Joint Calibration of Caplet and Swaption Smiles: A Finite-Panel Impossibility Theorem and Its Payment-Timing Completion

Keywords: joint calibration; caplet and swaption smiles; CMS convexity; payment timing; support-function duality.

MSC (2020): 91G20; 91G30; 60G44; 90C46. JEL: G12; G13; E43.

1Introduction

Caplets and swaptions are two central vanilla option markets of the interest-rate world, written on overlapping segments of one discount curve. Parametric models can and do fit selected caplet and swaption panels jointly, while the literature documents both the modeling difficulty and economically relevant cap–swaption relative-value deviations (Henry-Labordère, 2008; Hunt et al., 2000; Longstaff et al., 2001). This paper does not claim those restricted calibration tasks are unsolved. It addresses the sharper nonparametric question: on an untruncated state space, retaining the natural numeraire and annuity weights, does there exist an arbitrage-free joint law matching both quoted smile families, is the weighted image closed, and which market information makes that problem well posed?

The equity analogue of this question—the joint S&P 500/VIX smile calibration puzzle—was solved by Guyon (2020, 2024): the set of martingale laws matching the SPX marginals, the VIX marginal, and the consistency constraint linking them is characterized by a duality, and, relative to a chosen reference law, a minimum-entropy member can be selected and computed by Sinkhorn iteration (Bourgey and Guyon, 2024). A structural feature supports that program: the prescribed coordinate marginals are ordinary, unit-weight marginals. They pin the corresponding laws and provide the tightness backbone; Guyon’s results then close the conditional constraints and prove the relevant duality under their printed moment assumptions. This is the familiar martingale-transport setting: existence under fixed marginals in convex order is classical (Strassen, 1965), and fixed marginals with suitable moments supply the main compactness input (Beiglböck et al., 2013).

The rates problem transposes differently, and the difference is the subject of this paper. Written in units of a terminal (anchor-maturity) bond $P(\cdot ,T_\star )$, the normalized price of a payer swaption at strike $K$ is

where $e$ is the exercise date, $S_e$ the swap rate, $q_j$ the fixed-leg payment dates with accrual factors $\delta _j>0$, and $\Zbar _t(u):=P_t(u)/P_t(T_\star )$ the price at $t$, in units of the anchor bond, of one currency unit paid at $u$; thus $\Abar _e$ is the annuity in anchor units. A caplet price is $\E _Q[\Zbar _\theta (p)(L-K)^+]$, carrying a bond weight at the fixing date $\theta$ for payment at $p$; a cash-settled swaption carries yet another curve-dependent weight. Differentiating twice in the strike à la Breeden and Litzenberger (1978), a quoted smile family determines not the law of the rate but the weighted marginal $(Y)_\#(W\,Q)$—the pushforward of the measure $W\,\dd Q$ under the observable $Y$, with a state-dependent, unbounded weight $W$. Weighted masses and weighted first moments are then pinned by martingality alone ($\E _Q[\Abar _e]$ is the initial annuity, for every candidate law), but an $L^1$ bound is not uniform integrability: nothing in the quoted data controls the joint tail of the weight and the state. The question this paper answers is whether a finite standard-timing rates book supplies that missing control, and which additional quote does so when it does not—equivalently, whether the weighted-smile image of the calibrated class can be made closed, so that feasibility has a duality characterization and calibration is well posed on the declared untruncated class.

1.1Results

Fix a finite tenor calendar $0=t_0<\dots <t_N=T<T_\star$, an exercise date $e$, annuity dates $q_1<\dots <q_m=T_m\le T$, and the anchor maturity $T_\star$; write $\zeta _t=1/P_t(T_\star )$ for the inverse currency price of the anchor bond and call $(T_m,T_\star )$ the numeraire segment. All definitions are given in Section 2; the calibrated class $\cM (\cB )$ consists of the martingale term-structure laws on a declared path space matching a finite quote book $\cB$. The results come in matched negative and positive sides, together with an implementation side. They instantiate one portable principle: state-dependent weighted marginal constraints need not control the weight itself, and a single quoted convexity functional identifying a coercive moment of the weight restores compactness and closedness. Everything below is that sentence made exact in the rates market.

