SANOS-Evolve: A Discrete Stochastic-Local-Volatility Model for European-Option Smile Dynamics

Keywords: transition-kernel identification; smile dynamics; skew-stickiness ratio; arbitrage-free marginals; martingale couplings; Gaussian-mixture kernels; skew-aware deltas.
JEL: G13; C61.

1Introduction

Understanding how the implied-volatility smile evolves is a central problem in derivatives pricing and risk management. A European-option surface can be fitted accurately across strikes and maturities and still leave forward-starting prices, path-dependent exposures, and smile-aware hedges undetermined. The reason is that vanilla prices pin down the distribution of the underlying at each maturity without pinning down how it travels between them.

That gap has a precise statement. At each maturity $T$, under the usual regularity and no-arbitrage conditions, the call surface determines a risk-neutral marginal [3]

Across maturities, however, the family $\{\mu _T\}$ constrains but does not determine the conditional transition law

Infinitely many martingale kernels carry the same marginal chain—the observation that underwrites the model-independent bounds of martingale optimal transport [20]. Every quantity listed above depends on which of them the market is taken to follow, and the vanilla surface does not choose. Selecting one is a second inverse problem, layered on top of static calibration.

Yet “smile dynamics” is not itself an observable calibration target. Before a parametric family is imposed, the missing transition law is infinite-dimensional, and the statement that spot and volatility move together stays qualitative until one names functionals that can both be estimated from data and evaluated on a candidate kernel. Those functionals fix two things at once: what agreement with observed dynamics is taken to mean, and how two otherwise admissible kernels are to be compared.

We therefore work with two complementary leading-order at-the-money readouts. The first is the skew-stickiness ratio, the normalised regression response of at-the-money implied volatility to a spot return,

Its level places the smile response on the sticky map—sticky-moneyness, sticky-strike, and the local-volatility regime above them—and its term structure measures how far that coupling persists with maturity. Being a ratio, though, it is comparatively insensitive to the amplitude of volatility’s own motion: to leading order in this normalisation the denominator carries the same scale as the numerator, so much of the amplitude cancels. A forward-variance dispersion readout supplies that missing scale, observed on SPX through the VIX at-the-money implied-volatility term structure, which is sensitive primarily to the dispersion and persistence of future variance. The two are coupling and amplitude coordinates rather than substitutes—each is dominated by one of them, neither is a pure measurement of it. Together they are a parsimonious and empirically checkable description of leading-order at-the-money dynamics—not a complete statistic for the transition law. Calibrating to them identifies the corresponding readout-equivalence class within the chosen kernel family, and nothing finer.

This paper takes that selection as its subject. We propose SANOS-Evolve, a discrete stochastic-local-volatility framework that treats the missing transition kernel as the object of calibration. An arbitrage-free marginal surface is supplied by the SANOS framework [1] and held fixed, while a tractable two-timescale Gaussian-mixture kernel is calibrated to those two readouts. The two layers share one representation: read as a density fit, SANOS delivers each $\mu _T$ as a finite Gaussian mixture in log-moneyness—via Black–Scholes evaluation of its basis components, restated in Appendix A—and propagating one through a Gaussian-mixture kernel $K_{T_1,T_2}$ as in (1) returns a finite mixture again. Before the state-dependent leverage overlay the intrinsic propagation stays exactly inside the class; the overlay retains it through the collocation and recompression of Section 3. Either way the dynamic readouts are deterministic fixed-resolution evaluations rather than simulations. A deterministic leverage overlay aligns the conditional variance level with the prescribed marginals, leaving the kernel to describe the relative volatility states, their persistence, and their dependence on the return innovation. The resulting finite-mixture construction admits explicit propagation and deterministic dynamic readouts. In the empirical implementation the realised skew-stickiness term structure is the primary target and the forward-variance block is softly weighted beside it.

The paper develops this identification architecture (Section 2), constructs the martingale kernel (Section 3) and its readouts (Section 4), and evaluates it across SPX and NDX regimes and in a smile-roll replay (Section 5); the approximation and stability results supporting the broader mixture framework are stated where they are used and proved in Appendix B, while target construction and the calibration protocol are collected in Appendices D–E.

1.1Kernel-first identification

Conditional on the fixed marginal surface, the transition kernel is the unknown and observed dynamic behaviour selects it (Figure 1).

The construction separates the marginal level from the relative stochastic-volatility structure. Conditional on the carried volatility state the intrinsic return law is translation-invariant in log spot—computationally valuable, but unable in general to connect an arbitrary prescribed marginal pair—so a deterministic state-dependent leverage overlay supplies the local scaling the fixed surface implies, while the intrinsic kernel keeps the dispersion, persistence and return–state dependence the readouts identify. At finite resolution the resulting marginal compatibility is measured by the digital residual rather than assumed.

The implemented kernel is a parsimonious realisation of that architecture: two carried persistence timescales and a renewed within-step return branch, with martingality imposed structurally by a log-sum-exp normalisation. Conditional on the volatility state and the branch the log-return is Gaussian, so the transition law is a finite Gaussian mixture whose weights and moments are explicit functions of the parameters, and one-step propagation of the intrinsic kernel is an analytic mixture update. Finite mixtures have a long history as a tractable smile-consistent family [12]; specifying the transition law directly rather than a driving process has an architectural analogue in Markov-functional interest-rate models [45], which share a low-dimensional Markov driver, a deterministic functional calibration map and an efficient implementation. That analogy is structural—what selects the kernel here is observed dynamic behaviour.

