Finite Gaussian-mixture martingale kernels: density for projected smile-jet readouts

Keywords: smile dynamics; skew-stickiness ratio; stochastic-local volatility; Gaussian mixtures; martingale transition kernels; adapted Wasserstein distance; arbitrage-free calibration.

JEL classification: G13; C65; C63; G12.

MSC 2020: 91G20 (primary); 60G42, 60J05, 60B10, 49Q22, 91G60 (secondary).

1Introduction

An arbitrage-free European surface fixes the risk-neutral marginal law of the underlying at each maturity, but not the martingale kernels joining those laws; kernels with identical marginals produce different continuation smiles, spot–volatility responses and path-dependent prices. For independent-increment models the continuation smile is even deterministic in advance, however well the marginals fit, which is one form of the surface-versus-dynamics problem [19, Proposition 11.2 and Section 14.3]. The kernel generating those dynamics is never observed. What a surface does reveal is a hierarchy of at-the-money characteristics — the skew-stickiness ratio and its higher-order extensions — so a modelling class is adequate only if no admissible kernel displays dynamics the class cannot express in those coordinates. We ask whether finite Gaussian-mixture martingale chains meet that test: can they reproduce a prescribed finite order of at-the-money smile dynamics to arbitrary accuracy while approximating the marginals and remaining exact asset martingales? They can, at every finite order, for homogeneous stochastic volatility (SV) and for local stochastic volatility (LSV) (Theorems 7.3 and 9.13).

Why martingale kernels and Gaussian mixtures. The kernel is the natural object for calibrating dynamics: it carries exactly the freedom European prices leave, lives on the grid of quoted dates, prices every maturity and path-dependent claim from one law, and combines local and stochastic volatility; as an exact martingale chain it excludes static and dynamic arbitrage by construction. Finite Gaussian mixtures make such kernels tractable: calls and their strike derivatives are sums of Black–Scholes terms; martingality is one exponential-barycentre equation per node, linear in the weights, so price fitting with fixed components is a linear or quadratic program; positive variances give smooth densities and stable smile derivatives; and chains of mixture kernels have mixture marginals. Mixtures are an approximation basis, not an economic mechanism, with a long history in smile fitting [13] and a discrete stochastic-local-volatility implementation in [29]. The costs are thin tails and component counts that grow with accuracy and order.

The observable dynamics. What the kernel settles beyond today’s prices is how the smile moves when the spot moves, and calibrating that motion needs a quantity the quotes show; the skew-stickiness ratio [10, 11] is the one desks use, and it is the lowest order of a family. Che and Das [18] grade the response of the smile to spot moves by a transport equation whose at-the-money coefficients extend the ratio to every order; we take that family in finite-step form, computed from the law of a chain on the grid rather than from a pathwise derivative. More specifically, on a grid $t_0<\cdots <t_J$ with log price $X_j$ and continuation state $\zeta _j$, let $w_{j,\ell }(k,\zeta )$ be the total implied variance at log-moneyness $k$ of the maturity-$t_\ell$ smile seen from $\zeta _j=\zeta$. The readout consists of at-the-money jets and their projected spot responses over the first step,

where $m$ is the order in log-moneyness and $\ell$ the maturity, collected over a finite maturity set $\mathcal L$ and truncated at an order $N$ in $\Rcal ^{\rm raw}_{N,\mathcal L}=\bigl ((a_m^\ell )_{m\le N+1},(d_m^\ell )_{m\le N}\bigr )_{\ell \in \mathcal L}$. At order zero, $a_0^\ell$ and $a_1^\ell$ are the level and the skew, and $v_0^\ell =d_0^\ell /a_1^\ell$ is the finite-step analogue of that ratio; on $a_1^\ell \ne 0$ the rows determine the transport coefficients $v_0^\ell ,\ldots ,v_N^\ell$ of [18] by triangular inversion (Section 2.4). The higher orders are not decoration: the same study rejects $v_1=v_2=0$ at every tenor from one month to twenty-four, on five years of SPX surfaces.

The approximation problem. Quantifying over unknown kernels is not possible, so we quantify instead over the classes practitioners posit. Given a target chain $P$ in such a class, an order $N$, a fixed $p\ge 1$ and $\varepsilon >0$, the problem is to find a finite Gaussian-mixture chain $Q$ with

Density in the topology generated by (1) and the marginals (Section 2) is the statement that this is always possible; if it is, calibrating in finite mixtures costs nothing the readout can see.

