zeta -1|,(y-\tfrac 34)^+)

. The moment constraint gives

\operatorname {Var}(\zeta )\le 1/4

, hence

\E |\zeta -1|\le 1/2

. Equality forces

\zeta \in \{1/2,3/2\}

with equal probabilities. The feasible laws

\delta _{(1,1/2)},\qquad \tfrac 12\delta _{(1/2,1/4)}+\tfrac 12\delta _{(3/2,3/4)},

and

\tfrac 12\delta _{(1/2,1/4)}+ \tfrac 14\delta _{(3/2,1/4)}+ \tfrac 14\delta _{(3/2,5/4)}

have panel values

(0,0),(1/2,0),(1/2,1/8)

, respectively. Consequently the body is two-dimensional and conditional choices of

y

realize the flat exposed facet

\{1/2\}\times [0,1/8]

. Without the budget a two-point escape law with mass

p_R=(1/2)/(R-1/2)

(R,R/2)

and the remaining mass at

(1/2,1/4)

has mean

r_0

and makes the first payoff tend to

1

. Since

|\zeta -1|\le \zeta

[1/2,\infty )

, the unbudgeted support in this direction is exactly

1

, whereas

|\zeta -1|\le (\zeta -1)^2+\tfrac 14 =\zeta ^2-2\zeta +\tfrac 54

has budgeted cost

1/2

. Thus every optimal budget multiplier is positive although the exposed face is flat.

The budget also creates a growth tier of its own. For claims growing at most like the budgeted moment, the next proposition keeps duality and an attained optimal hedge under a relative-interior condition, while Remark 2.19 shows that the supremum over models need not be attained.

Proposition 2.18 (Critical quadratic tier: attained hedge, possible model nonattainment) . Assume $r_0\in \ri \Omega$ . Let $f$ be a finite exact book with quote $c$ , and let $X,f_1,\ldots ,f_n:\Omega \to \R$ be Borel functions satisfying $|X|+\sum _i|f_i|\le C(1+q)$ . Define the attainable parameter domain

\mathcal D:=\{(z,u):\exists \mu \in \cP (r_0),\ \E _\mu q\le z,\ \E _\mu f=u\}.

(\beta ,c)\in \ri \mathcal D

, then

\begin{equation} \begin {split} &\sup _{\substack {\mu \in \cP (r_0)\\ \E _\mu q\le \beta ,\ \E _\mu f=c}}\E _\mu X\\ &\quad =\min _{\substack {a\in \R ,\ b\in \R ^k,\ \nu \ge 0,\ \ell \in \R ^n\\ a+b\cdot x+\nu q(x)+\ell \cdot f(x)\ge X(x)\ \forall x\in \Omega }} \bigl (a+b\cdot r_0+\nu \beta +\ell \cdot c\bigr ). \end {split} \label {eq:pc2-critical-dual} \end{equation}

The minimum is attained. No maximizing-law assertion is made at this growth tier. If

X

and the exact book are continuous and, for every

\varepsilon >0

, satisfy

|X|+\sum _i|f_i|\le C_\varepsilon +\varepsilon q

, then the primal supremum is attained as well.

Proof.Let ∎

\Psi (z,u):=\sup \{\E _\mu X:\mu \in \cP (r_0), \ \E _\mu q\le z,\ \E _\mu f=u\}.

It is finite and concave on

\mathcal D

. At the relative-interior point

(\beta ,c)

it has a supergradient

(\nu ,\ell )

, with

\nu \ge 0

by monotonicity in

z

. Put

W(m):=\sup _{\substack {\mu \ \mathrm {Borel\ probability\ on}\ \Omega \\ \E _\mu x=m,\ \E _\mu q<\infty }} \E _\mu [X-\nu q-\ell \cdot f].

The supergradient inequality and calibrated near-optimizers give

W(r_0)=\Psi (\beta ,c)-\nu \beta -\ell \cdot c<\infty

. If

W(m)=\infty

for some

m\in \Omega

, relative interior supplies

y\in \Omega

and

\theta \in (0,1)

with

r_0=\theta m+(1-\theta )y

; mixing the exploding laws at

m

with

\delta _y

contradicts finiteness at

r_0

. Hence

W

is finite and concave on

\Omega

and has a supergradient

b

r_0

. Point masses yield

X(x)-\nu q(x)-\ell \cdot f(x) \le W(x)\le W(r_0)+b\cdot (x-r_0),

which constructs the minimizing hedge. Weak duality proves (7). In the subcritical continuous case, the moment bound gives weak compactness, while the displayed

\varepsilon q

estimate gives uniform tail control and hence continuity of all constrained expectations; the feasible set is closed and the maximum is attained.

Remark 2.19 (The distinction is sharp) . On $\Omega =[1/2,\infty )$ take $r_0=1$ , $\beta =2$ and

X(\zeta )=(\zeta -1)^2\frac {\zeta }{1+\zeta }.

Every feasible nondegenerate law has

\E X<\E (\zeta -1)^2\le 1

. For

R\ge 3

, set

a_R=1-1/(R-1)

and

p_R=1/((R-1)^2+1)

. The law

(1-p_R)\delta _{a_R}+p_R\delta _R

has mean one, second moment two, and expectation of

X

tending to one. Thus the value is one but no probability law attains it. The strip hedge

X(\zeta )\le (\zeta -1)^2=\zeta ^2-2\zeta +1

attains cost one. The market is completed in the price-and-hedge sense even though an ordinary extremal model is absent.

Remark 2.20 (Two horizon scales) . An atom of mass $c/R$ at $x+Rd$ carries an $O(1)$ PL defect, but its quadratic budget cost is asymptotic to $cR\,d_\zeta ^2$ and therefore diverges. The strip eliminates every such first-order horizon generator, which is why (4) is exact. Mass $c/R^2$ at the same point has vanishing first and PL moments but an $O(1)$ quadratic moment. This second-order concentration is invisible to $o(q)$ claims and may survive for $O(q)$ claims, causing the model nonattainment above. It does not reopen the companion paper’s market gap: the robust value remains finite, and under Proposition 2.18’s relative-interior hypothesis the optimal strip hedge exists. Over compact support neither escape mechanism is present.

3Quote completion and attained hedges

We now move from one date to a finite calendar $t_0<\cdots <t_N$ , in the market of Huang (2026), and ask what the quoted carrier buys once hedging is dynamic. Let $\cM$ be the martingale laws on the ambient-closed path space, with pinned initial curve and the currency-price bound. For post-initial dates fix accruals $\tau _i>0$ and write

s_i:=\frac {\zeta _{t_i}^2-\zeta _{t_i}}{\tau _i},\qquad \chi :=1+\sum _{i=1}^N\zeta _{t_i}^2, \qquad \cM _\chi :=\{Q\in \cM :\E _Q\chi <\infty \}.

The in-advance asks are $\beta _i$ . It is essential to distinguish the two growth tiers

\cC _\chi ^0:=\{f\in C(\Omega ): \forall \varepsilon >0\ \exists C_\varepsilon <\infty , \ |f|\le C_\varepsilon +\varepsilon \chi \}, \qquad \cC _\chi :=\{f\in C(\Omega ):|f|\le C_f\chi \}.

The first is the subcritical class used by the companion paper; the second is the critical quadratic class. The distinction cannot be removed: the example following Proposition 2.18 lies in $\cC _\chi \setminus \cC _\chi ^0$ and has no maximizing model.

3.1The traded gauge and dynamic support

Two ingredients are needed before any attainment statement can be made on the calendar. The first is a notion of admissible semistatic hedge whose running gains are bounded below by a multiple of the gauge (Definition 3.1). The quoted strips make that bound tradable, because cash, the strips and a buy-and-hold bond position replicate $\chi$ exactly (Proposition 3.2). The second ingredient is a way to realize the robust value of a claim by an actual strategy. Lemma 3.3 supplies the one-step building block, Assumption 3.4 lists the measurable structure needed to glue the steps together, and Theorem 3.5 produces the strategy. That theorem is used in the proofs of both attainment theorems of the next subsection, of the carrier completion in Section 3.3, and of the certificate classification in Section 6.

Definition 3.1 (Semistatic hedges) . A semistatic hedge consists of cash, finitely many static exact or one-sided book positions, and a predictable dynamic gain

D=\sum _{i,u}\phi _{i,u}(h_i) \bigl (\Zbar _{t_{i+1}}(u)-\Zbar _{t_i}(u)\bigr ).

Write

D^{(k)}:=\sum _{i<k,u}\phi _{i,u}(h_i) \bigl (\Zbar _{t_{i+1}}(u)-\Zbar _{t_i}(u)\bigr ),\qquad \chi _k:=1+\sum _{j=1}^k\zeta _{t_j}^2.

It is admissible when

D^{(k)}\ge -C\chi _k

pathwise for every

k

and one constant

C

; unpaid static claims are not included in this running-gain condition. A long ask position has nonnegative coefficient and is charged its ask. Superreplication is pointwise on

\Omega

Proposition 3.2 (The quoted strip trades the gauge) . For each $i$ , holding the $t_i$ -bond from $t_0$ to maturity produces the zero-cost gain

G_i:=\sum _{k<i} \bigl (\Zbar _{t_{k+1}}(t_i)-\Zbar _{t_k}(t_i)\bigr ) =\zeta _{t_i}-\Zbar _0(t_i).

Consequently

\begin{equation} 1+\sum _{i=1}^N\bigl (\tau _i s_i+\Zbar _0(t_i)+G_i\bigr )=\chi \quad \text {pathwise}. \label {eq:pc2-gauge-replication} \end{equation}

This admissible hedge costs

M_{\rm quoted}=1+\sum _i(\tau _i\beta _i+\Zbar _0(t_i))

. Hence every

X\le C\chi

C\ge 0

, has a finite explicit superhedge of cost at most

CM_{\rm quoted}

Proof.The gain telescopes and $\Zbar _{t_i}(t_i)=\zeta _{t_i}$ . Since $\tau _i s_i+\zeta _{t_i}=\zeta _{t_i}^2$ , summing gives (8). Every partial long-bond gain is bounded below by $-\Zbar _0(t_i)$ because bond prices are nonnegative, so the strategy is admissible. Scaling proves the last assertion.∎

Dynamic hedges are assembled one step at a time, and the next lemma is the one-step building block.

Lemma 3.3 (Finite concave-envelope dominator) . Let $S\subseteq \R ^d$ be convex, $m\in \ri S$ , and $g:S\to \R$ . Set

W(r):=\sup \Bigl \{\int g\,\dd \mu :\mu \text { is a finitely supported probability on }S,\ \int x\,\dd \mu =r\Bigr \}.

W(m)<\infty

, then

W

is finite on all of

S

and there is

b\in \R ^d

such that

g(x)\le W(m)+b\cdot (x-m)\qquad (x\in S).

No maximizing probability measure is asserted.

Proof. $W$ is concave and $W(r)\ge g(r)>-\infty$ . If $W(r)=\infty$ for some $r$ , relative interior supplies $y\in S$ and $\theta \in (0,1)$ with $m=\theta r+(1-\theta )y$ ; mixing would give $W(m)=\infty$ . Thus $W$ is finite. A finite concave function has a supergradient at a relative-interior point. The supergradient inequality, first on the segment from $m$ to $x$ and then at its endpoint, together with $g\le W$ , gives the claim.∎

Gluing these one-step dominators along the calendar requires the following standing structure.

Assumption 3.4 (Rectangular sections and tradable carried means) . At each step $x$ denotes only the vector of live traded normalized zero-bond prices; tape, stopped, or other nontraded coordinates may be part of the history $h_i$ but not of this one-step vector. Every full successor history is determined by a universally measurable update map $T_{i+1}(h_i,x)$ ; in particular, all next nontraded marks are measurable functions of $(h_i,x)$ . The support is generated recursively by nonempty convex sections $S_{i+1}(h_i)$ of the live-price vector and these update maps, and every path obtained by choosing successively from the sections belongs to $\Omega$ (rectangularity). The carried mean $m_{i+1}(h_i)$ lies in $\ri S_{i+1}(h_i)$ ; every affine row $b\cdot (x-m_{i+1}(h_i))$ is therefore the gain of an available dynamic curve position. Mean continuation belongs to the support and satisfies a uniform carry bound: its terminal gauge is at most $K(1+\chi _i(h_i))$ . Finally the backward values and the correspondences of affine dominators in Lemma 3.3 admit universally measurable selectors. Universal measurability, analytic sets and the selection theorems available for them are used throughout in the sense of Bertsekas and Shreve (1978, Ch. 7). These are explicit standing hypotheses; analyticity alone is not claimed to imply them.