The readout vector implements those two coordinates and carries a third block that is reported rather than optimised. $\mathcal S$ is the realised skew-stickiness term structure, read at five tenors from one week to three months. $\mathcal V$ is the forward-variance dispersion block, whose market observation is the VIX at-the-money implied-volatility term structure on SPX and a corrected realised strip off index—related observation operators occupying one slot, not the same quantity. Only these two enter the objective. The third, $\mathcal D$, is a fixed digital band comparing the propagated and target marginals: a diagnostic of the leverage overlay’s finite-resolution performance, entering the loss at zero weight. Section 4 gives each observation operator, the measure each lives under, and the identification assumption linking the pooled physical-measure regression to its within-regime risk-neutral analogue.

The static layer used in the empirical implementation is SANOS [1]. It fits weights only, over a fixed grid of Gaussian bumps in log-moneyness sharing one bandwidth, each priced by Black–Scholes; its linear program enforces convex order. The fitted marginal is therefore itself a finite Gaussian mixture—a smoothed grid of point masses, which the bumps become as the bandwidth vanishes.

That representation is why this engine and not another. The marginal layer arrives in the same class the kernel propagates, so a mixture updated by a Gaussian-mixture kernel is again a mixture, and the local variance and its Breeden–Litzenberger density are closed forms on that same object. We retain those marginals and replace the canonical discrete-local-volatility transition associated with them. The architecture itself asks only for a convex-ordered marginal engine with an explicit tractable representation—arbitrage-free surface interpolation [41] and its mixture-preserving variants [46] supply alternatives—but it is the closure under propagation that keeps every readout a finite sum.

1.2How existing methods select the kernel

Other constructions select the same missing law by other means. Process-first models—Heston [9], SABR [10], the Bergomi family [22, 23], rough volatility [25, 26]—specify state dynamics and derive the kernel, so smile and smile dynamics are encoded together. Local projection draws the dynamics from the marginals: Dupire local volatility [2] and the contemporaneous implied trees [4, 5] are the limiting case in which the static surface determines the dynamic law—so its skew-stickiness is pinned by the static skew, near the Doeff–Kamal value under an empirical $\tau ^{-1/2}$ scaling [39], while the measured SPX term structure is materially maturity-dependent. Stochastic-local volatility [7, 8, 13] fixes a backbone and identifies a leverage function through Gyöngy’s projection [6], by particle or learning-based schemes [14, 16], with existence known in the regime-switching case [17]; the leverage recovers the marginal projection, while the remaining conditional dynamics stay backbone-dependent. Transport and entropy methods—Bass constructions [42, 44], martingale optimal transport [43, 15], martingale Schrödinger bridges [35]—impose marginal or market-price constraints and select an admissible law by a variational or entropic criterion.

A fourth family models the observable surface directly: market models of implied volatility posit surface dynamics and derive the no-arbitrage drift restrictions those dynamics must satisfy [32, 33], and recent work learns the joint dynamics of liquid option prices under no-arbitrage constraints [34]. Selecting dynamics from observed surface behaviour is therefore not new; what differs is the object carried. Those models evolve the surface, so static no-arbitrage is a constraint their dynamics must respect at every date, whereas the marginals here are fixed once by a convex-ordered engine and the unknown is the kernel between them.

In each family the smile response follows from the chosen process, projection, or optimality principle, and the empirical consequence is measurable. Calibrating two-factor Bergomi, rough Bergomi, rough Heston and Heston as closely as possible to the same SPX smile, [29] finds SSR term structures that all deviate materially from the market’s, and concludes that roughness alone does not change the joint spot–volatility dynamics enough to close the gap; [27] supplies the analytic counterpart, a model-free expression for the SSR whose short-time limit is $H+\tfrac 32$ rather than the diffusive $2$. The statistic can also be targeted directly: the two-factor Quintic OU model [30] calibrates to SPX at-the-money volatility, skew and SSR term structures, and separately to the joint SPX–VIX smiles under a penalty holding the model SSR inside $[0.9,2.0]$. One qualification shapes what is claimed below: such comparisons are drawn against point estimates of a windowed realised SSR carrying no sampling band, whereas every fit here is reported against its own displayed precision scale, which is the target’s joint-HAC standard error at every date but one (Appendix C).

Adjacent joint SPX–VIX calibration. Joint SPX–VIX calibration addresses a related but different inverse problem. Guyon imposes SPX-smile, VIX-future and VIX-smile constraints and selects a minimum-relative-entropy law [37, 35]; Guo, Loeper, Oblój and Wang formulate finite SPX and VIX price constraints through semimartingale optimal transport [36]; and Che et al. derive first-order risk perturbations around an entropically calibrated SPX–VIX coupling, using the SSR to transmit SPX shocks to the VIX side [38]. SANOS-Evolve does not seek to reproduce a variance-index market. It fixes each underlying’s marginal chain and calibrates a restricted explicit kernel to skew-stickiness and a softly weighted forward-variance-dispersion readout. VIX at-the-money implied volatility supplies that observation for SPX; off index, a corrected realised proxy built from the underlying’s own option strip fills the same slot, and variance-index futures are not separate calibration constraints. The distinction that matters is therefore joint cross-market calibration versus portable readout-based kernel calibration, not competing solutions to one fitting problem.