The difficulty, and the topology that resolves it. Static density of finite Gaussian mixtures is classical [2], and quantizing a Markov kernel yields weak convergence of finite-dimensional laws; (2) reduces to neither, because its requirements pull against one another. Exact martingality represents every cell of a finite approximation by its exponential barycentre, leaving atoms where the readout asks for derivatives of a conditional density; and a finite chain cannot meet a generic marginal exactly (Proposition 2.5). What has to be settled first is therefore not how to approximate but what approximation should mean, and the ready-made answers do not serve. Weak or Wasserstein closeness of path laws ignores the flow of information, so models close in those topologies can call for very different hedges; the adapted Wasserstein distance repairs that, makes hedging Lipschitz stable [3], and controls the conditional laws the rows $d_m^\ell$ depend on. It is the natural candidate but still not enough: finite Gaussian-mixture chains can converge to a Black–Scholes target adaptedly while their at-the-money smile curvature diverges. No Wasserstein distance on path laws, adapted or not, makes the readout continuous (Remark 2.4). The notion must therefore come from the readout itself, together with the marginals, which is what (2) asks for and what Section 2 constructs: martingality is kept exact, the marginals are allowed to move, and the readout is required to converge. Placing each demand where it can be met is the first contribution here, and to our knowledge this topology has not been isolated before.

Main results. Theorems 7.3 (SV) and 9.13 (LSV) give, for every target $P$ in the respective class, one sequence $Q_n$ of exact-martingale finite Gaussian-mixture chains whose errors in (2) tend to zero for every $N$ simultaneously, with the velocity coefficients converging on $a_1^\ell \ne 0$; in the topologies of Section 2,

Both classes assume a compact latent state, Lipschitz kernels with exponential moments, $a_0^\ell$ in a compact subset of $(0,\infty )$ and $\Var (X_1-X_0)>0$. The SV class adds the Gaussian factor of Assumption 3.1, supplied by a variance floor with correlation bounded away from $\pm 1$ (Proposition 3.3), and contains capped Heston (Proposition 3.4); the LSV class adds one-step return densities with bounded, uniformly continuous derivatives, of which order $N$ uses the first $\max \{N-1,0\}$ (Remark 9.15). Theorem 8.3 makes the SV statement quantitative: with $M$ components per node and a $d$-dimensional latent state the readout error is $O(M^{-1/(d+1)})$ up to logarithms, only that dimension entering the exponent, and the quantization exponent is optimal (Proposition 8.4). An LSV target supplies no factor, so the construction adds Gaussian innovations of variance $\beta _n\downarrow 0$. Adapted approximation forces innovations preceding a price-dependent kernel to vanish (Proposition 9.18), and when that dependence is visible the construction’s adapted error is of exact order $\sqrt {\beta _n}$ (Proposition 9.19); a final-edge factor may stay (Proposition 9.20). Two results mark the limits of the class: finite mixtures have zero implied-variance wing slopes [33], so density fails for a target with a finite moment index once the wing slope is added to the readout (Proposition 7.4); and with the marginals held fixed rather than approximated, the finite class caps the dynamics (Proposition 7.5). On an SPX surface, the vanilla quotes leave the at-the-money forward-start volatility undetermined by $7.0$ points, and matching the first three readout coordinates as well narrows this to $4.0$, both ranges exact; over fifty-five further monthly surfaces the three coordinates remove a median $39\%$ of that range (Section 10.2).

Method. Both proofs quantize each transition by exponential barycentres $\log \E [e^L\mid \text {cell}]$ and couple target and approximant bicausally, which carries conditional laws along the sequence. Three obstacles follow. The martingale-preserving representative is not the optimal one: quantization rates are stated for $L^2$ centroids, which violate martingality, and martingale-preserving quantization is a problem in its own right [30], so the estimates are rebuilt around exponential barycentres (Lemmas 4.1 and 8.1). The labels move: the cells chosen at one date create the label set of the next, so the continuation jets must converge uniformly over sets that change along the sequence, and errors compound through the recursion (25). And the matched quantity is a jet of implied variance, recovered by an inversion whose denominator, vega, must stay away from zero along the sequence (Assumption 6.3). For an LSV target, density derivatives of order $r$ converge once the quantization error $\varepsilon _n$ vanishes faster than $\beta _n^{(r+2)/2}$, and a diagonal in $r$ serves every order at once; this is sufficient rather than necessary, since the readout itself needs only $\beta _n^{-m/2}$ at order $m$ (Corollary 9.17).