The following realization statement plays the role that optional decomposition plays in nondominated discrete-time duality (Bouchard and Nutz, 2015): there the dual object is a family of measures with an analytic-graph structure and the strategy is obtained quasi surely, whereas here the class is pinned by a quoted gauge budget and the strategy is obtained pointwise on $\Omega$ under the standing selection hypotheses.

Theorem 3.5 (Dynamic support realization) . Under Assumption 3.4, let $Y$ be real and universally measurable with $Y\ge -C\chi$ , and put $h(Y):=\sup _{Q\in \cM _\chi }\E _QY$ . If $h(Y)<\infty$ , then an admissible universally measurable strategy $D^*$ exists such that

h(Y)+D^*\ge Y\quad \text {on }\Omega , \qquad h(Y)=\min \{a:a+D\ge Y\text { pathwise},\ D\text { admissible}\}.

Proof.Set $V_N=Y$ . Backward, let $V_i(h_i)$ be the finite-support concave-envelope value of $x\mapsto V_{i+1}(T_{i+1}(h_i,x))$ at $m_{i+1}(h_i)$ . Every admissible finite history can be assigned positive mass by a finite martingale tree: at each step relative interiority writes the carried mean as a strict mixture of the chosen next state and a compensating state. If some $V_i(h_i)$ were infinite, assign positive mass to that history and attach finite continuations with arbitrarily large value; the other finitely many branches have finite lower bounds because $Y\ge -C\chi$ . This would force $h(Y)=\infty$ . Hence all backward values are finite.

By Lemma 3.3 and the selector hypothesis, choose $b_i(h_i)$ with ∎

V_{i+1}(T_{i+1}(h_i,x)) \le V_i(h_i)+b_i(h_i)\cdot (x-m_{i+1}(h_i)).

Unwinding gives

V_0+D^*\ge Y

. Following deterministic mean continuation from a history shows

V_i(h_i)\ge -C K(1+\chi _i(h_i))

. The preceding inequalities then give

V_0+D^{*(i)}\ge V_i(h_i)

, and hence bound every cumulative dynamic gain from below by one gauge-order constant. Thus

D^*

is admissible. Finite trees approximate

V_0

from below, and rectangularity glues them into laws in

\cM _\chi

, giving

V_0\le h(Y)

. Conversely, fix

Q\in \cM _\chi

and take Borel versions of the selectors under the completed

Q

-history filtrations. Let

\tau _n

be the first date at which either the next selected position or the current gauge exceeds

n

, and stop before taking that increment. The stopped integrand is bounded, so its terminal gain

D_N^{(n)}

is integrable and has

Q

-expectation zero by the martingale rows. It is one of the running gains of the original strategy and therefore

D_N^{(n)}\ge -C\chi _{\tau _n}\ge -C\chi _N.

Finite calendar depth gives

D_N^{(n)}\to D^*

pathwise. Fatou applied to

D_N^{(n)}+C\chi _N

yields

\E _QD^*\le 0

. Thus

h(Y)\le V_0

and weak duality proves minimality.

3.2Subcritical and critical attainment

With the gauge traded and dynamic support realized, the calendar problem admits two attainment results that differ in the growth of the claim. For subcritical claims, Theorem 3.6 attains the maximizing model, the multipliers and the hedge. At critical growth, the growth of the carrier itself, Theorem 3.7 keeps the attained multipliers and hedge but makes no model assertion, and Remark 3.8 explains why strip bids must be excluded from the first result.

Encode exact book quotes by $\E _QH=c$ and every bid–ask side by $\E _QF\le q$ ; a band contributes both $g\le u$ and $-g\le -\ell$ . In the attained-model theorem below, however, the only strip components of $F$ are the ask rows $s_i$ : strip bids, lower quadratic-moment bounds, and exact strip rows are critical-order constraints and are excluded.

Theorem 3.6 (Attained subcritical semistatic duality with strip asks) . Let Assumption 3.4 and the companion paper’s weak compactness hypotheses hold. Write $F=(s_1,\ldots ,s_N,F^0)$ , where the first $N$ rows are exactly the asks $\E _Qs_i\le \beta _i$ , one for every $i=1,\ldots ,N$ , so they bound $\E _Q\chi$ uniformly; no strip bid or exact strip row is imposed. Let $X$ , $H$ , and every component of $F^0$ lie in $\cC _\chi ^0$ . Let

\mathcal A:=\{(\E _QH,r):Q\in \cM _\chi ,\ \E _QF\le r\}

and assume the relative moment-Slater condition

(c,q)\in \ri \mathcal A

. This is a joint condition on one attainable moment domain, not a collection of separate Slater witnesses. Then

\begin{equation} \begin {split} P&:=\max _{\substack {Q\in \cM _\chi \\ \E _QH=c,\ \E _QF\le q}}\E _QX\\ &=\min _{\lambda \in \R ^{\dim H},\,\eta \ge 0} \{\lambda \cdot c+\eta \cdot q+ h(X-\lambda \cdot H-\eta \cdot F)\}\\ &=\min \{\text {cost of an admissible semistatic superhedge of }X\}. \end {split} \label {eq:pc2-subcritical-dual} \end{equation}

The maximizing model, the finite-dimensional multipliers, and the superhedge are all attained. No strict-positivity assumption on the strip multipliers is needed; as ask multipliers they are merely nonnegative.

Proof.The strip asks bound $\E _Q\chi$ , and their sublevel constraints are weakly closed by lower semicontinuity of the quadratic moments. That compactness and the defining $\varepsilon \chi$ tail estimate make the feasible set compact and all subcritical expectations continuous; hence the primal maximum exists.

Let $P(c',q')$ denote the same constrained value at nearby right-hand sides. It is finite, concave and nondecreasing in $q'$ on its domain. Relative moment-Slater places $(c,q)$ in that domain’s relative interior, so $P$ has a supergradient $(\lambda ,\eta )$ there, with $\eta \ge 0$ . Applying its inequality to the moment vector of an arbitrary $Q\in \cM _\chi$ and then to calibrated near-optimizers gives ∎

P=\lambda \cdot c+\eta \cdot q+ h(X-\lambda \cdot H-\eta \cdot F).

The residual on the right is real, bounded below by

-C\chi

, and has finite support value. Theorem 3.5 supplies an attained dynamic dominator; adding the static positions gives the middle-to-last equality in (9). For the reverse inequality, localize any admissible dynamic gain, use the true martingale rows, and pass with Fatou; exact and one-sided static quotes then give ordinary weak duality.

At the carrier’s own growth rate the model side can fail, but the hedge side of the duality survives.

Theorem 3.7 (Critical calendar tier: attained hedge, no model assertion) . Let Assumption 3.4 hold and $\cM _\chi \ne \emptyset$ . Let $X:\Omega \to \R$ , $H:\Omega \to \R ^p$ , and $F:\Omega \to \R ^m$ be real universally measurable and satisfy

|X|+|H|_1+|F|_1\le C\chi .

The rows of

H

are quoted exactly and those of

F

are one-sided asks. Assume that

F

contains the ask-side strip row

s_i

at every post-initial date, so finite values of its corresponding quote coordinates bound

\E _Q\chi

. Put

\mathcal A_{\rm crit}:= \{(\E _QH,r):Q\in \cM _\chi ,\ \E _QF\le r\}.

(c,q)\in \ri \mathcal A_{\rm crit}

, then

\begin{equation} \begin {split} P&:=\sup _{\substack {Q\in \cM _\chi \\ \E _QH=c,\ \E _QF\le q}}\E _QX\\ &=\min _{\lambda \in \R ^p,\,\eta \in \R _+^m} \bigl \{\lambda \cdot c+\eta \cdot q +h(X-\lambda \cdot H-\eta \cdot F)\bigr \}\\ &=\min \{\text {cost of an admissible semistatic superhedge of }X\}. \end {split} \label {eq:pc2-critical-calendar-dual} \end{equation}

The finite-dimensional multipliers and the semistatic hedge are attained. No maximizing-model assertion is made.

Proof.For $(u,r)\in \mathcal A_{\rm crit}$ define ∎

\Phi (u,r):=\sup \{\E _QX:Q\in \cM _\chi , \ \E _QH=u,\ \E _QF\le r\}.

This is finite and concave on its domain. Indeed, the carrier rows bound

\E _Q\chi

at every finite

r

, and

X=O(\chi )

. At the relative-interior point

(c,q)

, a supergradient

(\lambda ,\eta )

exists; monotonicity in the ask vector gives

\eta \ge 0

. Applying its inequality at

(\E _QH,\E _QF)

for arbitrary

Q\in \cM _\chi

yields

\E _Q[X-\lambda \cdot H-\eta \cdot F] \le P-\lambda \cdot c-\eta \cdot q.

Conversely, calibrated near-optimizers and

\eta \ge 0

give the reverse inequality after taking the supremum over

Q

. Hence, for

Y:=X-\lambda \cdot H-\eta \cdot F

h(Y)=P-\lambda \cdot c-\eta \cdot q<\infty .

The growth assumption gives

Y\ge -C'\chi

, so Theorem 3.5 supplies an attained admissible

D^*

with

h(Y)+D^*\ge Y

pathwise. Adding the exact positions

\lambda \cdot H

and the nonnegative ask positions

\eta \cdot F

gives a superhedge of cost

P

. Weak duality gives the reverse inequality for every pair

(\lambda ,\eta )

and every admissible semistatic superhedge, proving both minima and their attainment.

Remark 3.8 (Why strip bids are excluded from model attainment) . The ask-only qualification is sharp. At one date let $\Omega =[1/2,\infty )$ , $\E \zeta =1$ , $\tau =1$ , $s=\zeta ^2-\zeta$ , and impose the band $1/2\le \E s\le 1$ . Its two inequality rows have joint moment domain

\mathcal A=\{(r_1,r_2):r_1\ge 0,\ r_1+r_2\ge 0\},

(1,-1/2)\in \operatorname {int}\mathcal A

. The continuous claim

X=-|\zeta -1|

is in

\cC _\chi ^0

. Put

v=1/2

and, for

R\ge 2

a_R=1-\frac {v}{R-1},\qquad p_R=\frac {v}{(R-1)^2+v},\qquad \mu _R=(1-p_R)\delta _{a_R}+p_R\delta _R.

Then

\E _{\mu _R}\zeta =1

\E _{\mu _R}s=v

, and

\E _{\mu _R}X=-\frac {2v(R-1)}{(R-1)^2+v}\longrightarrow 0.

Every feasible law has

\operatorname {Var}(\zeta )\ge v

, while equality

X=0

almost surely would force

\delta _1

. Hence the subcritical supremum is zero but is not attained. A strip bid can therefore be added to an attained-model theorem only with a further condition making the critical moment row expectation-continuous (for example, a stronger tail budget). The finite carrier hedge and ask-side market completion of Propositions 3.2 and 3.9 are unaffected.

3.3What the carrier completes

We now return to the object that motivated the completion: the square of the annuity weight, whose robust value the standard book leaves infinite. The companion paper’s hypotheses enter in one of two regimes, fixed in the setup below. Proposition 3.9 then records the obstruction, the explicit carrier hedge and its cost, and Corollary 3.10 draws the attainment consequences of Theorems 3.6 and 3.7. The two closing remarks place the result relative to the companion paper and delimit what a second-moment carrier can price.

Completion setup. For the next proposition and its attainment corollary, fix one of the following two regimes.

Base regime. Let $e=t_1$ . The calibrated book is finite and standard-timing, every book claim is $\cF _e$ -measurable, and the companion paper’s price-bound, escape-richness and quote-Slater conditions hold. Choose its calibrated centre and relative-interior correcting laws with finite $\chi$ -moment, and set $r=e$ .

Multi-date regime. Let a finite standard-timing book fix and pay across the calendar, and fix a surgery date $s=t_j$ with $t_1\le s\le T_m$ , where $T_m$ is the last book date. Write $X_1,\ldots ,X_n$ for the book’s anchor payoffs, $x^s$ for the date- $s$ state, and

\cM _{\rm f}:=\{Q\in \cM :\E _Q|X_i|<\infty ,\ i\le n\}, \qquad c(Q):=(\E _QX_i)_{i\le n}.