1.3Contributions and findings

The paper makes three contributions.

1.

Dynamic-readout identification of a stochastic lift. We develop a selection principle for smile dynamics: conditional on an independently fitted marginal surface, the latent transition kernel is calibrated directly to the two complementary leading-order readouts motivated above, and what it selects is a readout-equivalence class. A deterministic leverage overlay supplies the local variance level, while the kernel controls the relative stochastic-volatility states, their persistence, and their dependence on the return innovation.

2.

An explicit finite realization and approximation foundation. We construct a two-timescale Gaussian-mixture martingale kernel with structural fibrewise normalization, analytic one-step mixture propagation, and deterministic finite-sum SSR and variance-index readouts. Two propositions accompany the construction—leverage and finite-kernel stability (Proposition 2) and dynamic-readout regularity (Proposition 3)—proved in Appendix B.

3.

Cross-regime evidence and identified limits. Across nine SPX regimes from 2012 to 2024, one uniform protocol places the realised SSR term structure within every year’s HAC sampling band ($0.9$–$3.1\%$ RMS) and fits the VIX ATM implied-volatility term structure to $1.8$–$9.2\%$ RMS. An NDX study using own-strip and realised forward-variance observations preserves the SSR fit but exposes a steep-decay limitation in the variance-of-variance term structure. A smile-roll replay—a residual-variance proxy, not traded profit and loss—suggests that the benefit of a smile-aware adjustment is stability relative to a trailing realised regression rather than superior short-horizon forecasting; a fixed ratio performs similarly.

The baseline is an ATM-dynamics model: it targets the level response summarized by SSR and one forward-variance distributional readout, with VIX entering as a softly weighted observation rather than as a joint calibration target. Its diffusive finite-factor structure does not reproduce rough $T\downarrow 0$ scaling, and translation invariance excludes spot-level dependence of conditional smile shape beyond the latent-regime channel. Section 6 evaluates those limits against the evidence.

2Identification architecture and guarantees

The marginal engine supplies an arbitrage-free spot law at every listed maturity. The dynamic unknown is a transition kernel on a lifted spot–volatility state, whose role is not to refit the vanilla surface but to select a conditional coupling whose observable dynamics agree with the chosen readouts. This section defines those objects and separates the ideal admissible problem from the finite-resolution construction used in calibration.

2.1The identification problem

Fix a pricing date and maturities $T_0<T_1<\dots <T_M$ with forwards $F_j$ and discount factors $D_j$, and work in log-moneyness $z=\log (S/F)$. Four objects.

A fixed marginal chain. An external convex procedure returns arbitrage-free spot laws $\{\mu ^Z_j\}_{j=0}^{M}$, increasing in convex order, with an explicit finite mixture representation. They are fixed once and are not re-fitted when the dynamic targets change.

A lifted source law. The kernel acts on a lifted state $X_j=(Z_j,U_j)$ carrying a latent volatility regime, with law $\widehat \mu _j(\dd z,\dd u)$ whose spot projection is the engine’s output, $\pr ^Z_{\#}\widehat \mu _j=\mu ^Z_j$. What is transported from one maturity to the next is $\widehat \mu _j$; the engine pins only its projection. A spot marginal cannot be pushed through a kernel whose source carries a latent volatility state.

The ideal family, and the implemented object. The ideal family $\Kfam$ is the set of Markov kernels $\cK _j(x,\dd y)$ on the lifted state that are positive, normalised, martingale in the forward-normalised spot on every fibre, and—together with a deterministic leverage overlay—propagate $\mu ^Z_j$ to $\mu ^Z_{j+1}$. The implementation does not sit in it, so the implemented object is not a kernel at all. It is a law-evolution pipeline $P^{\mathrm {prod}}_\theta$: an overlaid finite kernel applied to a mixture, followed by the recompression projection of Section 3.4, which maps laws to laws and is not a Markov kernel. The pipeline is indexed by $\theta \in \Theta _{\mathrm {prod}}$, the box of Section 5. Section 2.2 says what $P^{\mathrm {prod}}_\theta$ retains of admissibility and what it does not.

A dynamic readout. A map $\Rcal$ returning the finite vector of quantities the calibration actually measures, defined on a kernel or on a pipeline alike. Identification is the inverse problem

in the ideal statement, over kernels, and

as implemented—over parameters indexing the pipeline, and over the calibrated blocks only, since $\mathcal D$ carries zero weight. Every fitted object in this paper solves (4); (3) is the problem it approximates. Here $\Rcal ^{\star }$ is the observed targets and $W$ a fixed weighting; Section 4 writes $\Rcal$ out in full. Figure 1 sets this architecture beside the instance implemented here and against the standing answers to the selection question.