Negative side. Definition 3.1 introduces the escape family: a one-parameter family of admissible curve deformations supported in the numeraire segment, which multiplies every swap-segment anchor price by a factor $\Lambda \ge 1$ while leaving every swap-segment currency price, hence every swap-segment swap rate and forward rate, and every swap-segment cash-settlement functional, unchanged. Proposition 3.5 shows that the anchor payoff of every standard-timing claim—bounded currency payoffs, arbitrary functions of swap-segment curves, and natural-timing rate claims of linear growth; together these cover caplets, floorlets, physical and cash-settled swaptions, CMS caplets and coupons of every tenor and payment lag, and bond options—grows at most linearly in $\Lambda$ along the family. Theorem 3.6 then proves: for every finite standard book with quote-Slater room whose support contains the escape family, and for every superlinear convex $\Phi$,

The proof is constructive: a vanishing-probability escape atom is recalibrated exactly by a fixed finite mixture, so the two laws agree on every quote while the superlinear annuity moment diverges. Three corollaries sharpen the statement. Corollary 3.11: any finite panel of CMS and cash-settled quotes at arbitrarily chosen strikes cannot repair the failure—their convexity is convexity in the rate, while the required control is a tail property in the numeraire direction. Corollary 3.9: on escape-rich supports, whenever the quote-Slater correctors can be chosen with finite budget for the gauge under test, no admissible coercive gauge is observably proper—the natural compactness hypothesis for feasibility on the unbudgeted class $\cM$ is not merely unverifiable but false. And Theorem 3.16 extends both statements from single-fixing books to books fixing and paying across the whole calendar—under explicit integrable surgery envelopes, with broad sufficient cases covering bounded-currency and natural-timing claims and exact-date swap-segment claims—by a conditional-mean-preserving surgery: the diagonal-linear deformation $D_\Lambda$ is paired with the compensator $S_{\Lambda ,p}=(I-pD_\Lambda )/(1-p)$, in which $p$ is the probability carried by the deformed branch, so that $(1-p)S_{\Lambda ,p}+pD_\Lambda =I$ as an identity of maps and every conditional martingale row survives pointwise, leaving only finitely many book values for the Slater correction.

Positive side. The obstruction is exactly priced by payment-timing convexity. The in-advance claim $\mathrm {IA}(b)$ pays at $b$ the simple rate fixed at $b$ for the full segment $[b,T_\star ]$—payment at the start of the accrual it references. Lemma 4.2 is an exact, model-free identity within the class:

for every calibrated law, where $\tau _b>0$ is the contractual accrual factor of $[b,T_\star ]$ and $\pi _b$ the claim’s normalized ask; one quoted number is affinely one second moment, with no convexity adjustment or expansion. Theorem 4.3 converts a strip of such asks into a closed-form second-moment bound for every linearly dominated pricing weight (Cauchy–Schwarz plus conditional Jensen; no rate floor); Theorem 4.6 builds from the strip a continuous, inf-compact gauge whose budget is a market quote; and Theorem 4.7 proves that on the strip-calibrated class the weighted-smile image is compact, convex, and closed, with an exact support-function characterization of feasibility. The well-posedness input thus becomes a price, not a model parameter. Proposition 5.2 locates the boundary: the in-advance claim is statically equivalent to a natural-timing FRA plus an infinite-strike cap/floor strip, and every finite-strike truncation restores the negative theorem (Proposition 5.3); a finite-strike implementation therefore requires a wing extrapolation or another tail assumption.

Implementation side. Section 6 removes the need for a strip at every date. The rolled overnight deposit on the published compounded fixing tape has a martingale anchor value $V^{(i)}$ with $V^{(i)}_{t_i}=\zeta _{t_i}$, so conditional Jensen chains every calendar date’s budget to a single settlement date (Theorem 6.2); the account-convexity claims $\mathrm {AC}(t_i)$, Borel functions of the published tape and one terminal bond fixing, carry the entire gauge budget in one settlement family (Corollary 6.5). A designed margined settlement index prices these claims exactly with no sign or correlation condition (Corollary 6.7), whereas plain compounded-rate futures quotes are provably uninformative for the upper budget: they are exactly invariant under a terminal escape surgery that blows the budget up, and—given quote-Slater room for the panel augmented by the futures quotes themselves—the surgery-plus-correction leaves any standard book and all futures quotes exact while the budget diverges (Proposition 6.8); an imposed positive-association condition bounds the required moment from the wrong side. A deterministic numerical certificate (Section 7) exhibits two exactly calibrated laws, indistinguishable on a six-instrument standard book to ten digits, whose second annuity moments differ by a factor $3.2\times 10^4$; the strip separates them by the same order.