Assume every hypothesis of the companion paper’s multi-date negative theorem: the frozen-envelope condition; its price-bound, ambient-support and multi-date-richness assumptions; and a law $Q^*\in \cM (\cB )\subseteq \cM _{\rm f}$ such that $Q^*(x^s\in S_s)=1$ for its declared surgery region $S_s$ and $c(Q^*)\in \ri c(\cM _{\rm f})$ . Assume in addition that $Q^*$ and all correcting laws have finite $\chi$ -moment, so the escaping surgery sequence lies in $\cM _\chi$ , and set $r=s$ .

In either regime, the companion paper’s inventory convention places every retained coupon date on the master calendar. Define

\begin {aligned} I_r&:=\{d:d=q_j<r\text { for some }j\}\cup \{r\}, &\gamma _d&:=\sum _{j:q_j=d}\delta _j\quad (d<r),\\ \gamma _r&:=C_P\sum _{q_j\ge r}\delta _j, &K_r&:=\sum _{d\in I_r}\gamma _d. \end {aligned}

Proposition 3.9 (Standard-book obstruction and explicit carrier completion) . Under this completion setup, the following hold.

(i): The standard book leaves the annuity square uncontrolled:
$\sup _{Q\text { calibrated to the standard book}} \E _Q\Abar _r^{\,2}=\infty .$
Consequently no finite-cost admissible semistatic hedge over that book can superreplicate $\Abar _r^{\,2}$ .
(ii): If the in-advance strip ask is quoted at every $d\in I_r$ with $\gamma _d>0$ , then the explicit pathwise bound
$\begin{equation} \Abar _r^{\,2} \le K_r\sum _{d\in I_r}\gamma _d\zeta _d^2 =K_r\sum _{d\in I_r}\gamma _d \bigl (\tau _ds_d+\Zbar _0(d)+G_d\bigr ), \label {eq:pc2-annuity-completion} \end{equation}$
is an admissible hedge of finite cost
$K_r\sum _{d\in I_r}\gamma _d \bigl (\tau _d\beta _d+\Zbar _0(d)\bigr ).$

The hedge exists whether or not the augmented quotes are consistent. Before the first coupon, in particular at the base date $e$ , $I_e=\{e\}$ and $K_e=\gamma _e=C_A:=C_P\sum _j\delta _j$ , so this reduces to the one-strip hedge $\Abar _e^2\le C_A^2\zeta _e^2$ .
(iii): If the class calibrated to the standard book and the strip asks in (ii) is nonempty, then the robust upper value of $\Abar _r^{\,2}$ over that augmented class is finite and real.

Proof.The divergence in (i) is precisely the applicable base or multi-date negative theorem of the companion paper. In either regime, the assumed finite-gauge laws and the stopped-gain/Fatou argument used in Theorem 3.5 give dynamic weak duality; the standard-book anchor payoffs are absolutely integrable on the finite-moment domain. A finite-cost hedge would therefore bound the divergent expectations, a contradiction.

For (ii), the stopped-coordinate convention and the price bound give ∎

\Abar _r \le \sum _{q_j<r}\delta _j\zeta _{q_j} +C_P\!\sum _{q_j\ge r}\delta _j\zeta _r =\sum _{d\in I_r}\gamma _d\zeta _d.

Weighted Cauchy–Schwarz gives the first inequality in (11), and Proposition 3.2 gives its equality and cost. Part (iii) follows immediately.

Corollary 3.10 (Attainment after carrier completion) . Continue under either regime of the completion setup.

(i): Suppose the augmented finite-calendar problem satisfies every hypothesis of Theorem 3.6: the companion paper’s weak-compactness hypotheses, Assumption 3.4, gauge strips appearing only as ask rows, one at every post-initial date, with no strip bid or exact strip row, continuity and subcritical growth of the objective and every non-strip calibration row, and the joint relative moment-Slater condition. Then any continuous subcritical weighted-smile objective has both an attained maximizing model and an attained optimal semistatic hedge.
(ii): At the critical tier, take the objective $X=\Abar _r^2$ and suppose instead that the augmented finite-calendar problem satisfies Assumption 3.4, that $X$ and every calibration row are universally measurable of $O(\chi )$ growth, that the ask-side carrier row is present at every post-initial date, and that the target quote lies in the relative interior of the joint exact/ask domain $\mathcal A_{\rm crit}$ of Theorem 3.7. Then $\Abar _r^2$ has an attained optimal finite-calendar critical hedge. No maximizing-model assertion is made.

Proof.Part (i) is Theorem 3.6. For (ii), (11) gives $\Abar _r^2=O(\chi )$ in both completion regimes, so Theorem 3.7 applies.∎

Remark 3.11 (Position relative to the completion result, and minimality) . Under the corresponding base-date or multi-date hypotheses, Huang (2026) proves necessity relative to the finite standard-timing instrument class: that class leaves the annuity-tail direction unpriced. Proposition 3.9 proves explicit sufficiency of the designed carrier: the infinite-price/no-hedge obstruction becomes a finite superhedging bound, and a finite real robust value whenever the augmented class is nonempty. Under the joint relative-interior hypotheses of Corollary 3.10, the critical hedge is optimal on the finite calendar. For a nonempty augmented class, this is market-data completion even when critical second-order concentration prevents a maximizing model. The result is not quote-count or uniqueness minimal. For example, one aggregate ask on $\sum _i s_i$ also bounds each $\E s_i$ because $s_j\ge -1/(4\tau _j)$ . “Canonical gauge-completing family” is therefore the precise claim.

Four literatures border this pairing, and it is the pair, not either half, that is specific here. Attainment by reformulation: in martingale optimal transport, attainment is restored by changing the formulation of the dual (Beiglböck et al., 2017, 2019), whereas here the notion of hedge is held fixed and the market data are changed by one quoted ask per date. Completion by options in the spanning sense: Romano and Touzi (1997) show a single call completes a one-factor stochastic-volatility market and Davis and Obłój (2008) extend completion-by-options inside diffusion classes; those are sufficiency statements within a parametric class, with no necessity claim over an instrument universe. Why an equity market needs no designed carrier: there, superlinear payoffs are synthesizable from the call strip (Breeden and Litzenberger, 1978; Carr and Madan, 1998), the log contract prices realized variance (Neuberger, 1994), and weighted variance swaps admit robust bounds from finitely many puts (Davis et al., 2014); the rates analogue fails for annuity weights, which is why a carrier must be designed rather than synthesized. Augmentation without minimality: adding instruments, including dynamically traded ones, refines model-free bounds (Neufeld and Sester, 2021; Neufeld et al., 2023), but no minimality is claimed there. That the annuity-convexity gap is economically material, and not an artifact, is visible in the cash-settled swaption literature (Mercurio, 2008; Henrard, 2010).

Remark 3.12 (Reach of a second-moment carrier) . On an escape-rich unbounded section with strict second-moment room, a vanishing-mass atom can keep the second moment bounded while making a fourth moment diverge. Thus a second-moment carrier does not in general price quartic growth. On compact support the quartic is of course bounded. The statement is a tail-order example, not a support-free theorem that every quartic value is infinite.

4A cross-date primitive: perspective-calendar monotonicity

The static inequalities of Remark 2.10 relate quotes fixing at a single date and cannot, by construction, see relations that couple quotes across fixing dates. The first such family is nevertheless available directly, independently of the multi-date structure theory of Section 5, by a perspective-convexity argument.

Lemma 4.1 (Perspective-calendar monotonicity) . Fix an accrual $[a,b]$ with accrual factor $\delta _{ab}>0$ and a convex $f:\R \to [0,\infty )$ . Assume $\Zbar _t(b)>0$ on the declared support for every $t\le a$ . For calendar dates $t\le a$ define the natural-timing weighted payoff

\Phi _t:=\Zbar _t(b)\,f\bigl (L_t(a,b)\bigr ), \qquad L_t(a,b)=\frac {\Zbar _t(a)-\Zbar _t(b)}{\delta _{ab}\,\Zbar _t(b)} .

Then

\Phi _t=g\bigl (\Zbar _t(a),\Zbar _t(b)\bigr )

with

g(u,w):=w\,f\bigl ((u-w)/(\delta _{ab}w)\bigr )

jointly convex on

\R \times (0,\infty )

, and for every

Q\in \cM

with

a

and

b

carry-closed and

\E _Q[\Phi _{t'}]<\infty

\E _Q[\Phi _t]\ \le \ \E _Q[\Phi _{t'}] \qquad \text {for all calendar dates } t\le t'\le a :

quoted natural-timing prices on a fixed accrual and fixed convex payoff are nondecreasing in the fixing date, and a quoted violation is a calendar arbitrage against the class. Under the same carry and later-date integrability conditions, the same monotonicity holds for physical swaptions on a fixed underlying segment. For exercise date

e

, underlying start

a_0

, terminal date

q_m

, strike

K

, and annuity

\Abar _e=\sum _{j=1}^m\delta _j\Zbar _e(q_j)

, the payoff

\bigl (\Zbar _e(a_0)-\Zbar _e(q_m)-K\Abar _e\bigr )^+

is the maximum of zero and a linear functional of the date-

e

coordinates, hence is convex. The physical receiver payoff is the positive part of the negative of the same linear functional and is therefore convex as well.

Proof.Write $s=u-w$ . A convex $f$ is the supremum of its affine minorants $f(x)=\sup _i(\alpha _ix+\beta _i)$ , so for $w>0$ ∎

g(u,w)=w\,f\!\Bigl (\frac {s}{\delta _{ab}w}\Bigr ) =\sup _i\Bigl (\frac {\alpha _i}{\delta _{ab}}\,s+\beta _i\,w\Bigr ),

a supremum of linear functions of

(s,w)

, hence jointly convex; and

(u,w)\mapsto (u-w,w)

is linear, so

g

is jointly convex in

(u,w)

. By carry closure the pair

(\Zbar _t(a),\Zbar _t(b))_{t\le a}

is a

Q

-martingale, so conditional Jensen gives

\Phi _t=g(\E [\,\cdot \mid \cF _t])\le \E [\Phi _{t'}\mid \cF _t]

, and taking expectations proves the monotonicity; integrability at the later date suffices since

\Phi \ge 0

Remark 4.2 (The weight is essential) . Without the weight $\Zbar _t(b)$ the statement is false: the rate $L_t(a,b)$ is not a $Q$ -martingale, and $f(L_t)$ alone need not be a submartingale. The perspective weight is exactly what restores convexity in the martingale coordinates—the same perspective structure that makes the anchor-unit payoffs piecewise linear in Section 2. In the multi-date polytopes of Theorem 5.2 this family supplies calendar-spread supporting inequalities; a particular inequality is a relative facet only when it is nonredundant and exposes a codimension-one face. This is the rates analogue, with state-dependent weight, of maturity monotonicity of call prices for martingales.

5Multi-date polyhedrality: a sufficient grammar and its boundary

Section 4 produced one cross-date relation directly. This section asks the global question: when does the feasibility body of a multi-date panel remain a polytope with explicit certificates?

We now isolate a checkable sufficient class. A separated panel is polyhedral under the rectangular linear-carry hypotheses below. A path-mixed forward-start pair then shows that curvature is possible outside that class. Since every non-path-mixed PL panel of linear growth is separated, path mixing is necessary for curvature within this PL class; it is not sufficient.

Fix a finite calendar $t_1<\cdots <t_N$ with polyhedral sections $\Omega _i\subseteq \R ^{k_i}_+$ and full linear carry: the class $\cM$ consists of laws of $(x^1,\ldots ,x^N)$ , $x^i\in \Omega _i$ , with $\E [x^1]=r_0$ and $\E [x^{i+1}\mid x^1,\ldots ,x^i]=A_ix^i$ for fixed linear carry maps $A_i$ . A separated PL panel is a finite family of payoffs, each of the form $f(x^i)$ for a single date $i$ , continuous PL of linear growth on $\Omega _i$ .

Lemma 5.1 (Polyhedral value graph, with fiber identification) . Let $\widetilde \Omega \subseteq \R ^K$ be a polyhedron, $P$ a continuous PL payoff vector, and $B:\R ^K\to \R ^p$ linear. For $m\in B\widetilde \Omega$ define

\widehat \cV _B(m):=\operatorname {cl}\Bigl \{\textstyle \int P\,\dd \kappa :\ \kappa \ \text {prob on }\widetilde \Omega ,\ \int |\tilde x|_1\dd \kappa <\infty ,\ B\!\int \tilde x\,\dd \kappa =m\Bigr \}.