2.2Ideal and finite-resolution admissibility

The natural metric would be $W_1$ on the forward-normalised marginals, which in one dimension is

the second equality because $\partial _K C_\mu (K)=F_\mu (K)-1$ for undiscounted calls—so $W_1$ is the total variation across strikes of the call-price difference. A finite set of quoted strikes does not resolve that total variation, so we report the discrepancy we can measure and make no claim about $W_1$.

The content is that the no-arbitrage claim attaches to the admissible set, and that the set is nonempty precisely because SANOS delivers marginals in convex order (Strassen). The discrete-local-volatility transition associated with the SANOS parameterisation [1] becomes an element of $\cA _j$ once embedded in the lifted state—for instance with latent labels drawn independently of the return—and it is the rigid element the introduction set out to replace. The two calibrated readouts of Section 4 select a different, controllable element of $\cA _j$ in the ideal statement, and they never replace the set. What is fitted is not that element: the production pipeline is not in $\cA _j$ at all, since it satisfies the martingale condition at its own numerical state and before recompression, and replaces exact marginal propagation by the finite-feature discrepancy of Definition 2. $\cA _j$ is the ideal object the construction targets, not a set the implementation is claimed to sit in.

One question remains before any particular kernel is written down: is the class we are about to restrict to rich enough to be worth restricting? The useful form of that question is not density among admissible martingale kernels but whether the family’s readout image covers the behaviour the market shows—a finite-dimensional question, answered by the attainable-range experiment of Section 5.1 and the nine-regime panel of Section 5.3. Table 1 records which layer every later claim attaches to.

2.3Structural guarantees and approximation layers

Each layer of the construction carries a property that holds exactly, by structure, and a separate element whose accuracy is finite. Table 1 grades them, and every later use of “exact” names the row it belongs to.

What the construction avoids, relative to classical stochastic-local volatility, is the continuous-time leverage fixed point.

3A finite martingale realization

We now construct one tractable member of the admissible architecture. The design has four steps. A finite branch kernel represents the joint return–state transition; a two-timescale coefficient map supplies persistence and leverage; a structural normalization enforces martingality; and a deterministic local-variance overlay aligns the conditional variance level with the fixed marginal surface. The resulting kernel remains explicit enough to propagate as a finite Gaussian mixture.

3.1A finite martingale kernel

The state is the lifted pair of Section 2, written concretely as

with $Z_j$ log-spot and $U_j$ two continuous latent volatility factors, each standardised to unit stationary variance. The class of kernels we work in is

a finite mixture over a within-step branch index $\ell$ of Gaussian return increments, each branch carrying its own Gaussian transition $Q_\ell$ of the two factors. Three properties of (10) do the structural work, and the rest of the paper leans on them repeatedly.

Translation invariance of the intrinsic return law. The spot enters only as the additive centre of the Gaussian, so the conditional return law—and hence the conditional smile in moneyness—depends on the factor pair alone, not on the spot level. This is what makes the intrinsic propagation an exact mixture map, before the marginal overlay reintroduces a $z$-dependence (Section 3.4), and the dynamic readouts deterministic finite sums (Section 4).

The branch index mediates return–factor dependence. A common $\ell$ appears in both the Gaussian mean and the factor transition $Q_\ell$, so the joint law of return and next factor state is not a product of its marginals. This is precisely the dependence that leverage and skew-stickiness measure, and it is why an approximation theorem for this class must control the joint conditional law rather than the two conditional marginals separately (Appendix B).

Gaussian destination factor law. Each $Q_\ell (u,\cdot )$ is Gaussian, so a law carried as a Gaussian mixture in $(z,u)$ maps to another one and every readout reduces to a finite quadrature sum—not to a sum over a latent alphabet, of which there is none.

The two carried factors and the within-step branch play different roles, and the distinction matters throughout:

Spot marginal versus lifted law. The kernel acts on the joint state, so its one-step law is a lifted joint $\widehat \mu _j(\dd z,\dd u)$, whereas the marginal engine supplies only the spot marginal $\mu _j^Z(\dd z)=\int _{\R ^2}\widehat \mu _j(\dd z,\dd u)$. The first maturity is lifted with the stationary bivariate factor law $\pi$ of (17); thereafter the joint is carried by the propagation. Throughout, $\mu _j\cK _j=\mu _{j+1}$ abbreviates the match of spot marginals after propagation of the lifted law, $\pr ^Z_{\#}(\widehat \mu _j\cK _j)=\mu ^Z_{j+1}$.

Equation (10) admits unrestricted weights, drifts and variances; the implementation below restricts them to an eight-coefficient image, seven of them fitted. Two scope points belong here rather than in the appendix. Appendix B establishes stability and readout continuity for the overlaid finite kernel, and no approximation-theoretic claim is made for the family: what that image reaches is answered by measurement (Sections 5.1 and 5.4), not by a density theorem. And the two operations the production chain performs at every step—evaluating the leverage overlay by collocation at finitely many price abscissas (Section 3.3), and recompressing the propagated mixture back to a fixed component budget (Section 3.4)—appear in no result of that appendix, which is stated throughout at a fixed numerical pattern (Remark 1). Their effect is measured (Section 5) rather than bounded.