The finite-panel boundary. The negative and positive sides meet at a sharp three-regime boundary, and the impossibility is directional, not a growth count. Finite panels fail at any strikes: the standard class includes payoffs of arbitrary growth in the swap-segment rates (class (S2) of Definition 3.4), and they fail identically, because every standard claim weights its convexity by a numeraire-linear factor (Proposition 3.5). The idealized all-strike far-segment smile does span the missing moment (Proposition 5.2), and every finite-strike truncation of it restores the failure (Proposition 5.3); one payment-timing quote per date, or one settlement family at a single date, replaces the continuum exactly (Lemma 4.2 and Theorem 6.2). The impossibility theorem therefore concerns practical finite quote panels—the data that exist—while the completion closes the full weighted-smile image; neither statement reduces to the other.

1.2Related literature

Robust pricing and martingale transport. In the model-independent literature, existence under fully prescribed unit-weight marginals goes back to Strassen (1965), and the compactness of the calibrated class is obtained from those marginals with suitable moments (Beiglböck et al., 2013); finitely many co-maturing quotes are treated by Davis and Hobson (2007). None of this applies verbatim to rates smiles, whose constraints carry unbounded state-dependent weights; the present paper can be read as the statement that, on the declared escape-rich supports and under the printed Slater hypotheses, the weighted-marginal analogue of that compactness theory fails, and cannot be repaired by a finite standard-timing quote book. Closest in mechanism is the observation of Acciaio et al. (2017), refined by Nutz et al. (2023), that semistatic outcome spaces can fail to be closed for integrability reasons; the annuity escape is an integrability failure of the same species, realized inside the term structure and shown here, under the printed support and quote-Slater hypotheses, to be immune to every finite standard-timing remedy. Campi et al. (2017) study how martingale transport interacts with a numeraire change in the symmetric two-marginal case; the rates problem is the asymmetric case in which each constraint family lives in its own numeraire and no change of measure removes the weights simultaneously.

The equity precedent. The joint SPX/VIX literature (Guyon, 2020, 2024; Bourgey and Guyon, 2024) supplies existence, duality, reference-dependent entropy selection, and computation in its ordinary-marginal setting. Our negative theorem identifies the rates obstruction at the closedness step, and the positive theorems recover compactness, a closed weighted image, and the support-function feasibility dual once the missing tail budget is quoted. Entropy dual potentials and a non-atomic numerical scheme are not proved here. The instrument analogy is structural rather than literal: VIX derivatives reveal a conditional log-contract dispersion quantity, whereas the rates completion must control the numeraire/annuity tail directly. A variance-style contract written only on a rate can remain blind to the escape (Remark 3.12).