Then

\{(m,v):v\in \widehat \cV _B(m)\} =\{(m,v):(1,m,v)\in G_B\},

where

G_B:=\operatorname {cl}\cone \{(1,B\tilde x,P(\tilde x)): \tilde x\in \widetilde \Omega \}

is a polyhedron. If additionally

B\widetilde \Omega \subseteq \R _+^p

and

|P(\tilde x)|\le K(1+|B\tilde x|_1)

, then

|v|\le K(1+|m|_1)

throughout the graph. Under these additional hypotheses, for every direction

y

the scalar fiber support is the finite concave polyhedral function

\begin{equation} H_y(m):=\sup _{v\in \widehat \cV _B(m)}y\cdot v =\min _{\substack {a\in \R ,\ b\in \R ^p\\ a+b\cdot B\tilde x\ge y\cdot P(\tilde x)\ \forall \tilde x}} (a+b\cdot m), \label {eq:pc2-projected-dual} \end{equation}

the minimum is attained, and

H_y

is the minimum of finitely many affine functions on

B\widetilde \Omega

Proof.Cellwise Motzkin decomposition shows that $G_B$ is a finite sum of finitely generated cones, hence polyhedral. Barycenters place the fiber closure inside its slice. A continuous finite PL map on a convex polyhedron is globally Lipschitz: a line segment crosses only finitely many cells and every affine piece has uniformly bounded slope.

For the reverse inclusion, let $F$ be the minimal face of the polyhedron $B\widetilde \Omega$ containing $m$ . Fix an atom-plus-horizon representation of $(1,m,v)$ . For every active face inequality, its nonnegative slack on each projected atom and the negative directional slack on each projected horizon sum to zero. Hence every projected atom lies in $F$ , and every projected horizon direction $Bd$ belongs to $\rec F$ .

There is one further point to check: the original base point of a cell horizon need not project into $F$ . For a generator $(Bd,A_jd)$ with $d\in \rec C_j$ , choose instead $\bar x_j\in \widetilde \Omega$ with $B\bar x_j\in F$ . A recession direction of the nonempty cell is a recession direction of the convex polyhedron $\widetilde \Omega$ , so $\bar x_j+td\in \widetilde \Omega$ and $B\bar x_j+tBd\in F$ . If $x_j\in C_j$ , global Lipschitzness gives

\frac {|P(\bar x_j+td)-P(x_j+td)|}{t} \le \frac {L|\bar x_j-x_j|}{t}\longrightarrow 0,

while

P(x_j+td)/t\to A_jd

. Thus a far atom of mass

s/t

on the rebased ray has the same limiting payoff horizon

sA_jd

, and its projected state lies in

F

for every

t

Replace all horizon generators this way and normalize the resulting finite measure. Its $B$ -moment $m_t$ lies in $F$ and tends to $m$ , while its payoff tends to $v$ . Since $m\in \ri F$ , choose finitely many points $z_l\in \widetilde \Omega$ whose $B$ -images affinely span $F$ and strictly positive weights $\eta _l$ with $m=\sum _l\eta _lBz_l$ . Choose $\varepsilon _t\downarrow 0$ so slowly that $|m_t-m|/\varepsilon _t\to 0$ . The affine system ∎

(1-\varepsilon _t)m_t+\sum _l\gamma _l(t)Bz_l=m,\qquad \sum _l\gamma _l(t)=\varepsilon _t

is a perturbation, of size

o(\varepsilon _t)

, of its strictly positive solution

\gamma _l=\varepsilon _t\eta _l

m_t=m

. Because all moments stay in

\operatorname {aff}F

, it therefore has a nonnegative solution for all large

t

. Scaling the approximating measure by

1-\varepsilon _t

and adding the atoms

\gamma _l(t)\delta _{z_l}

restores the

B

-moment exactly to

m

and preserves payoff convergence. This proves fiber equality, including on boundary faces. Under the final hypotheses,

\int |B\tilde x|_1=|m|_1

for every representing law, and the growth bound follows; it passes to the closure. Finally, maximize

y\cdot v

over the polyhedral slice

(1,m,v)\in G_B

. Its linear-programming dual is exactly the affine-domination problem in (12); the growth bound makes the value finite, so polyhedral duality gives equality and attainment. Parametric polyhedral linear-programming duality gives a finite lower envelope of affine dual objectives, proving the last assertion.

Applied backward along the calendar, the lemma gives polyhedrality of the whole separated body.

Theorem 5.2 (Multi-date polytope theorem, separated panels) . Assume $r_0\in \Omega _1$ and nonempty polyhedral state spaces $\Omega _i\subseteq \R _+^{k_i}$ with the price-bound geometry. Let the linear carry maps satisfy $A_i\Omega _i\subseteq \Omega _{i+1}$ . Assume rectangularity: every finite sequence of kernels supported on $\Omega _{i+1}$ with conditional mean $A_ix^i$ is admissible. Let the panel be separated and PL of linear growth. Then the closure of the multi-date feasibility body is a polytope, the relative-interior sandwich $\ri \widehat \cV \subseteq \cV \subseteq \widehat \cV$ holds, and every relative facet is certified by an explicit semistatic portfolio: cash, a date-one curve position, and per-date dynamic coefficients obtained by unwinding the slices.

Proof.Group all date- $i$ panel payoffs into a vector $F_i(x)$ . Starting from the last date, construct recursively

\widetilde \Omega _i:=\{(x,w):x\in \Omega _i, \ (A_ix,w)\in \Gamma _{i+1}\},\qquad P_i(x,w):=(F_i(x),w),

where

\Gamma _{i+1}

is the value graph supplied by Lemma 5.1, with

B

the projection onto the base state. At the terminal date use

\widetilde \Omega _N=\Omega _N

and

P_N=F_N

. Inductively,

\widetilde \Omega _i

is polyhedral and

P_i

is PL. The preceding graph bound gives

|w|\le K(1+|A_ix|_1)

, so the explicit growth hypothesis of the lemma propagates.

Every admissible law embeds in this recursion by taking conditional future panel values and using the tower property. Conversely, a point of a lifted fiber can be approximated by a finite atomic kernel by the fiber identification in Lemma 5.1. A point of the outer slice also has a finite atom-plus-horizon representation. After the lemma’s exact moment correction, only finitely many nodes remain; choose an $\varepsilon$ -realizing finite kernel at each node and glue the resulting finite tree. Rectangularity makes that tree admissible, and its panel value converges to the lifted value. No global disintegration or measurable selection is needed. Hence the recursively lifted slice is exactly $\overline \cV$ .

The final slice is a bounded polyhedron by the propagated growth bound and pinned first mean, hence a polytope. Since $\cV$ is convex, $\ri \overline \cV =\ri \cV \subseteq \cV$ . The certificate assertion is proved by the explicit scalarized recursion and telescoping identity in Theorem 5.3 below.∎

The unwinding step of the preceding proof is worth isolating as a statement about the taxonomy itself: it says that the certificate grammar of separated panels is closed under calendar depth.

Theorem 5.3 (Certificate-grammar completeness for separated panels) . In the setting of Theorem 5.2, every supporting inequality of the separated feasibility body at any finite calendar depth is realized by a composed stage certificate: cash, static positions in the panel payoffs, and a per-date dynamic part assembled by backward induction from attained affine one-step dominators, one per date and reachable history cell of the lifted state. The stage engine is the polyhedral one-step duality of (12), attained because polyhedral programs have no duality gap (Rockafellar, 1970, Section 28). Conversely, every such composed chain is a valid inequality. Consequently the taxonomy grammar consisting of (i) single-date static families and (ii) dynamic compositions of stage certificates is complete at every finite depth, and each facet carries a certificate with polyhedral stage data.

Proof.Split the supporting direction by fixing date as $y=(y_1,\ldots ,y_N)$ , with date- $i$ payoff vector $F_i$ . Put $V_{N+1}\equiv 0$ and, backward for $i=N,\ldots ,1$ , define

g_i(x):=y_i\cdot F_i(x)+V_{i+1}(A_ix),\qquad V_i(m):=\sup _{\substack {\kappa \ {\rm prob.\ on}\ \Omega _i\\ \int x\,\dd \kappa =m}} \int g_i\,\dd \kappa ,

where the term

V_{N+1}(A_Nx)

is omitted. Backward induction using Lemma 5.1 shows that every

g_i

is continuous PL of linear growth and that every

V_i

is finite concave polyhedral. More precisely, (12) supplies finitely many globally feasible affine dominators

g_i(x)\le a_{i\ell }+b_{i\ell }\cdot x, \qquad V_i(m)=\min _\ell (a_{i\ell }+b_{i\ell }\cdot m).

At a history through $t_{i-1}$ let $m_i=r_0$ for $i=1$ and $m_i=A_{i-1}x^{i-1}$ otherwise. Choose the least-index active affine piece at $m_i$ ; its active region is a polyhedral history cell, so this is a predictable finite-valued choice. Along every path,

y_i\cdot F_i(x^i)+V_{i+1}(m_{i+1}) \le V_i(m_i)+b_i(h_{i-1})\cdot (x^i-m_i).

Summing these inequalities cancels all continuation values and gives the explicit telescoping certificate

\begin{equation} \sum _{i=1}^Ny_i\cdot F_i(x^i) \le V_1(r_0)+\sum _{i=1}^N b_i(h_{i-1})\cdot (x^i-m_i). \label {eq:pc2-grammar-telescope} \end{equation}

The final sum is precisely a predictable curve gain under full linear carry (the

i=1

term is the date-zero curve position).

Conditional integration proves that every admissible law has support value at most $V_1(r_0)$ . Conversely, choose finite $\varepsilon /N$ -optimal kernels in the definitions of the $V_i$ and glue their finite tree; rectangularity makes it admissible and its value approaches $V_1(r_0)$ . Hence $V_1(r_0)=h(y)$ and (13) has exact support cost. Conversely, any chain of globally valid stage dominators telescopes in the same way and is a valid inequality. Facet normals are special supporting directions, so their certificates inherit the finite polyhedral stage data.∎

Remark 5.4 (Relation to multistage stochastic programming, and the failure boundary) . The engine of Theorem 5.3—polyhedral cost-to-go functions composed backward and represented by finitely many cuts—is the classical mechanism of nested Benders decomposition and stochastic dual dynamic programming for linear multistage stochastic programs (Pereira and Pinto, 1991; Birge and Louveaux, 2011; Philpott and Guan, 2008). The theorem is its transplant to calibration bodies over martingale kernels with weighted, possibly unnormalized quote measures, where the composed-cut description becomes a certificate grammar, and where the object of interest is the finite-dimensional body of quote vectors rather than a single optimal value. The curved example below loses the separated stage structure: a path-mixed payoff couples the stage problems through the conditioning state, the analogue of non-rectangular ambiguity in robust dynamic programming (Epstein and Schneider, 2003; Iyengar, 2005; Nilim and El Ghaoui, 2005), and by Proposition 5.5 no finite grammar of affine stage certificates can describe the resulting curved boundary—a strictly convex arc has no finite facet description. Whether some richer finite certificate grammar describes particular curved bodies is open; the example rules out the affine grammar only.

Proposition 5.5 (Sharp failure for path-mixed payoffs) . On $\Omega =[0,1]$ with two dates, $r_0=K=\tfrac 12$ , the feasibility body of the path-mixed PL panel

f_1(x,y)=(y-x)^+\ \ \text {(forward start)}, \qquad f_2(x)=(x-\tfrac 12)^+\ \ \text {(vanilla)}

is not a polytope: its support function in the directions

(1,\theta )

equals

h(\theta )\;=\;\tfrac 14+\tfrac {\theta ^2}{16}, \qquad 0\le \theta \le 2,

smooth and strictly convex. The body remains semialgebraic.

Proof.Reduction. For fixed $x$ , the concave envelope on $[0,1]$ of the convex hinge $y\mapsto (y-x)^+$ is its chord $y(1-x)$ , attained by the two-point law on $\{0,1\}$ with mean $x$ . Thus

\sup _{\substack {\nu \text { probability on }[0,1]\\ \int y\,\dd \nu (y)=x}} \int (y-x)^+\,\dd \nu (y) =x(1-x).