3.2The two-timescale specialization

The volatility state is a baseline plus a fast and a slow mean-reverting factor, the two-factor Bergomi picture [22, 23]. Conditional on that state the one-step log-return is Gaussian, but its mean and variance are functions of a continuous latent state, so the kernel is an integral over the factors rather than a finite mixture. The remedy is quadrature— applied to the factors’ effect on the log-variance, not to the factors themselves, which stay continuous throughout. An $n$-point Gauss–Hermite rule represents $\cN (0,1)$ by nodes $\zeta _l$ and weights $w_l$ reproducing its moments through order $2n-1$,

so replacing each continuous latent quantity by its nodes turns the integral into a finite sum, each node a volatility scenario. The nodes, base log-weights $\eta ^{\mathsf F},\eta ^{\mathsf S}$ and within-step weights $w_l=\softmax (\eta ^{\mathsf L})$ are fixed quadrature constants.

Eight structural knobs $\bth =(\nu _{\mathsf {F}},\nu _{\mathsf {S}},\nu _{\mathsf {L}},\lambda _{\mathrm {skew}}, \rho _{\mathsf {F}},\rho _{\mathsf {S}},\kappa _{\mathsf {F}},\kappa _{\mathsf {S}})$ determine every coefficient in (10); the level $\bar \gamma$ is a ninth coefficient but is solved rather than fitted, from the reference-variance condition (18) below.

Each carried factor is a standardised Gaussian AR(1),

with standard normal innovations, so each has unit stationary variance and $\kappa$ is exactly the one-step autocorrelation, $|\kappa |<1$. This is the continuous-shock idealisation. Production replaces the branch part of each innovation by the finite Gauss–Hermite rule of (11): conditional on a branch the factor step is Gaussian, but marginally the carried law is a finite Gaussian mixture, and every “Gaussian” statement below—the bivariate stationary law of (17), the $k$-step law of (40)—is that idealisation, realised by matching moments and closing the recursion in the Gaussian family. The closure is a quadrature approximation, not an identity, and its error is not measured here: the refinement study of Section 5.2 varies node counts and the recompression budget, which is a different comparison. Quantifying the closure would mean propagating the full mixture against the moment-closed recursion, and we do not report it. The leverage is a correlation between those innovations and the branch variable $\zeta ^{\mathsf L}_\ell$ that drives the return,

leaving each factor an idiosyncratic innovation variance $(1-\kappa ^2)(1-\rho ^2)$. This is the return–factor channel of (10): a branch carrying a large negative return also tilts both factors, and it does so through the innovation rather than through log-spot.

The two factors enter the return law only through the scalar combination $x_j=\nu _{\mathsf F}u^{\mathsf F}_j+\nu _{\mathsf S}u^{\mathsf S}_j$, so the branch variance and the within-step skew tilt are

the drift $d_\ell$ of (10) following from $\widetilde m_\ell$ by the martingale lock (19). Only $x_j$ is integrated by quadrature, at $n_{\mathsf X}$ nodes; the factors themselves are never placed on abscissas. One derived quantity recurs throughout: the full one-step increment variance at a factor state,

the law of total variance applied across branches—the within-branch term plus the mean spread the skew tilt creates. It is what the level condition, the leverage overlay and the forward-variance readout all measure.

Under (12)–(13) the stationary factor law is bivariate normal with unit marginals and correlation

writing $s_\bullet =\sqrt {1-\kappa _\bullet ^{2}}$ for the innovation scales. The only channel coupling the two factors is their shared leverage on the return branch: if either $\rho$ vanishes they are stationarily independent. The level is then fixed by requiring the stationary expected one-step variance to match the reference,

which determines $\bar \gamma$ and removes it from the fitted vector.

The parameters separate by role: $\nu _{\mathsf F},\nu _{\mathsf S}$ set the spread of the log-variance across the two factors and $\nu _{\mathsf L}$ its within-step dispersion; $\lambda _{\mathrm {skew}}$ the return skew; $\kappa _{\mathsf F},\kappa _{\mathsf S}$ the persistence; and $\rho _{\mathsf F},\rho _{\mathsf S}$ the return–factor dependence, the innovation leverages. A down move tilts the factors toward higher volatility through $\rho _{\mathsf F},\rho _{\mathsf S}$. Because the leverage enters the innovation around a fixed stationary target rather than shifting that target, the equilibrium volatility is not a deterministic function of spot: this is genuine stochastic volatility, and the kernel stays translation-invariant in $z$.

Since $\kappa$ is an autocorrelation it carries an effective rate directly, through the one-step half-life $-\log 2/\log \kappa$, and needs no separate interpretation. Across the nine SPX fits of Section 5.3 the ordering $\kappa _{\mathsf S}>\kappa _{\mathsf F}$ holds at every date, with median half-lives of $0.7$ weeks fast and $52$ weeks slow, so the two timescales the construction posits are the two the calibration finds.

The finite approximation is not a discretisation of the two persistent factors. They are carried continuously—each step propagates their means and covariance, and the state passed between steps is $(\text {weight},\ \mu ,\ \sigma ,\ m_{\mathsf F},\ m_{\mathsf S},\ V_{\mathsf {FF}},\ V_{\mathsf {SS}},\ V_{\mathsf {FS}})$ per component—so what is finite is the quadrature that integrates their combined effect on the log-variance, the within-step branch count $n_{\mathsf L}$, the price sub-abscissas, and the recompression budget. Production values and the measured readout sensitivity are reported with the empirical protocol in Section 5.