CMS convexity and the practitioner folklore. The classical CMS literature prices convexity by replication against the swaption smile under an annuity-mapping (terminal swap rate) assumption (Hagan, 2003; Mercurio and Pallavicini, 2006; Andersen and Piterbarg, 2010); it is folklore that the mapping is a model input not identified by vanilla quotes. More specifically, Cedervall and Piterbarg (2012) model the joint covariance of the swap rate, its annuity, and the payment-delay bond, linking CMS and payment convexity to swaption volatilities across relevant tenors and CMS-spread prices while controlling potential arbitrage. Their construction supplies a dependence model. Building directly on that line, Bang and Daboussi (2022, Proposition 3.1 and Corollary 3.1.1), see also Bang and Daboussi (2023), introduce an annuity-due measure for a common-expiry family of RFR swap rates. In their single-curve setting, idealized full swaption smiles determine the individual rate marginals, while a specified dependence structure supplies a common joint law. Their Corollary 3.1.1 prices the receipt of a swap rate at reset against payment of the same rate at maturity: its present value is model-independent and equals the initial annuity times the annuity-measure second moment of that rate. Applied to the one-period swap spanning $[b,T_\star ]$, that contract is the in-advance claim minus its natural-timing FRA, and their identity is then, after accrual and anchor normalization, exactly Lemma 4.2 below: our one-period identity is a specialization and re-expression of their Corollary 3.1.1 in anchor units, not a novelty claimed here. Those works supply dependence constructions and a contract-level identity; the result here instead concerns the full nonparametric class calibrated to a finite standard-timing panel and, under the printed escape-rich-support, common-integrability, and quote-Slater hypotheses, proves that the panel does not control the numeraire-weight tail. Theorem 3.6 and Corollary 3.11 turn that folklore into a theorem about the calibrated class: the non-identified direction carries, quantitatively, an unbounded superlinear annuity moment, and no finite family of CMS or cash-settled quotes (Mercurio, 2008) at any strikes can bound it. The hinge decomposition behind the in-advance replication is classical, while the identity used here is the pathwise anchor-unit form of the payment-timing precursor just described. Its economic motivation is the same marking-to-market convexity studied in the classical forward–futures literature (Cox et al., 1981); the contribution here is its exact placement as a quoted uniform-integrability budget, together with a recession-direction theorem showing why finite standard-timing books do not play that role.

Transport methods in rates. Roome (2026) calibrates the joint law of two CMS rates to CMS and spread options by a constrained Schrödinger problem and computes no-arbitrage bounds for bivariate exotics. Posed under a single payment measure with unit weights, finitely many strikes, and a truncated compact state grid, it does not meet the weighted-tail obstruction studied here. The two meet at the bounds: for payoffs growing superlinearly in the annuity weight, the untruncated escape-rich class gives infinite bounds (Theorem 3.6), so the grid is the tail assumption, not merely a computational device.

1.3Scope and organization

Scope. The analysis is single-collateral on a fixed finite calendar; Bermudan exercise, collateral optionality, and moving calendars lie outside it. The negative results require supports rich enough to contain the escape family—a declared hard cap on the numeraire coordinate evades them by construction, at the price of becoming the tail assumption itself. No liquidity or execution claim is made for the completing instruments; whether sufficiently close structures are quoted and usable is empirical, so absent that binding the honest output of an exact joint calibration is relative to a declared envelope. Each boundary is stated precisely in Section 8.

Organization. Section 2 sets the market, the calibrated class, and the two properness notions. Section 3 proves the negative results. Sections 4 to 6 develop the in-advance strip, its replication boundary, and the chaining to a single settlement family, with the lower-power and capped contract variants recorded in Section 4.4. Section 7 reports the numerical certificate, Section 8 collects boundaries and open problems, and Section A discusses adoption economics and contract design of the budget carrier; the three longest proofs are deferred to Section B.

2The finite-calendar market and the properness target

Every result in the paper plays out on a single fixed arena, which this section constructs. We declare the finite calendar and path space, express all prices in the anchor units $\Zbar _t(u)$ and $\zeta _t$ in which term-structure laws become martingale families, and define the calibrated class $\cM (\cB )$ of laws matching a finite quote book, together with the carry-closure convention without which conditional-mean arguments fail outright (Lemma 2.5). The section culminates in the target that both sides of the paper address: observable properness for linearly dominated pricing weights (Definition 2.8)—a uniform-integrability property that free first moments cannot supply (Remark 2.11), that Section 3 proves no standard-timing book supplies, and whose market-implied restoration is the business of Sections 4 and 6. The section closes by exhibiting a canonical host class—curves with a fixed log-forward-rate floor—realizing the support and one-date richness hypotheses outright, and the multi-date branch hypothesis in the localized form the surgery uses (Section 2.3). A separate finite-dimensional lemma explains the genericity of relative-interior targets; it does not assert that the observed target is interior.