Scope and related work. The theorems are existence statements about risk-neutral capacity on a finite grid; fixed component budgets, calibration selection, instantaneous limits and physical-measure use are separate problems (Section 11). The readout meets two requirements that other gradings fail (Section 2.1): (A) order zero recovers the skew-stickiness ratio and higher orders refine it; (B) every coordinate is defined from the grid law, without a Brownian driver or perturbation parameter. Like [18], which confines transport to compact moneyness intervals and defers the wings to [33], the readout here is at-the-money by design. Model-side work computes such quantities for given models rather than asking which classes can match them: a representation formula for the skew-stickiness ratio [24], a second-order volatility-of-volatility expansion [12], and a quintic Ornstein–Uhlenbeck model tested against the skew-stickiness term structure [1]. Arbitrage-consistent frameworks for dynamic surfaces characterize admissible dynamics [34, 35, 16, 17], and [20] derives the motion of the surface inside local volatility.

Organization. Section 2 defines the readout and the topology it generates, and Section 3 the SV class; Sections 4–7 prove Theorem 7.3 and Section 9 proves Theorem 9.13. Section 8 adds a convergence rate and Section 10 the closed-form readout, a cubature bound, two numerical checks of the rates and a constrained band on an SPX surface; Section 11 discusses scope.

AI-use disclosure. The authors 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.

2The jet readout and the topology it generates

Smile dynamics are compared in a finite-step form of the transport coordinates of Che and Das [18], computed from the law of a chain on the grid, and the topology is the one those coordinates generate. Static jets are read off conditional smiles by Black–Scholes inversion, dynamic coordinates are their finite-step projections on the spot increment, and velocity jets follow by triangular inversion.

2.1The transport construction and what we use

Che and Das [18] write the action of spot moves on total implied variance $w(k,u,T)$, with $k$ the log-moneyness and $u$ the log forward, as a transport

in which $v\equiv 0$, $v\equiv 1$ and $v\equiv \beta$ are sticky delta, sticky strike and the skew-stickiness rule. With $\tilde v(k;u,T):=v(k,u,T;w(k,u,T))$ and $v_j:=\partial _k^j\tilde v(0;u,T)$, differentiating (3) at the money gives a triangular identity whose projected form is (7). The family is graded, each $v_n$ being a residual after lower-order transport.

We use (7) only to define coordinates. As an identity, (3) holds with $v=\partial _uw/\partial _kw$ wherever $\partial _kw\ne 0$ and degenerates where $\partial _kw=0$; one trajectory fixes the composite jets $v_j$, not the dependence of $v$ on $w$ [18, Remark 5.2]; the jets are coordinates rather than invariants (Remark 2.2); and they see only the spot-spanned response. We therefore replace the pathwise $\partial _u$ by the finite-step projection of Section 2.3 and state the theorems in the raw coordinates $(a_m,d_m)$, which remain defined where the $v_j$ are not. The theorems concern realizability: every projected jet profile generated by a target in the classes of Sections 3 and 9 is realized to arbitrary accuracy by exact-martingale finite chains. Attainability of an arbitrary empirical jet vector is not claimed, nor a characterization of the transport flows compatible with martingale dynamics. Other gradings fail requirement (A) or (B) of the introduction: Bergomi–Guyon functionals [12], Wiener-chaos order and martingale-expansion coefficients [23] need a forward-variance representation or a perturbation parameter, so they are not computable from the grid law of a finite mixture; marginal moments and cumulants are not nested descriptors of the at-the-money spot response; and the number of time points [27] is a complementary axis.

2.2Conditional smiles and static jets

For any chain considered below, let $\zeta _j$ denote a sufficient continuation state: a state variable that determines the conditional law of normalized future returns. For $j<\ell$ and a state value $\zeta$, define

Here and below the expectation denotes the Markov continuation started from $\zeta$, fixing one continuation version on the state space. Let $B(k,w)$ denote the normalized Black–Scholes call with log-strike $k$ and total variance $w$,

The conditional implied total variance $w_{j,\ell }(k,\zeta )$ solves

and its ATM jets are

For the homogeneous SV class of Section 3, $\zeta _j=Y_j$, the latent state; for a local-SV target, $\zeta _j=(X_j,Y_j)$; for the lifted approximants of Section 9, $\zeta _j=(\Xi _j^n,\widehat Y_j^n)$, the finite proxy label on which their normalized continuation law depends. Quantities of the $n$th approximant carry a subscript $n$, as in $a_{m,n}^{j,\ell }$.