3.3Structural martingality and the marginal overlay

Martingality is enforced twice: once on the intrinsic kernel, and again after the market-level overlay that rescales its variance. The first is stated for the ideal construction, at a fixed latent state; the second is what the implementation performs, and it conditions on the numerical state instead. The difference is spelled out below and is not cosmetic.

Stage 1: the intrinsic kernel (ideal). The within-step drift is locked to the forward on every fibre—the kernel’s conditional law at a fixed state $(z,u)$, with $u=(u^{\mathsf F},u^{\mathsf S})\in \R ^2$ the continuous factor pair. Writing $t_\ell (u)=\lambda _{\mathrm {skew}}\zeta ^{\mathsf L}_\ell \sqrt {V_\ell (u)}$ for the skew tilt of (14), so that the raw mean is $\widetilde m_\ell (u)=-\tfrac 12V_\ell (u)+t_\ell (u)$, subtract the per-fibre constant

so that $\E [e^{Z_{j+1}}\mid z,u]=e^{z}$ exactly at a realised $u$ (Proposition 4). The shift is a single per-fibre constant, independent of $z$, stable and differentiable, and it preserves the relative branch tilts that carry the leverage and the skew-stickiness.

Stage 2: the market-level overlay. A translation-invariant kernel cannot, on its own, carry a prescribed marginal chain: its unleveraged propagation has convolution form $\mu _j\cK _j=\mu _j\ast \eta _j$, which connects only marginal pairs admitting a common increment-law representation. Matching is restored by one deterministic ingredient, a leverage overlay $\sigma _{\mathrm {LV}}(z,T_j)$ that rescales the conditional variance. Writing the increment as a local scale times a unit stochastic-volatility factor,

with $\bar V_u=\overline V(u)$ of (16): the first two factors are the target decomposition, exact in the continuous-time limit, and $R^{u}_\Delta$ is what the weekly step leaves at a fixed latent state. The implementation does not condition there, so a second remainder $R^{c}_\Delta$—same algebra, coarser conditioning—appears at (24) below and is the one measured; the two are not the same number, and only $R^{c}_\Delta$ is reported. Taking $\E _\ell [V_\ell ]$ alone mis-scales the leverage by $\sim 3\times$ at the calibrated $\lambda _{\mathrm {skew}}$. The overlay itself is the finite-kernel analogue of the stochastic-local-volatility leverage relation,

with $\sigma _{\mathrm {Dupire}}$ read from the fixed marginals: a closed-form Gaussian-mixture call, a Breeden–Litzenberger density, and a calendar finite difference. That difference is well posed only on a convex-ordered surface, which is what the constraint in Algorithm 1 is for; the overlay inherits that hypothesis from the marginal engine (Assumption 1). Rescaling the variance changes $\E [e^{\Delta z}]$, so the drift must be re-locked after the overlay rather than carried over. Writing $L(z)=\sigma _{\mathrm {LV}}(z,T_j)/\sigma _{\mathrm {ref}}$, the overlay scales

the variances by $L^2$ and the skew tilt by $L$, and then re-locks. Two things follow, and only the first is exact.

The forward is restored exactly, at the state the implementation carries. The production lock is taken over the branches and the component’s own factor quadrature together,

with $x_{ck}$ the component’s factor nodes and $\varpi _k$ their weights (Section 3.4), so that $\E [e^{Z_{j+1}}\mid c,z]=e^{z}$ identically (Proposition 5). That is martingality at the numerical information state—the component index and the price abscissa—which is the filtration the propagation actually carries. It is not the same as martingality conditional on a realised $u$: the two coincide only where the component’s factor law is degenerate.

The conditional variance is matched to leading order, not exactly. Because variances scale by $L^2$ while the tilt scales by $L$, the branch dispersion of the drift does not scale as $L^2$. At the state the implementation conditions on—the component $c$ and the price abscissa $z$ of (23)—

both terms vanishing at $L=1$, with the moments taken over the joint factor-node/branch law $\varpi _kw_\ell$ that the lock normalises over. The same algebra at a fixed $(z,u)$ gives the same expression with branch-only moments, but that is a finer information set than the implementation uses and understates the discrepancy by roughly a factor of two: $R^{u}_\Delta$ and $R^{c}_\Delta$ are different quantities and only the latter is reported. Since $V_\ell =O(\Delta )$ and $t_\ell =O(\Delta ^{1/2})$ the covariance term dominates and both remainders are $O(\Delta ^{3/2})$, so the relative error is $O(\Delta ^{1/2})$—at a weekly step that is not automatically negligible, and Section 5.2 measures it rather than bounding it. Equation (21) is therefore the continuous-time relation the weekly construction targets, and the finite step realises it to leading order. The re-lock of (23) shifts every branch by one constant, so it restores the forward without touching $R^{c}_\Delta$. The complete per-step pipeline is