2.1Calendar, states, and anchor prices

Fix a finite master calendar

containing the exercise date $e$, the annuity payment dates $q_1<\cdots <q_m=T_m\le T$, and every other fixing, settlement, and payment date of the finite book. The date $T_\star$ is the maturity of the selected anchor numeraire. The last calendar date is kept strictly before the anchor: every segment $[t_i,T_\star ]$ referenced by the strip below then has positive accrual, whereas at $b=T_\star$ the in-advance claim of Definition 4.1 would degenerate ($\zeta _{T_\star }\equiv 1$ with zero accrual). Nothing is lost: the boundary date’s budget is trivial, and the boundary configuration is recovered by omitting the degenerate terminal claim (Remark 5.6). We call $(T_m,T_\star )$ the numeraire segment: the maturity range strictly between the last annuity payment and the anchor.

At each calendar date $t$ the market state contains the currency discount curve $u\mapsto P_t(u)$ on $[t,T_\star ]$, with $P_t(t)=1$ and $P_t(u)>0$. All pricing is in anchor units. The anchor price at $t$ of one currency unit paid at $u\ge t$ is

Only finitely many maturities enter the book, its carries, and the strip below. The master calendar is enlarged, if necessary, so that every retained maturity not later than $T$ is itself a calendar date. Let $U_t\subset [t,T_\star ]$ be the finite set of live maturities at date $t$, with $t\in U_t$, and let the date-$t$ price state be the vector

Two standing conventions govern the maturity inventory. Initial completeness and carry closure: every maturity that appears in some $U_t$ already belongs to $U_0$, and, for every such $u$, one has $u\in U_s$ at each calendar date $s\le \min \{u,T\}$. Thus adding a maturity needed by a later book or strip only enlarges the finite state vector; it cannot create an unconstrained coordinate midway through its carry chain. Stopped coordinates: if $u<t$, notation is extended by

This is an $\cF _u$-measurable historical coordinate frozen from $u$ onward, not a member of the live vector $x^t$; maps declared to act on the live coordinates leave it untouched. In particular, formulas such as an annuity weight at a date after an early payment use this stopped extension, never a newly resampled coordinate. The path space is a set $\Omega \subseteq \prod _{i=0}^NK_{t_i}$ with each $K_t\subseteq \R _{>0}^{k_t}$. The deterministic initial state is part of the support declaration itself: every $\omega \in \Omega$ has $x^{t_0}(\omega )=x^0$. Thus $\Omega$ lies in the corresponding closed initial-state cylinder; it is Polish, carries the raw filtration $(\cF _{t_i})_i$ generated by the price states, and encodes the declared contractual support identities of the market (for example, the tape–curve consistency identities of Section 6). Non-price production coordinates (fixing tape, settlement marks) are suppressed from the notation, and their status must be stated precisely, because the gauge of Section 4 is coercive in the price coordinates only. Three regimes are compatible with the arguments below: (i) compact-valued coordinates, such as marks confined to a declared bounded settlement domain, change nothing; (ii) coordinates that are continuous functions of the retained price coordinates through the support identities inherit pathwise control from the price state—the compounded fixing tape is of this type, each realized period rate being a ratio of price coordinates with positive denominator on $\Omega$ (Section 6); (iii) a genuinely unbounded coordinate not controlled by the price state would require its own coercive gauge term with its own quoted budget, and no claim of this paper covers it.

Since $\Omega$ lies in the open positive orthants, ambient closedness is a genuine support declaration: it forbids admissible states from accumulating, at bounded price levels, at degenerate curves with a vanishing bond price. It plays, pathwise, the role that inverse-price terms in coercive gauges ( $\Psi =1+M^p+m^{-q}+\cdots$ ) play in expectation: control of the zero-price boundary. It does not bound $\zeta$ above, and the escape family of Section 3, whose coordinates diverge rather than degenerate, respects it. Closedness of $\Omega$ merely in the relative topology of the open orthant would not suffice for the gauge theorem below.

This is a mild deterministic floor on negative rates (for example $C_P=e^{|r_{\min }|T_\star }$ under a rate floor $r_{\min }$). No upper rate bound is imposed: $\zeta _t$ is unbounded on the supports of interest.