Since the optimal kernel is explicitly measurable in

x

h(\theta )=\sup _{Q}\E \bigl [f_1+\theta f_2\bigr ] =\sup _{\mu :\ \text {mean }1/2}\int \bigl [\,\underbrace {x-x^2}_{=:p(x)}+\theta (x-\tfrac 12)^+\bigr ] \dd \mu ,

the single-date moment value of

\varphi _\theta :=p+\theta (\cdot -\tfrac 12)^+

, which equals the concave envelope

\widehat \varphi _\theta (\tfrac 12)

Tangency. $\varphi _\theta$ is the concave parabola $p$ on $[0,\tfrac 12]$ and $p+\theta (x-\tfrac 12)$ on $[\tfrac 12,1]$ , with an upward kink at $\tfrac 12$ . The tangent line to $p$ at $x_-$ is $\ell (x)=(1-2x_-)x+x_-^2$ ; tangency to the right piece at $x_+$ forces equal slopes, $1-2x_-=1+\theta -2x_+$ , i.e. $x_+=x_-+\theta /2$ , and equal values, which reduce to $\theta (\tfrac 12-x_--\theta /4)=0$ , i.e. $x_-=\tfrac 12-\theta /4$ . For $0\le \theta \le 2$ both tangency points lie in $[0,1]$ ; $\ell$ dominates both concave pieces globally (a tangent from above to a concave function), so $\ell$ is an affine majorant, while the two-point measure on $\{x_-,x_+\}$ with mean $\tfrac 12$ attains $\ell (\tfrac 12)$ since $\varphi _\theta =\ell$ at both atoms. Hence

h(\theta )=\ell (\tfrac 12) =\tfrac 12(1-2x_-)+x_-^2 =\tfrac 14+\tfrac {\theta ^2}{16}.

A polytope’s support function is piecewise linear along every segment of directions;

h

is strictly convex on

[0,2]

, so the body is not a polytope.

Semialgebraicity of the whole body. The envelope computation above describes only the upper boundary; the body itself is recovered by a conditional slack coordinate. Given the date-one state $x$ , the attainable conditional values of the forward-start coordinate under mean- $x$ kernels form the full interval $[0,\,x-x^2]$ : the lower endpoint is Jensen ( $(y-x)^+$ is convex, minimized by $\delta _x$ ), the upper endpoint is the chord kernel computed above, and every intermediate value is a mixture of the two. Hence the two-payoff body equals the single-date moment body ∎

\Bigl \{\bigl (\textstyle \int s\,\dd \mu ,\ \int (x-\tfrac 12)^+\dd \mu \bigr ):\ \mu \ \text {prob on }\widehat S,\ \int x\,\dd \mu =\tfrac 12\Bigr \}, \qquad \widehat S:=\{(x,s):x\in [0,1],\ 0\le s\le x-x^2\},

whose support

\widehat S

is compact semialgebraic and whose payoff pair is semialgebraic (indeed PL) in

(x,s)

; the Richter–Tarski argument of Theorem 2.15 applies to this lifted problem verbatim, and the body is semialgebraic.

Remark 5.6 (The dichotomy, and where the curvature comes from) . (i) Within the continuous PL, linear-growth panel class under the rectangular full-linear-carry hypotheses of this section, path-mixing is necessary for curvature, and the pair witness shows how it becomes sufficient: the one-step value of a payoff depending on the conditioning state evaluates a state-dependent envelope on the diagonal, producing polynomial value functions ( $x-x^2$ above), while separated payoffs evaluate a fixed envelope at the carried mean, staying PL. Necessity is not sufficiency: a path-mixed payoff in isolation can still generate a polyhedral body when no transverse direction exposes the curvature—projecting the pair body of Proposition 5.5 onto its forward-start coordinate alone gives the interval $[0,\tfrac 14]$ , a polytope. Curvature requires a path-mixed payoff and a panel direction coupling it to a transverse coordinate, exactly as in the pair body. (ii) In the example, the reduced continuation value is quadratic in the conditioning state; this curvature is endogenous and is distinct from the exogenous quoted budget in Theorem 2.16. (iii) Tape and compounded products (the account-convexity claims of the companion paper, cliquets, forward starts) are path-mixed and therefore fall outside the separated theorem; their exact body geometry is not classified here.

6The ordered certificate boundary after quote completion

The grammar of Section 5 says how supporting certificates for separated PL panels are assembled. We now answer the complementary question: for an inconsistent exact/ask quote vector, when can the dynamic gain be omitted? The question is a finite-panel, one-sided form of the classical distinction between inconsistencies that a fixed portfolio exhibits and those that only a trading strategy detects. What makes the answer below specific to the completed market is that it uses the one-sided carrier as quoted, admits critical $O(\chi )$ rows, and operates on the finite calendar through Theorem 3.5.

6.1The ordered pathwise and martingale quote sets

Two sets organize the answer. One collects the quotes that some admissible law reproduces, with asks allowed to lie above the model price; the other collects the quotes that pathwise support alone cannot rule out. This subsection defines both, together with the certificates that separate them, and relates their support functions.

Fix finite vectors $H:\Omega \to \R ^p$ of exact-price rows and $F:\Omega \to \R ^m$ of one-sided ask rows. A bid for $g$ is encoded as the ask row $-g$ ; a two-sided band contributes both rows. In this subsection the rows are real and universally measurable, and

|H|_1+|F|_1\le C\chi .

Let

\mathcal C_\uparrow :=\{0\}^p\times \R _+^m,\qquad G:=(H,F),

and define the two ordered quote sets

\begin{align} \mathcal A &:=\{(\E _QH,r):Q\in \cM _\chi ,\ \E _QF\le r\}, \label {eq:pc2-ordered-model-set}\\ \cK _{\rm stat}^{\uparrow } &:=\cl \bigl (\conv G(\Omega )+\mathcal C_\uparrow \bigr ). \label {eq:pc2-ordered-static-set} \end{align}

Thus $\mathcal A$ imposes the martingale rows and finite-gauge integrability, whereas $\cK _{\rm stat}^{\uparrow }$ remembers only pathwise support and the right to pay more for an ask.

Definition 6.1 (Ordered semistatic certificate) . At a proposed quote $z=(c,q)$ , an ordered semistatic certificate is a quadruple $(a,\alpha ,\eta ,D)$ , with $\alpha \in \R ^p$ , $\eta \in \R _+^m$ , and $D$ admissible in the sense of Definition 3.1, such that

a+D-\alpha \cdot H+\eta \cdot F\ge 0\quad \hbox {on }\Omega , \qquad a-\alpha \cdot c+\eta \cdot q<0.

It is static when

D=0

. The sign

\eta \ge 0

is the execution constraint: ask rows may be bought but not shorted at their asks. The static leg is cash together with the exact and ask positions; the dynamic leg is

D

alone.

For $\alpha \in \R ^p$ and $\eta \in \R _+^m$ , put

k(\alpha ,\eta ):=\sup _{\omega \in \Omega } [\alpha \cdot H(\omega )-\eta \cdot F(\omega )], \qquad h(\alpha ,\eta ):=\sup _{Q\in \cM _\chi } \E _Q[\alpha \cdot H-\eta \cdot F].

Lemma 6.2 (Ordered nesting and support functions) . The set $\mathcal A$ is convex,

\begin{equation} \mathcal A\subseteq \cl \mathcal A \subseteq \cK _{\rm stat}^{\uparrow }, \label {eq:pc2-ordered-nesting} \end{equation}

and, in the extended-real sense,

\begin{equation} \sigma _{\cK _{\rm stat}^{\uparrow }}(\alpha ,-\eta ) =k(\alpha ,\eta ),\qquad \sigma _{\cl \mathcal A}(\alpha ,-\eta ) =h(\alpha ,\eta ). \label {eq:pc2-ordered-supports} \end{equation}

Every finite upper-support normal of either ordered set has ask component

-\eta

for some

\eta \ge 0

Proof.Convexity follows from convexity of $\cM _\chi$ and of $\mathcal C_\uparrow$ . For $Q\in \cM _\chi$ , $G\in L^1(Q)$ , so the barycenter $\E _QG$ lies in the closed convex hull of $G(\Omega )$ . Adding the nonnegative slack $(0,r-\E _QF)$ proves the inclusions.

Both sets are invariant under addition of $\mathcal C_\uparrow$ . A linear functional bounded above on such an upper set must therefore be nonpositive on every ask coordinate, hence has the displayed form $(\alpha ,-\eta )$ . For this normal, maximization over the order slack chooses zero slack, giving both identities in (17).∎

6.2The ordered four-region certificate classification

The nesting of Lemma 6.2 splits every exact/ask quote into four regions, according to where the quote sits relative to the three sets. The next theorem shows that each region has its own certificate behaviour: a static certificate, a certificate whose dynamic leg is indispensable, a zero-gap supporting hedge with no strict certificate, or feasibility. The corollaries that follow identify when the boundary region disappears and when static certificates suffice.

Theorem 6.3 (Static, dynamically essential, boundary, and feasible regions) . Let Assumption 3.4 hold and $\cM _\chi \ne \emptyset$ . For every proposed exact/ask quote $z=(c,q)$ , exactly one of the following holds.

1.: $z\notin \cK _{\rm stat}^{\uparrow }$ . A static certificate exists, and conversely a static certificate exists only in this region.
2.: $z\in \cK _{\rm stat}^{\uparrow }\setminus \cl \mathcal A$ . No static certificate exists, but an ordered semistatic certificate exists. Its admissible dynamic leg is attained by Theorem 3.5 and is indispensable.
3.: $z\in \cl \mathcal A\setminus \mathcal A$ . The quote is infeasible and is a limit of feasible quotes, but no strict finite-cost admissible semistatic certificate exists.
4.: $z\in \mathcal A$ . The quote is feasible.

The first three cases form the certificate trichotomy for infeasible quotes; the fourth records feasibility. Every point in the third region nevertheless admits a zero-gap supporting semistatic hedge. Thus this is a nonattainment region, not an absence of supporting hedges.

Proof.If $z\notin \cK _{\rm stat}^{\uparrow }$ , strong separation and the last assertion of Lemma 6.2 give $\alpha \in \R ^p$ and $\eta \ge 0$ with

\alpha \cdot c-\eta \cdot q>k(\alpha ,\eta ).

The static portfolio

k(\alpha ,\eta )-\alpha \cdot H+\eta \cdot F\ge 0

then has strictly negative quoted cost. Conversely, any static certificate is a strict separating affine functional for

\cK _{\rm stat}^{\uparrow }

, so it can exist only outside that set.

Now let $z\notin \cl \mathcal A$ . Separation of this closed convex upper set gives $\alpha \in \R ^p$ , $\eta \ge 0$ , and

\alpha \cdot c-\eta \cdot q>h(\alpha ,\eta ). \label {eq:pc2-dynamic-separation}

For

Y:=\alpha \cdot H-\eta \cdot F

, the growth assumption gives

Y\ge -C'\chi

, and (6.2) makes

h(Y)=h(\alpha ,\eta )<\infty

. Theorem 3.5 supplies an admissible

D^*

with

h(\alpha ,\eta )+D^*\ge \alpha \cdot H-\eta \cdot F.

After rearrangement this is an ordered semistatic certificate whose cost is strictly negative by (6.2). If

z\in \cK _{\rm stat}^{\uparrow }

, the first part excludes every static certificate, so

D^*

is indispensable.

Suppose instead that $z\in \cl \mathcal A$ . For any ordered semistatic portfolio with nonnegative terminal payoff, localization and Fatou as in Theorem 3.5 give $\E _QD\le 0$ . Its quoted cost is therefore nonnegative at every point of $\mathcal A$ , and by continuity also at every point of $\cl \mathcal A$ . It cannot be a strict certificate at $z$ . Membership in $\cl \mathcal A\setminus \mathcal A$ gives infeasibility and an approximating sequence of feasible quotes.

Finally, for every convex set $\ri (\cl \mathcal A)\subseteq \mathcal A$ . Hence a point of $\cl \mathcal A\setminus \mathcal A$ lies on its relative boundary and has a nonzero supporting normal. Upper-set invariance writes that normal as $(\alpha ,-\eta )$ , $\eta \ge 0$ , with $\alpha \cdot c-\eta \cdot q=h(\alpha ,\eta )<\infty$ . The preceding dynamic-support construction now has quoted cost zero, which proves the last assertion.∎

The boundary region is the only place where an infeasible quote has no strict certificate. In the completed market, with every strip ask present and subcritical rows, that region is empty.