The division of labour. This is the decoupling that organises the paper, and it is stated here once, with the moment it applies to named. The marginal engine determines the local variance level; the intrinsic kernel determines the relative regime dispersion and the return–regime dependence. The overlay is a variance rescaling, so what it carries is the second moment: the per-step enforcement is (21) followed by the martingale re-lock (23), which fixes a variance and a mean and constrains nothing beyond them. The propagated skew is therefore not carried by the overlay but generated by the kernel, through $\lambda _{\mathrm {skew}}$ and the innovation leverages, and Section 5.3 measures how far short of the target marginal’s skew it falls. The $\pi$-average of (20) equals $\sigma ^2_{\mathrm {LV}}\Delta _j$ up to the finite-step remainder of (24), so the level is the leverage’s responsibility and the kernel enters only through the unit-mean ratio $\nu _u$. Positivity and normalisation are structural in the coefficient map and martingality holds at the numerical state before recompression; marginal compatibility is the overlay’s; and calibration therefore fits only dynamic parameters. The kernel’s own level $\bar \gamma$ nearly divides out of $\nu$ and is reset by $\sigma _{\mathrm {LV}}$, leaving it close to inert; what survives is the relative structure of the factor state—the $\nu _{\mathsf F},\nu _{\mathsf S}$ spread and, through (12) and (13), the autocorrelations and the innovation leverages—which is exactly what the two calibrated readouts interrogate jointly: the leverages and the spread set the coupling that $\mathcal S$ measures, the spread and the autocorrelations set the amplitude and persistence that $\mathcal V$ measures.

The overlay is a composition of three maps—conditional variance, leverage ratio, overlaid kernel—and each must be stable for approximation of a target-compatible limit to transfer to the finite-mixture implementation.

Proof idea. Stability of the conditional mean is not derived here—it is Assumption 1(iv), for the reason given there. Granting it, the ratio is stable because $\nu$ is bounded away from zero on the nondegenerate region; and the overlaid kernel depends on $(\theta ,\ell )$ through locally Lipschitz coefficient maps, a locally Lipschitz log-sum-exp re-lock, and a Gaussian-mixture $W_1$ bound. The proof is Proposition 6, at the fixed numerical pattern stated there.

Here $\E [\nu \mid z]$ is the spot-conditional average of the unit factor, distinct from the stationary $\E _\pi [\nu ]=1$ and tilted only by the spot–volatility correlation. For marginal matching—each step re-seeded from the target law—it is close to $1$ and the residual is discretisation rather than bias. Under multi-step concatenation from a fixed spot (forward densities, hedging) it is not: the negative leverage tilt correlates high-$\nu$ regimes with low $z$, so $\E [\nu \mid z{=}0]\approx 0.7$ on SPX, and ignoring it under-reads the accumulated ATM level by ${\sim }10\%$ in volatility over a multi-month chain. The full spot-conditional correction of (21) is therefore retained for absolute-level outputs.

3.4Propagation and component control

One-step propagation of the intrinsic kernel is exact. Translation invariance means the discretised kernel adds, conditional on a regime, a fixed Gaussian increment—constant in the integration variable—and fixed Gaussians convolve in closed form:

A law carried per component as a Gaussian in $(z,u)$ therefore propagates to a finite mixture of such Gaussians,

where $\varpi _k$ are the $n_{\mathsf X}$ Gauss–Hermite weights integrating that component’s own log-variance combination $x=\nu _{\mathsf F}u^{\mathsf F}+\nu _{\mathsf S}u^{\mathsf S}$—itself Gaussian, since the component is—and $(\mathrm A,\mathrm b_\ell ,\Sigma _\ell )$ is the affine factor update of (12)–(13), with $\mathrm A=\operatorname {diag} (\kappa _{\mathsf F},\kappa _{\mathsf S})$ and the branch-dependent mean shift carrying the leverage. The coefficient variability lives entirely in the outer sum over $(c,k,\ell )$ and never inside a convolution, so propagation of the intrinsic kernel is an exact finite Gaussian-mixture update, needing no state-space integration, PDE solve or Monte Carlo.

The production chain is not that object, and the difference should not be collapsed. The leverage overlay of Section 3.3 scales the branch variances by $L(z)^2$, and that $z$-dependence is exactly what breaks (25): fixed Gaussians convolve, $z$-dependent ones do not. Each overlaid step is therefore closed by a first-order collocation in $z$ rather than by the identity above. What survives exactly is structural—positivity and normalisation at every state and parameter value, and martingality at the numerical state (23)—while the local-variance match carries the remainder (24) and marginal propagation is approximate; Table 1 grades which is which. The collocation residual is moreover a property of the overlay rather than of the component budget: the Gyöngy match is re-imposed at every step, so this is a first-order approximation error and not a quantity that refining the recompression drives to zero. (A Monte-Carlo cross-check showed the same collocation biases the multi-step skew-stickiness when it is applied to the intrinsic branch law instead.)

Why the component count must be controlled. Equation (26) multiplies the component count by $n_{\mathsf L}n_{\mathsf X}n_{\mathsf P}$ at each step, where $n_{\mathsf X}$ is the quadrature over the combined factor state and $n_{\mathsf P}$ the price sub-abscissas of the leverage evaluation, so composing $\cK _j\circ \cK _{j+1}\circ \cdots$ over $J$ steps—needed for forward densities, exotics and intermediate maturities—would leave $(n_{\mathsf L}n_{\mathsf X}n_{\mathsf P})^J$ components. A single kernel application needs no reduction; concatenation does.