2.2Quotes, weights, and the properness target

A European claim of the book is a pair $(C,b)$: normally, an $\cF _b$-measurable Borel currency payoff $C$ paid at the calendar date $b$. We also admit the terminal-settlement convention used only by the static replication in Section 5: an $\cF _T$-measurable payoff may be paid at the anchor maturity $b=T_\star$, in which case its anchor payoff is $C$ because $\zeta _{T_\star }=1$. This convention adds neither a state date nor a martingale row. Its anchor payoff is $\zeta _bC$, and a normalized bid–ask quote $[\ell ,u]$ imposes the affine constraint $\ell \le \E _Q[\zeta _bC]\le u$. Currency prices at time $0$ are normalized by $\zeta _0$, i.e. quoted in time-$0$ anchor units. A finite book $\cB$ is a finite family of such constraints together with the martingale rows (3); its calibrated class is

a band constraint presupposing that its expectation is absolutely convergent. Thus the negative results may use arbitrary Borel book payoffs, but only on laws for which those finitely many anchor payoffs are in $L^1(Q)$. Weak-continuity and closed-image claims are made only for the separate regular observation/quote class specified in Assumption 2.3 and Theorem 4.7; absolute integrability alone does not imply weak continuity.

Quoted smiles are, through the lens of Breeden and Litzenberger (1978), weighted marginals: twice differentiating a payer-swaption smile in the strike recovers the pushforward of the annuity-weighted law of the swap rate, and likewise for caplets and cash-settled swaptions with their own curve weights. We therefore take as observation data a finite family of weighted pushforwards

with real continuous observables $Y_a$ and nonnegative weights $W_a$, each $\Gamma _a(Q)$ a finite nonnegative Borel measure on $\R$. The observation topology on each output coordinate is the metric compounding a bounded-Lipschitz distance with the differences of total masses and of first absolute moments (Lemma 3.8 below); $\Gamma =(\Gamma _a)_{a\in \cA }$ maps into the finite product. Representative weights, measurable at their respective observation dates, are

with $\delta _j>0$ the accruals and $H^a_e$ the strike-independent cash multiplier on the declared settlement domain. For a cash observable $\widehat S_e$ we require, on that domain, $0\le H^a_e\le C_H$ and $H^a_e(1+|\widehat S_e|)\le C_{H,1}<\infty$; the latter is the precise condition controlling the cash-rate first-moment test.

All weights in (4) are linearly dominated: the annuity with $c_j=\delta _j$ and equality; the caplet weight trivially; the cash weight with the single term $C_H\Zbar _e(\sigma )$. The associated observables $Y_a$ satisfy $W_a|Y_a|\le (\text {another linear domination})$ for the vanilla book (for the annuity, $W|Y|=|\Fbar _e|$ is a signed combination of anchor bond prices, dominated by the sum of their absolute coefficients), so control of second moments of linearly dominated weights controls the full test family $W_a(1+|Y_a|)$. For the cash observation the second domination uses $H^a_e(1+|\widehat S_e|)\le C_{H,1}$ explicitly.

2.3A canonical host class

The support and richness hypotheses used below are structural declarations, and one may ask whether any recognizable nonparametric class satisfies them. The following rate-floor host realizes the support and one-date richness hypotheses outright, and the multi-date branch hypothesis in the localized, interior-margin form that the surgery actually uses. The quote-Slater condition remains a separate target hypothesis. Fix a floor $r_{\min }\le 0$ and let $K^{\rm host}_t$ be the set of date-$t$ anchor states $(\Zbar _t(u))_{u\in U_t}$ with $\Zbar _t(T_\star )=1$ and

the set of curves whose average continuously compounded, or log-forward, rates

between retained maturities are at least $r_{\min }$. Let $\Omega _{\rm host}$ be the initial-state cylinder over $\prod _iK^{\rm host}_{t_i}$ with a fixed admissible initial curve. Tape coordinates, when retained, are appended through their defining identities and fall under regime (ii) of the trichotomy above.

The following finite-dimensional fact shows that relative-interior targets are dense in any chosen attainable affine image. It neither asserts that the observed target is interior nor preserves separately fixed target coordinates unless it is applied to the corresponding convex slice.