Corollary 6.4 (The completed subcritical market has no certificate gap) . Assume the companion paper’s weak-compactness hypotheses, Assumption 3.4, and $\cM _\chi \ne \emptyset$ . Let

F=(s_1,\ldots ,s_N,F^0),

where the first

N

rows are exactly the ask-side strip rows, one at every post-initial date, and let

H

and every component of

F^0

belong to

\cC _\chi ^0

. Then

\mathcal A

is closed. Consequently every infeasible exact/ask quote has a strict attained admissible semistatic certificate. It has a static certificate exactly outside

\cK _{\rm stat}^{\uparrow }

; at every point of

\cK _{\rm stat}^{\uparrow }\setminus \mathcal A

, the dynamic leg is indispensable. No relative moment-Slater condition is required for this infeasibility alternative.

Proof.Let $z_n=(c_n,q_n)\in \mathcal A$ converge to $z=(c,q)$ , and choose $Q_n\in \cM _\chi$ realizing the exact rows and satisfying the ask bounds. The strip coordinates of $q_n$ are uniformly bounded. Since $\E _{Q_n}\zeta _{t_i}=\Zbar _0(t_i)$ , ∎

\E _{Q_n}\chi =1+\sum _i\bigl (\tau _i\E _{Q_n}s_i+\Zbar _0(t_i)\bigr )

is uniformly bounded. That weak compactness gives, along a subsequence,

Q_n\Rightarrow Q\in \cM _\chi

. The defining

\varepsilon \chi

bound makes the expectations of

H

and

F^0

continuous along this sequence. Lower semicontinuity of each quadratic strip expectation gives

\E _Qs_i\le \liminf _n\E _{Q_n}s_i\le q_i.

Thus

Q

realizes

z

, proving that

\mathcal A

is closed. The rest is Theorem 6.3.

More generally, whether strict or static certificates are always available is decided by which of the three sets coincide.

Corollary 6.5 (Exact and robust static saturation) . Under Theorem 6.3’s hypotheses,

\begin{equation} \begin {aligned} &\text {every infeasible quote has a strict semistatic certificate} &&\Longleftrightarrow \quad \mathcal A=\cl \mathcal A,\\ &\text {every infeasible quote has a static certificate} &&\Longleftrightarrow \quad \mathcal A=\cK _{\rm stat}^{\uparrow },\\ &\text {every quote outside \(\cl \mathcal A\) has a static certificate} &&\Longleftrightarrow \quad \cl \mathcal A=\cK _{\rm stat}^{\uparrow }. \end {aligned} \label {eq:pc2-saturation} \end{equation}

Proof.A strict semistatic certificate exists exactly outside $\cl \mathcal A$ , while a static certificate exists exactly outside $\cK _{\rm stat}^{\uparrow }$ , by Theorem 6.3. Since $\mathcal A\subseteq \cl \mathcal A\subseteq \cK _{\rm stat}^{\uparrow }$ , equality of the corresponding complements gives the three equivalences.∎

6.3Computable and calendar specializations

The classification becomes computable in two settings. At one date, with a piecewise-linear book and the strip quoted at an ask, both ordered sets are projections of explicit second-order-cone systems rather than of a linear-growth augmentation, so the classification reduces to two conic feasibility problems (Corollary 6.6); Example 6.7 works through a minimal case. Across dates, for separated piecewise-linear panels, the ordered boundary complements the certificate grammar of Section 5 (Corollary 6.8), and Examples 6.9 and 6.11 exhibit a quote that needs dynamic trading and the zero-gap boundary case.

Corollary 6.6 (One-date ordered SOC test) . Let Assumption 2.1 hold, fix $r_0\in \ri \Omega$ and $\tau >0$ , and let the exact book $f$ be PL on cells $C_j=\{x:E_jx\le e_j\}$ as in Definition 2.3. Let

s(x):=\frac {x_\zeta ^2-x_\zeta }{\tau }

be quoted only at ask

\beta

, and let

v

denote the exact-book quote. The ordered model set of pairs

(v,\beta )

is the projection of

\begin{equation} \begin {gathered} t_j,\rho _j\ge 0,\qquad \sum _jt_j=1,\qquad \sum _jm_j=r_0,\qquad E_jm_j\le e_jt_j,\\ m_{j,\zeta }^2\le t_j\rho _j,\qquad v=\sum _j(\alpha _jt_j+A_jm_j),\qquad \sum _j(\rho _j-m_{j,\zeta })\le \tau \beta . \end {gathered} \label {eq:pc2-ordered-model-soc} \end{equation}

The ordered static set

\cK _{\rm stat}^{\uparrow }

has the identical lift with only the mean row

\sum _jm_j=r_0

deleted. Both projections are closed. Therefore the dynamic curve position is indispensable exactly when the raw system is feasible and the mean-constrained system is infeasible.

Proof.Split a law cellwise as in Theorem 2.16, with $\rho _j$ dominating its cellwise second moment. Since $\E s\le \beta$ is represented in the lift by $\sum _j(\rho _j-m_{j,\zeta })\le \tau \beta$ , this gives the displayed systems. Conversely, one atom $m_j/t_j$ in every positive-mass cell realizes the PL value and no larger strip expectation; when $t_j=0$ , the SOC row and the price-bound recession property force $m_j=0$ .

For closedness, consider any convergent sequence of projected pairs $(v_n,\beta _n)$ and corresponding lifted variables, and put $R_n=\sum _j\rho _{j,n}$ and $M_n=\sum _jm_{j,\zeta ,n}$ . Positivity and the rotated-SOC rows give $M_n^2\le R_n$ , while the strip row gives $R_n-M_n\le \tau \beta _n$ . Hence $R_n-\sqrt {R_n}\le \tau \beta _n$ , which bounds $(R_n)$ because $(\beta _n)$ is bounded; the price bound then bounds every component of every $m_{j,n}$ . Passing to a lifted subsequence proves that its limit projects to $(v,\beta )$ . Thus both projections are closed.∎

Example 6.7 (The three completed-market outcomes from one strip) . Let $\Omega =[1/2,\infty )$ , $r_0=1$ , $\tau =1$ , take no book, and quote $s(x)=x^2-x$ at ask $\beta$ . Then

\cK _{\rm stat}^{\uparrow }=[-1/4,\infty ), \qquad \mathcal A=[0,\infty ).

For

\beta <-1/4

, the certificate

s+1/4=(x-1/2)^2\ge 0

is static. For

-1/4\le \beta <0

, no static certificate exists, but

s-(x-1)=(x-1)^2\ge 0

has negative cost

\beta

; the zero-cost curve gain

-(x-1)

is indispensable. For

\beta \ge 0

, the law

\delta _1

is feasible.

Corollary 6.8 (The ordered boundary complements the PL grammar) . In the setting of Theorem 5.2, split a separated PL panel $G=(H,F)$ into exact rows and one-sided PL ask rows, and define

\mathcal A_{\rm PL} :=\{(\E _QH,r):Q\in \cM ,\ \E _QF\le r\}.

Then

\cl \mathcal A_{\rm PL} =\widehat \cV +\mathcal C_\uparrow ,

a polyhedron, and the ordered four-region classification holds with

\mathcal A_{\rm PL}

in place of

\mathcal A

. Every quote in the dynamically essential region admits a certificate with the finite composed-stage form of Theorem 5.3.

Proof.Write $\cV _{\rm ft}$ for the panel values of the finite martingale trees constructed in the proof of Theorem 5.2. That proof gives $\cl \cV _{\rm ft}=\cl \cV =\widehat \cV$ . Since $\mathcal A_{\rm PL}=\cV +\mathcal C_\uparrow$ , while $\widehat \cV$ is compact and $\mathcal C_\uparrow$ is a closed polyhedral cone,

\cl (\cV +\mathcal C_\uparrow ) =\widehat \cV +\mathcal C_\uparrow .

The pathwise static-separation part of Theorem 6.3 is unchanged. Separation outside

\cl \mathcal A_{\rm PL}

gives a direction

(\alpha ,-\eta )

, with

\eta \ge 0

, whose residual

\alpha \cdot H-\eta \cdot F

is again separated PL of linear growth, and Theorem 5.3 supplies its attained finite polyhedral stage certificate. The same construction at equality supplies the zero-gap supporting certificate at a point of

\cl \mathcal A_{\rm PL}\setminus \mathcal A_{\rm PL}

It remains to justify the no-strict-certificate assertion without importing the $\cM _\chi$ localization argument. Let $(a,\alpha ,\eta ,D)$ , with $\eta \ge 0$ , have nonnegative terminal payoff on $\Omega$ . Under any finite martingale tree $Q$ used above, $D$ is integrable and the martingale increments give $\E _QD=0$ . Hence, for every ask slack $u\in \R _+^m$ , ∎

a-\alpha \cdot \E _QH+\eta \cdot (\E _QF+u) =\E _Q[a+D-\alpha \cdot H+\eta \cdot F]+\eta \cdot u \ge 0.

Thus the affine quoted-cost functional is nonnegative on

\cV _{\rm ft}+\mathcal C_\uparrow

, and by continuity also on

\cl (\cV _{\rm ft}+\mathcal C_\uparrow ) =\widehat \cV +\mathcal C_\uparrow =\cl \mathcal A_{\rm PL}.

No point of that closure therefore admits a strict certificate. Together with the two separation arguments, this proves the ordered four-region classification. Finally, the grammar certificate has only finitely many stage slopes; the currency-price bound makes every partial gain bounded below by a constant multiple of the running gauge, so the composed strategy is admissible in the sense of Definition 3.1.

Example 6.9 (A calendar quote for which dynamics are indispensable) . Take two traded-coordinate states

x^1=(S_1,\zeta _1)\in [2,4]\times \{5\},\qquad x^2=(S_2,\zeta _2)\in [1,5]\times \{5\},

initial state

r_0=(3,5)

, and identity carry. The state spaces are positive convex polyhedra, every carried mean lies in the relative interior of the next section, and the currency-price bound holds. Let the two exact separated claims be

C_1=(S_1-3)^+,\qquad C_2=(S_2-3)^+,

quoted at

(1/2,0)

. With

\tau _1=\tau _2=1

, both strip payoffs equal

20

and are quoted at asks

(20,20)

The nonmartingale path law putting probability $1/2$ on $((2,5),(3,5))$ and $1/2$ on $((4,5),(3,5))$ realizes the quote vector pathwise, so it belongs to $\cK _{\rm stat}^{\uparrow }$ and no static certificate exists. No martingale realizes it: $C_2=0$ and $\E S_2=3$ force $S_2=3$ almost surely, whence $S_1=\E [S_2\mid S_1]=3$ almost surely, contradicting $\E C_1=1/2$ . The convex subgradient inequality gives the pathwise certificate

C_2-C_1-\one _{\{S_1>3\}}(S_2-S_1)\ge 0.

Its quoted cost is

-1/2

, its bounded predictable dynamic leg is admissible, and that leg is indispensable. Here the strip multipliers vanish, so the certificate is exactly of the finite PL grammar in Theorem 5.3.

Remark 6.10 (Where the polyhedral grammar stops) . The preceding corollary is the exact complement to Theorem 5.3: the ordered hull decides whether $D$ is needed, and the grammar constructs $D$ when the active rows remain separated PL. In the completed market an active carrier multiplier adds $-\eta _i s_i$ , a concave quadratic term, to the residual. Theorem 3.5 still attains the dynamic certificate, and Corollary 6.6 gives a one-date conic test, but no finite polyhedral SOC grammar on the calendar is asserted.

This also clarifies the two closure mechanisms used here. Prescribed terminal marginals provide uniform integrability in Remark A.2; here the finite all-date strip asks provide the gauge bound that closes the finite ordered quote set.

Example 6.11 (Critical boundary: zero-gap hedge, no strict certificate) . In the one-date example of Remark 2.19, use the actual strip $s(\zeta )=\zeta ^2-\zeta$ , quote the exact claim

X(\zeta )=(\zeta -1)^2\frac {\zeta }{1+\zeta }

v=1

, and quote the strip at ask

\beta =1

. The laws displayed there have

\E s=1

and

\E X\uparrow 1

, but no feasible law attains

\E X=1

. Thus

(1,1)\in \cl \mathcal A\setminus \mathcal A.

As predicted by Theorem 6.3, there is no strict certificate, but the attained zero-gap supporting hedge is explicit:

s(\zeta )-X(\zeta )-(\zeta -1) =\frac {(\zeta -1)^2}{1+\zeta }\ge 0.

The closure conclusion of Corollary 6.4 does not apply because the exact row

X

is critical rather than subcritical.