Recompression, and what it does not preserve. Each composed step therefore ends with a deterministic reduction that holds the budget flat: bands on the two factor means and then on the price mean, and within each cell a match of mass, mean and variance. It is a law-level projection, not a kernel operation, and the distinction matters. A cell merges descendants originating in different source states, and the re-lock that follows is a single scalar per chain, chosen so that the reduced mixture’s forward equals the pre-reduction mixture’s. What the projection preserves is therefore mass and the unconditional forward; it does not preserve the forward conditional on each preceding numerical state, and the reduced object is consequently not itself a martingale kernel. We therefore claim no Strassen or convex-order property along the recompressed chain: exact martingality is a statement about the pre-projection kernel (23) only, and what the chain carries past the projection is monitored rather than guaranteed. Restoring the pathwise statement would need conditional recompression—a per-source-cell re-lock—which the production path does not perform. Multi-step composition therefore stays tractable as an exact mixture update followed by a deterministic reduced-mixture approximation whose law and martingale residuals are measured and reported. The per-step pipeline is propagate $\to$ recompress (budget and global re-lock) $\to$ leverage $\to$ re-lock; the recompression is the projection step. The reduction is never applied across the carried factor bands.

Intermediate maturities. Because the carried factors are Ornstein–Uhlenbeck, the latent transition operators form a semigroup, $P^U_sP^U_t=P^U_{s+t}$ with mean-reversion rate $a=-\log \kappa /\Delta$; the overlaid and recompressed production chain does not. Writing the latent log-variance as $x_t=g_0(t)+X^{\mathsf F}_t+X^{\mathsf S}_t$ separates a time-inhomogeneous forward-variance level $g_0$ from a time-homogeneous generator, so one interpolates the one-dimensional curve $g_0$ and constructs intermediate kernels by evaluation, with no matching optimisation. The carried volatility state is what makes the construction compose at all; a spot-only kernel, marginalising the volatility state at the intermediate step, does not, which is why a memoryless kernel needs a matching optimisation.

3.5Summary of the construction

4Dynamic readouts and identification

Section 1 motivates $\mathcal S$ and $\mathcal V$ as complementary coupling and amplitude readouts of leading-order at-the-money dynamics. This section is the formal counterpart: it defines their observation operators, names the measure each lives under, states the calibration map, and gives the equivalence that map induces. One point carries over and is worth stating once as a design principle rather than a claim—the readout is not a summary of the kernel but the topology in which it is identified, so a family is rich enough, and an approximation good enough, only relative to the functionals one intends to read off it.

4.1The identification map

The calibration reads these three blocks and optimises two, so two readout vectors have to be kept apart,

the first what the objective sees, the second what is reported. Exact equivalence is a statement about the calibrated pair alone,

an idealisation no fit attains. What a fit delivers is membership of a production tolerance class, stated on the pipeline and with the two blocks kept apart because their tolerances are not the same kind of quantity:

Only $\varepsilon _{\mathcal S}$ carries a sampling-error reading: it is set by the skew-stickiness target’s estimated standard error. The forward-variance target is a single-day $\mathbb Q$ read with no sampling band, so $\varepsilon _{\mathcal V}$ is a chosen calibration tolerance—the discrepancy we are willing to leave—and nothing about it is statistical. The weightings $W_{\mathcal S},W_{\mathcal V}$ are the corresponding diagonal blocks of the $W$ introduced below. Every downstream identification claim is a claim about (28); densities and exotic prices are computed from one representative of it and remain representative-specific. Write $\theta \in \Theta \subset \R ^{n_\theta }$ for the free parameters, $n_\theta =7$ in production: the eight structural coefficients of Section 3 less $\rho _{\mathsf S}$, which is pinned, and less the level $\bar \gamma$, which is not a coefficient of $\theta$ at all but solved from $\sigma _{\mathrm {ref}}$ by the level match of Section 3.3. The three blocks are read on grids that the design fixes but the model does not: a set $\mathcal {K}_{\mathcal {D}}$ of log-moneyness thresholds and a set $\mathcal {T}_{\mathcal {S}}=\{T_1,\dots ,T_{d_{\mathcal {S}}}\}$ of tenors, both stepped at $\Delta t$, and a set $\{\tau _1,\dots ,\tau _{d_{\mathcal {V}}}\}$ of forward-variance expiries. The readout vector is then

of total dimension $d$ as in (3), with the blocks

Only the last two enter the loss. The instance realises the production problem (4) with a diagonal $W$ that vanishes on the $\mathcal D$ block and normalises the surviving residuals by their own targets, plus one term outside that form—a ridge on $\theta$ rather than on the readout—so the objective is best given explicitly. With targets $\mathcal {S}^{\star },\mathcal {V}^{\star }$, stage anchor $\theta ^{0}$, per-tenor skew-stickiness weights $\varsigma \in \R ^{d_{\mathcal {S}}}$ normalised to $\overline {\varsigma ^2}=1$, ridge weights $\varrho \in \R ^{n_\theta }$ and a floor $\varepsilon >0$, the residual vector is