3Failure of standard-timing books

Remark 2.11 closed the market section with a precise question: which quoted convexity upgrades the free first-moment pins to a genuine tail control? The negative side of the paper, occupying this section, answers: none within the standard-timing universe. The escape family of Definition 3.1 supplies the mechanism, Proposition 3.5 isolates the structural reason—every standard claim weights its rate convexity by a numeraire-linear factor—and Theorem 3.6 converts the two into unbounded superlinear annuity moments over the exactly calibrated class, with observable properness itself failing in its wake (Corollary 3.9) and CMS and cash-settled panels offering no way out (Corollary 3.11).

Throughout this section $e=t_1$ is the first calendar date, the initial state $x^0$ is deterministic, and every book claim is $\cF _e$-measurable. This covers the configuration of the joint problem itself—a caplet–swaption panel referencing one common fixing/exercise date after a deterministic initial segment—and is the scope of the base theorem below. The general case—books fixing and paying across the whole calendar—is proved in Section 3.4 by a different device: a conditional-mean- preserving surgery exploiting the diagonal linearity of the escape deformation. We write $p^0$ for a base currency curve at $e$ and construct from it a one-parameter family of admissible deformations.

3.1The numeraire-segment escape family

The family is a single ramp in maturity: it steepens the curve only beyond the last annuity payment, and leaves everything the swap market quotes untouched.

The frozen continuation is admissible for Assumption 2.2: the implied later currency prices are $P_t(u)=\Zbar _e(u)/\Zbar _e(t) =\bigl (P^0(u)/P^0(t)\bigr )\Lambda ^{\kappa (t)-\kappa (u)} \le P^0(u)/P^0(t)$ since $\kappa$ is nondecreasing. Along $\omega ^\Lambda$ every post-$e$ price increment vanishes, so $\delta _{\omega ^\Lambda }$ satisfies every conditional martingale row after $e$ pathwise. A declared hard support that caps $\zeta _e$—a finite tree over bounded states, say—violates Assumption 3.3 by construction; such truncations are the standing alternative to everything below, and then the cap itself is the tail assumption (Section 8). Escape-richness is an explicit structural hypothesis: it asks that the declared support be closed under the particular terminal-segment steepenings above. An unbounded-rate class need not have that closure property, and unboundedness alone does not imply the hypothesis. It is satisfiable in a canonical class: by Proposition 2.12 (iii), the rate-floor host contains the escape family launched from every base path, and indeed every capped-ramp and terminal-ramp variant used later.

3.2Standard-timing claims are numeraire-linear

With the escape admissible, what matters is how quoted payoffs behave along it. One growth computation settles that, once the traded universe is delimited.

Class (S1) contains every physical swaption: indeed $(F^{ccy}_e-KA^{ccy}_e)^+\le |F^{ccy}_e|+|K|A^{ccy}_e$, and both legs are finite linear combinations of currency bond prices bounded by Assumption 2.2. It also contains every cash-settled swaption ($\mathfrak a(\widehat S)(\widehat S-K)^+ \le \sum _j\delta _j\widehat S/(1+\delta _j\widehat S)^{\,j}\le m$ on $\widehat S\ge 0$; more generally use the declared multiplier bound following (4)), and every bond option. Class (S2) contains every CMS caplet, floorlet, or linear CMS coupon of any tenor contained in the swap segment and any payment lag, and every swap-segment caplet. Class (S3) contains caplets and floorlets on arbitrary accruals, including accruals inside the numeraire segment, at their natural payment dates.

3.3The negative theorem

Let $\cB$ be a finite standard book: constraints $\ell _i\le \E _Q[\zeta _{b_i}C_i]\le u_i$, $i\le n$, with every $(C_i,b_i)\in \cS$, together with the martingale rows. Because $x^0$ is deterministic and $e=t_1$, the rows over $[0,e]$ reduce to the finitely many equalities $\E _Q[\Zbar _e(u)]=\Zbar _0(u)$, $u\in U_e$. It is convenient to treat these initial-curve rows as quote components. Let

i.e. the class obtained from Definition 2.4 by releasing only the finitely many unconditional rows over $[0,e]$. Write