6.4Finite-panel gauge invariance and the selection handoff

The ordered theorem is stated on $\cM _\chi$ , whereas a downstream selection problem may impose a stronger integrability gauge. Equality of the two law classes is neither necessary nor generally true. What is invariant is the image of every fixed finite integrable panel. The following raw-history formulation makes explicit the structural interface needed for that conclusion.

We first isolate the exact role of the all-date strip layout in Corollary 6.4. It is a concrete way to make bounded ask quotes control the pricing gauge, but it is not the only one.

Corollary 6.12 (Closure under a quote-controlling ask carrier) . Let Assumption 3.4 hold, let $\cM _\chi \ne \emptyset$ , and assume the bounded-gauge compactness property

\begin{equation} \{Q\in \cM _\chi :\E _Q\chi \le L\} \quad \text {is weakly compact for every }L<\infty . \label {eq:pc2-bounded-gauge-compactness} \end{equation}

This is the structural compactness conclusion proved in the companion paper from its ambient-closed support, currency-price, and observation hypotheses; no particular carrier quote is part of the assumption here. Write the finite ask panel as

F=(C,F^0)

, where

C=(C^1,\ldots ,C^\ell )

and every component

C^a

of the carrier block is lower semicontinuous and satisfies

\begin{equation} -K_a\le C^a\le K_a\chi \quad \text {on }\Omega . \label {eq:pc2-carrier-growth} \end{equation}

Let

H

and every component of

F^0

belong to

\cC _\chi ^0

. Suppose that the carrier asks control the gauge in the following exact sense: for each

b\in \R ^\ell

for which the set below is nonempty,

\begin{equation} \sup \{\E _Q\chi :Q\in \cM _\chi ,\ \E _QC\le b\}<\infty . \label {eq:pc2-carrier-control} \end{equation}

Let

\mathcal A

denote the ordered set (14) for these rows. Then

\mathcal A

is closed. Consequently every infeasible exact/ask quote has a strict attained admissible semistatic certificate, and its dynamic leg is indispensable exactly on

\cK _{\rm stat}^{\uparrow }\setminus \mathcal A

Proof.Let $z_n\in \mathcal A$ converge, and choose realizing laws $Q_n$ . The carrier coordinates of $z_n$ are bounded above by one vector $b$ , so (22) gives $\sup _n\E _{Q_n}\chi <\infty$ . The compactness assumption (20) yields a subsequence converging weakly to some $Q\in \cM _\chi$ . The subcritical bound passes the expectations of $H$ and $F^0$ continuously, while the lower bound in (21) and Portmanteau give $\E _QC^a\le \liminf _n\E _{Q_n}C^a$ componentwise. Thus $Q$ realizes the limiting exact/ask quote. Closedness follows, and Theorem 6.3 gives the certificate claims. The all-date strip asks satisfy (22) by the pinned anchor means and (8); a different finite spanning carrier may be used when it satisfies (21) and its chaining inequalities prove (22).∎

We now fix the structural market that the present problem and a downstream selection problem share.

Assumption 6.13 (Common raw-history market) . For this subsection, the finite-date state spaces $\mathsf E_i$ are standard Borel, the raw histories are $\mathsf H_i:=\prod _{j=0}^i\mathsf E_j$ , the initial history is deterministic, and $\Omega \subseteq \mathsf H_N$ is Borel. Every tape, accumulator, stopping, settlement, positivity, and production restriction is encoded as a pathwise identity or support restriction in $\Omega$ . At step $i<N$ , a finite Borel vector $\Delta _i:\mathsf H_{i+1}\to \R ^{d_i}$ contains every traded-price increment whose conditional-mean row is imposed over $[t_i,t_{i+1}]$ . The uncalibrated class $\cM$ consists exactly of the probability laws concentrated on $\Omega$ , with the declared initial history, for which

\begin{equation} \Delta _i\in L^1(Q),\qquad \E _Q[\Delta _i\mid \cF _i]=0, \qquad \cF _i:=\sigma (h_i), \label {eq:pc2-raw-martingale-rows} \end{equation}

where

h_i

is the raw-history projection. In the companion paper’s market,

\Delta _i

is the finite vector of all retained live normalized bond-price increments; stopped rows are constant after maturity.

Any downstream class compared below uses the same $\Omega$ , raw filtration, martingale rows, and pathwise identities. Only its integrability gauge may differ. Prescribed kernels, continuum marginals, conditional independence, atomlessness, and other distributional restrictions are therefore outside this gauge-only comparison.

On such a market, any law in the class can be replaced by a finitely supported law in the class with the same expectations of a fixed finite integrable vector.

Lemma 6.14 (Finite raw-history martingale cubature) . Under Assumption 6.13, let $Q\in \cM$ , let $A\subseteq \Omega$ be measurable for the $Q$ -completion with $Q(A)=1$ , and let $Z=(Z^1,\ldots ,Z^r):\Omega \to \R ^r$ be real, universally measurable, and in $L^1(Q)$ . There is a law $Q^{\rm fin}\in \cM$ , concentrated on a finite set of complete raw histories contained in $A$ , such that

\begin{equation} \E _{Q^{\rm fin}}Z=\E _QZ. \label {eq:pc2-raw-cubature-match} \end{equation}

Every retained node at date

i

may be chosen with at most

d_i+r+1

successors. In particular, the number of atoms is at most

\begin{equation} \#\{\omega :Q^{\rm fin}(\{\omega \})>0\} \le \prod _{i=0}^{N-1}(d_i+r+1). \label {eq:pc2-raw-cubature-size} \end{equation}

All pathwise tape and production identities remain exact. If

\rho :\Omega \to [1,\infty )

is universally measurable and finite at every point of

A

, then

Q^{\rm fin}\in \cM _\rho := \{P\in \cM :\E _P\rho <\infty \}

Proof.Under a fixed probability on a standard Borel space, a function measurable for the completion has a Borel version. Replace $Z$ by such a version and replace $A$ by a Borel subset of full $Q$ -measure on which the versions agree. It is therefore enough to work with Borel $A$ and $Z$ .

Let $Q_i$ be the raw-history marginal at date $i$ . Standard Borel disintegration supplies Borel conditional successor kernels $p_i(h_i,\dd e)$ , and conditional expectation supplies Borel versions

v_i(h_i):=\E _Q[Z\mid \cF _i](h_i)\in \R ^r, \qquad v_N=Z.

Define good-history sets backwards. Put

\mathsf G_N:=A

. For

i=N-1,\ldots ,0

, let

\mathsf G_i

consist of the histories

h_i

for which

\begin{align} p_i\bigl (h_i,\{e:(h_i,e)\in \mathsf G_{i+1}\}\bigr )&=1, \label {eq:pc2-cubature-good-support}\\ \int \bigl (|\Delta _i(h_i,e)|_1+ |v_{i+1}(h_i,e)|_1\bigr )p_i(h_i,\dd e)&<\infty , \label {eq:pc2-cubature-good-integrability}\\ \int \bigl (\Delta _i(h_i,e),v_{i+1}(h_i,e)\bigr ) p_i(h_i,\dd e)&=\bigl (0,v_i(h_i)\bigr ). \label {eq:pc2-cubature-good-barycenter} \end{align}

Kernel integration makes these sets Borel. Conditional integrability, (23), the tower property, and backward disintegration give $Q_i(\mathsf G_i)=1$ at every date. The deterministic root therefore belongs to $\mathsf G_0$ .

We use one elementary finite-dimensional fact. If $\mu$ is a probability, $Y\in L^1(\mu ;\R ^k)$ , and $B$ has full $\mu$ -measure, then

\begin{equation} \int Y\,\dd \mu \in \conv Y(B), \label {eq:pc2-integrable-barycenter} \end{equation}

and Caratheodory’s theorem represents this barycenter by at most

k+1

points of

Y(B)

. Indeed, intersect

B

with the inverse image of the support of

Y_\#\mu

; its image is dense in that support. Write

K:=\cl \conv Y(B)

. The barycenter belongs to

K

: otherwise strict separation of that point from

K

, followed by integration of the separating affine functional, would contradict

Y\in K

almost surely. If the barycenter lay on a proper relative boundary face of

K

, a supporting affine functional and equality after integration would force the whole support into that face, a contradiction to the definition of

K

. Hence the barycenter is in

\ri K

. For a nonempty finite-dimensional convex set

C

\ri (\cl C)=\ri C\subseteq C

, which proves (29). This is the integrable finite-vector form of Tchakaloff’s reduction (Bayer and Teichmann, 2006).

Start at the root. At each retained history $h_i\in \mathsf G_i$ , apply (29) under $p_i(h_i,\cdot )$ , on the full-measure successor set in $\mathsf G_{i+1}$ , to the $(d_i+r)$ -vector ∎

e\longmapsto \bigl (\Delta _i(h_i,e),v_{i+1}(h_i,e)\bigr ).

Choose at most

d_i+r+1

actual successors and weights whose barycenter is

(0,v_i(h_i))

. Repeat at the finitely many retained successors through date

N

, and multiply transition weights along each branch. This defines a finite-tree law

Q^{\rm fin}

. Every leaf lies in

\mathsf G_N=A

. The zero first block at each positive-mass raw-history node is exactly

\E _{Q^{\rm fin}}[\Delta _i\mid \cF _i]=0

; the second block telescopes from

v_N=Z

to the deterministic-root value

v_0=\E _QZ

. Thus (24) holds and (25) follows from the branching bounds. Every selected leaf is an actual admissible raw history, so every identity encoded by

\Omega

is preserved. Finally, a finite law concentrated where

\rho

is finite has finite

\rho

-moment.

The lemma is the raw-history version of martingale Tchakaloff cubature; compare Beiglböck and Nutz (2014, Theorem 5.1). The raw-history construction is essential because a direct projection onto the traded price path could lose martingale conditions given a larger raw tape, whereas adjoining nontraded tape coordinates to the martingale vector would incorrectly impose martingale rows on them.

Theorem 6.15 (Finite-panel gauge invariance) . Under Assumption 6.13, let $\rho _1,\rho _2:\Omega \to [1,\infty )$ be universally measurable and finite at every admissible path, and let $Z:\Omega \to \R ^r$ be a finite real universally measurable vector. Define

\begin {aligned} I_\rho (Z)&:=\{\E _QZ:Q\in \cM _\rho ,\ Z\in L^1(Q)\},\\ \cM _{\rm fin}&:=\{Q\in \cM : Q\text { is concentrated on a finite set}\}. \end {aligned}

Then

\begin{equation} I_{\rho _1}(Z)=I_{\rho _2}(Z) =\{\E _QZ:Q\in \cM _{\rm fin}\}. \label {eq:pc2-gauge-image-equality} \end{equation}

In particular, for finite exact rows $H$ and ask rows $F$ , put

\begin{equation} \mathcal A_\rho (H,F):= \left \{(\E _QH,r): \begin {array}{l} Q\in \cM _\rho ,\quad |H|_1+|F|_1\in L^1(Q),\\[-0.15em] \E _QF\le r \end {array}\right \}. \label {eq:pc2-gauge-ordered-body} \end{equation}

Define

\mathcal A_{\rm fin}(H,F)

by the same display with

Q\in \cM _{\rm fin}

in place of

Q\in \cM _\rho

. Then

\begin{equation} \mathcal A_{\rho _1}(H,F) =\mathcal A_{\rho _2}(H,F) =\mathcal A_{\rm fin}(H,F). \label {eq:pc2-gauge-ordered-equality} \end{equation}

For a fixed finite additional real payoff vector

X

, the attainable set of

\E _QX

over any exact/ask fiber is likewise identical across the two gauge classes whenever

(H,F,X)\in L^1(Q)

is imposed.

Proof.Every finite-tree law integrates both pointwise-finite gauges and every finite real vector on its support. Thus the finite-tree moment image is contained in both sides of (30). Conversely, take any moment vector in either side and apply Lemma 6.14 to the corresponding law with $A=\Omega$ . The resulting finite-tree law matches the vector exactly, proving (30). Apply this to $Z=(H,F)$ to preserve every ask slack and obtain (32); append $X$ to $Z$ for the last assertion.∎

Combined with the closure result of Corollary 6.12, gauge invariance transfers both feasibility and certificates to a downstream gauge.

Corollary 6.16 (Exact handoff to a downstream selector gauge) . Assume the hypotheses of Corollary 6.12 and Assumption 6.13. Let $\Psi :\Omega \to [1,\infty )$ be a pointwise-finite universally measurable downstream gauge with $\chi \le C_\Psi \Psi$ . Suppose the downstream problem uses the same structural market and the same finite compilation of exact and ask rows as the present setting. Then

\begin{equation} \mathcal A_\Psi (H,F)=\mathcal A_\chi (H,F)=\mathcal A, \label {eq:pc2-selector-body} \end{equation}

and this common ordered quote set is closed. Consequently, for every finite completed book

\cB =(c,q)

\begin{equation} \begin {split} &\{Q\in \cM _\chi :\E _QH=c,\ \E _QF\le q\}\ne \emptyset \\ &\quad \Longleftrightarrow \{Q\in \cM _\Psi :\E _QH=c,\ \E _QF\le q\}\ne \emptyset \\ &\quad \Longleftrightarrow \text {such a book is realized by a finite-tree law in } \cM _\chi \cap \cM _\Psi . \end {split} \label {eq:pc2-selector-feasibility} \end{equation}

Every infeasible downstream quote therefore has the same strict attained executable semistatic certificate as in Corollary 6.12; the ordered pathwise hull still decides exactly whether its dynamic leg is indispensable.

Proof.Because $\chi \le C_\Psi \Psi$ , every downstream law has finite $\chi$ -moment and the displayed rows are integrable. Apply Theorem 6.15 with $(\rho _1,\rho _2)=(\chi ,\Psi )$ , and then apply Corollary 6.12. The finite-tree statement comes from the middle equality in (30). The static hull depends only on $\Omega$ , $H$ , $F$ , and the ask order cone, all of which are common to the two problems.∎

Remark 6.17 (Compiling a selector’s data: a worked instance) . Suppose a downstream selector uses primitive-envelope exponents $p,r>1$ with $pr=2$ , and its leading primitive envelope contains every retained anchor $\zeta _{t_i}$ . Its gauge ledger then gives, with $W_{\rm anc}:=\max _i\zeta _{t_i}$ ,

\Psi \ge 1+W_{\rm anc}^2, \qquad \chi =1+\sum _{i=1}^N\zeta _{t_i}^2 \le \max \{1,N\}\Psi .

For a regular completed book, the finite weighted call, mass, and band rows belong to the

\chi

-subcritical observation class. A lower band on

g

is compiled as the ask row

-g

, and the selector’s quadratic ask

\E _Q\zeta _{t_i}^2 \le \tau _i\pi _i+\Zbar _0(t_i)

is exactly

\E _Qs_i\le \pi _i

, because

s_i=(\zeta _{t_i}^2-\zeta _{t_i})/\tau _i

and the anchor mean is pinned. Once the two problems are posed on the same raw-history structural market, Theorem 6.15 therefore identifies their finite quote bodies and feasibility decisions. If, in addition, Assumption 3.4 and equation (19) hold and the chosen completing carrier satisfies equations (20) and (21), then Corollary 6.16 supplies closedness and the executable certificate conclusion. The finite-panel feasibility identity itself uses only Assumption 6.13, pointwise finiteness of the gauges, and the row integrability built into (31).

Remark 6.18 (What the selector handoff does not supply) . The conclusion is an equality of finite observation images, not an equality $\cM _\chi =\cM _\Psi$ . The cubature law depends on the chosen finite vector. It need not preserve a continuum of marginal constraints, a prescribed kernel or tape distribution, atomlessness, absolute continuity relative to a reference prior, or finite relative entropy. Nor does pointwise finiteness of $\Psi$ produce a uniform bound on $\E _Q\Psi$ , compactness of $\cM _\Psi$ , an information projection, Gibbs potentials, or a numerical entropy algorithm.

The absence of uniform inverse-envelope control already appears in one period. Let

\Omega =\{(\zeta ,d):\zeta \ge 1,\ \zeta ^{-3}\le d\le \zeta \}, \qquad r_0=(2,1/2),

pin both coordinate means at

r_0

, and set

\chi =1+\zeta ^2

and

\Psi =\chi +d^{-2}

. The support is closed, convex, strictly positive, and satisfies the price bound

d\le \zeta

. For

R\ge 3

, define

p_R=R^{-2},\qquad a_R=\frac {2-p_RR}{1-p_R},\qquad b_R=\frac {1/2-p_RR^{-3}}{1-p_R},

and

Q_R=(1-p_R)\delta _{(a_R,b_R)} +p_R\delta _{(R,R^{-3})}.

Directly,

a_R>3/2

and

1/2<b_R<1

, so

a_R^{-3}<b_R<a_R

and

(a_R,b_R)\in \Omega

. Moreover,

\E _{Q_R}(\zeta ,d)=r_0

, and

\E _{Q_R}\zeta ^2<5

, but

\E _{Q_R}\Psi \ge p_R(R^{-3})^{-2}=R^4.

Thus the quadratic strip supplies no book-uniform

\Psi

-level. Moreover, with

R_n=2^n

and

w_n=12\cdot 4^{-n}

n\ge 2

, the mixture

Q_\infty :=\sum _{n\ge 2}w_nQ_{R_n}

belongs to

\cM _\chi \setminus \cM _\Psi

. This also proves directly that the law classes need not agree.

For the completed-book certificate conclusion, the common-market hypothesis is essential. A downstream change of path support, raw filtration, or distributional restrictions is not an integrability-gauge change and requires a separate interface theorem. Likewise, Corollary 6.12 states precisely what a downstream spanning or chaining construction must prove before the strict-certificate conclusion transfers.

7Conclusion

The companion paper establishes the necessity side: within the standard-timing instrument class, the standard rates book leaves one tail direction unpriced and identifies that direction. This paper describes what is achieved by adding one convexity-carrier type at the required dates.

The consistent-quote set acquires an exact description. At one date the closure of a piecewise-linear panel’s quote image is a polytope, while the image can omit boundary points; the quoted second moment removes the defect and replaces it by a second-order-cone body with attained support certificates, and piecewise-rational panels remain semialgebraic. On a finite calendar, under the companion paper’s weak compactness hypotheses, all-date ask-only carrier calibration, the stated recursive-selection conditions, joint relative moment-Slater, and subcritical objective and non-carrier calibration rows, both the maximizing model and the optimal hedge are attained. At exact carrier growth, under the stated recursive-selection, nonemptiness, all-date asks, and joint relative-interior hypotheses, the optimal hedge is still attained while a maximizing model need not exist: a second-order defect can remain, and removing it would require a stronger tail quote or a generalized defect optimizer. This is the paper’s main qualitative conclusion. A quoted moment completes the data—it converts an infinite price and an absent hedge into a finite bound and an attained one—without thereby producing a model, and the two should not be conflated in either direction.

The certificate side follows the same boundary. Under the closed completed-market hypotheses of Corollary 6.4, every subcritical inconsistency carries an attained strict semistatic certificate, and the ordered pathwise hull decides exactly whether that certificate needs its dynamic leg. Across dates the polyhedral description is sufficient rather than characteristic: Theorem 5.2 covers separated panels at every finite depth, Proposition 5.5 exhibits a curved path-mixed witness, and no universal taxonomy of the curved families is claimed.

Finally, the results delimit their interface with model selection. On a common raw-history market, every feasible finite quote book whose rows are integrable and compiled from the same exact and ask rows has a finite-tree witness in the selector’s own universally measurable, pointwise-finite gauge class. Thus changing only the integrability gauge leaves feasibility unchanged for the same finite compilation. Under the closed completed-market hypotheses, an infeasible quote also receives the same attained certificate. This interface equates only finite observation images. The construction of a suitable reference prior, finite-relative-entropy feasibility, uniform envelope control, and entropic selection remain separate questions.

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.

AThe finite-date constant-weight martingale image

This appendix records the finite-date companion to Theorem 2.12: again a full-terminal-marginal statement on its own canonical state space rather than a result about the panel body of this paper. The data-versus-finite-panel and product-versus-curve-support distinctions of Remark 2.11 persist, but compact Cartesian support is replaced here by the unbounded canonical component array. Its interest here is the source of its compactness, discussed after the proof.

Theorem A.1 (Finite-date constant-weight martingale image) . Write $\cP _1(\R )$ for the probability laws on $\R$ with finite first moment, equipped with the topology induced by the $1$ -Wasserstein distance $W_1$ . For $k=1,\ldots ,d$ fix an active terminal index $1\le r_k\le N$ and a law $\mu _k\in \cP _1(\R )$ with mean $x_0^k$ . Let $\cM (\boldsymbol \mu )$ be the set of laws on the canonical component array

\Omega _{\boldsymbol r}:=\prod _{k=1}^d\R ^{r_k+1}, \qquad \cF _i^X:=\sigma \bigl (X_j^\ell :\ 1\le \ell \le d,\ 0\le j\le \min \{i,r_\ell \}\bigr ),

with no unconstrained auxiliary coordinates, under which each

(X_i^k)_{i\le r_k}

is an

(\cF _i^X)

-martingale with

X_0^k=x_0^k

and

\Law (X_{r_k}^k)=\mu _k

. For finitely many dates

i_a

and constant weights

w_{ak}

with

i_a\le r_k

whenever

w_{ak}\neq 0

, put

B_a:=\sum _kw_{ak}X_{i_a}^k

. Then

\cM (\boldsymbol \mu )

is nonempty, convex and weakly compact; the map

Q\mapsto \bigl ((B_a)_\#Q\bigr )_a

is affine and continuous into the product

W_1

topology; and its image is compact and convex. Consequently a proposed tuple

(\nu _a)_a

, with every

\nu _a\in \cP _1(\R )

, is attainable if and only if

\sum _a\int \phi _a\,\dd \nu _a \ \le \ \sup _{Q\in \cM (\boldsymbol \mu )} \E _Q\Bigl [\sum _a\phi _a(B_a)\Bigr ]

for every finite tuple of continuous functions of at most linear growth.

Proof.Nonemptiness: keep each component constant at its mean until date $r_k$ and let it jump to $\mu _k$ there, independently across $k$ . Convexity is clear. For compactness, de la Vallée-Poussin gives increasing convex superlinear $\theta _k$ with $\int \theta _k(|x|)\,\dd \mu _k <\infty$ ; conditional Jensen and the martingale property propagate ∎

\E _Q\bigl [\theta _k(|X_i^k|)\bigr ] \ \le \ \int \theta _k(|x|)\,\dd \mu _k \qquad (i\le r_k)

backwards from the terminal date, uniformly over

\cM (\boldsymbol \mu )

. Hence every coordinate family is uniformly integrable and tight, so

\cM (\boldsymbol \mu )

is tight, and Prokhorov’s theorem gives relative compactness (Kallenberg, 2021, Thm. 16.3); weak limits preserve the martingale identities (bounded continuous test functions of the past times a coordinate increment, using uniform integrability) and the terminal laws, so the set is weakly closed, hence compact. Affinity of

Q\mapsto ((B_a)_\#Q)_a

is immediate, and continuity into

W_1

follows from weak convergence together with convergence of first absolute moments, which the same uniform integrability supplies. A continuous affine image of a compact convex set is compact and convex, and membership in it is equivalent to the displayed support-function inequality.

Remark A.2 (Where the compactness comes from) . Theorem A.1 needs no separate gauge inequality: the prescribed full terminal marginals themselves supply the uniform-integrability control, on a canonical component array with no unconstrained auxiliary coordinates and with tests of at most linear growth. In the body of this paper, by contrast, compactness of the admissible class is bought by a quoted budget (Section 3), and Theorem 2.16 and the critical-tier analysis show that even then a maximizing law may fail to exist at exact quadratic order. The contrast is about the source of compactness—prescribed marginals versus a quoted moment—and nothing more: the theorem above does not assert critical-tier model attainment, and it supplies no attained semistatic hedge.

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How to cite

Shaosai Huang (2026). Quote Completion at the Quadratic Tail Boundary for Joint Rates Panels. Working paper, version of September 2026. Kspectra Research. SSRN 7435978 (doi:10.2139/ssrn.7435978). https://kspectra.ai/papers/quote-completion-quadratic-tail-boundary/

@misc{huang2026quote,
  author = {Huang, Shaosai},
  title  = {{Quote Completion at the Quadratic Tail Boundary for Joint Rates Panels}},
  year   = {2026},
  month  = sep,
  note   = {Working paper, version of September 2026},
  doi    = {10.2139/ssrn.7435978},
  url    = {https://kspectra.ai/papers/quote-completion-quadratic-tail-boundary/}
}

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