2.3Finite-step projection

Fix the observation edge $[t_0,t_1]$ and an option maturity $t_\ell$ with $\ell \ge 2$. At the known root define

On the domain $\Var (X_1-X_0)>0$, the projected spot derivative of the $m$th smile jet is

Since $a_m^{0,\ell }$ and $a_m^{1,\ell }$ refer to the same maturity one step apart, the deterministic part of the calendar decay drops out of the covariance, leaving a spot response. The denominator is positive: $\Var (X_1-X_0)\ge \alpha _0$ under Assumption 3.1, and for LSV targets and their proxy approximants see (45) and Lemma 9.12.

In regression terms, $d_m^\ell$ is the population OLS slope of $A_m^\ell$ on $X_1-X_0$, with residual vanishing exactly when $A_m^\ell$ is affine in $X_1-X_0$. For a Gaussian-mixture edge with component label $I$, $\mathbb P(I=i)=p_i$ and $R:=X_1-X_0\mid \{I=i\}\sim \N (m_i,s_i^2)$, one has $\Var (R)=\sum _ip_i[s_i^2+(m_i-\bar m)^2]$ with $\bar m=\sum _ip_im_i$, and $\Cov (A,R)=\sum _ip_i(A_i-\bar A)(m_i-\bar m)$ if $A=A_i$ on $\{I=i\}$; if $A$ varies within components, the law of total covariance adds $\sum _ip_i\Cov (A,R\mid I=i)$.

2.4Triangular inversion and the pivot

The projected velocity jets are defined algebraically by

This is the pattern obtained by differentiating (3) at the money, with $d_n^\ell$ in place of the pathwise $\partial _ua_n$; identifying the two would require the instantaneous limit of Section 11.

If $a_1^\ell \ne 0$, the system is triangular:

More generally, if the hierarchy is compatible and $a_1^\ell =\cdots =a_{r-1}^\ell =0\ne a_r^\ell$, the rows impose $d_0^\ell =\cdots =d_{r-2}^\ell =0$ and, for $n\ge 0$,

so $v_0,\ldots ,v_N$ need $a_m$ for $m\le r+N$ and $d_m$ for $m\le r-1+N$, and near that stratum (9) is a continuous $r$-pivoted extension wherever $a_r\ne 0$. Approximation may perturb the vanishing jets, so the theorems use raw coordinates globally and velocity coefficients on the regular chart $a_1\ne 0$.

For finite $\mathcal L\subseteq \{2,\ldots ,J\}$ and $N\in \mathbb N_0$, the raw readout $\Rcal ^{\rm raw}_{N,\mathcal L}(P)$ collects $a_m^\ell (P)$, $m\le N+1$, and $d_m^\ell (P)$, $m\le N$, over $\ell \in \mathcal L$, as in (1). It recovers $v_0^\ell ,\ldots ,v_N^\ell$ on the chart $a_1^\ell \ne 0$, where the velocity coefficients are continuous functions of finitely many raw coordinates, so adjoining them leaves the topology unchanged; raw convergence also gives convergence of any fixed $r$-pivoted extension at a target with $a_r\ne 0$.

2.5The topology generated by the readout

Let $\mathfrak M$ be a class of rooted marked martingale chains. For maps $\Rcal _i:\mathfrak M\to Y_i$, $\sigma (\Rcal )$ is the initial topology, the coarsest making every $\Rcal _i$ continuous; a sequence converges in it exactly when every coordinate does. The construction is textbook, and the content is the choice of maps. Generated by the readout, the topology adapts to the quantities that describe the dynamics: two chains are close when their at-the-money jets and projected spot responses are close, whatever kernels produce them. Adjoining a coordinate refines the topology and can destroy density, as the wing slope does (Proposition 7.4). $\Pp _p(F)$ denotes the laws on a metric space $F$ with finite $p$th moment, with the distance $W_p$.

For an ambient class $\mathfrak M$ on which the raw readout $\Rcal ^{\rm raw}$ is defined, let $\mathsf m_j^X(P):=\Law _P(X_j)\in (\Pp _p(\R ),W_p)$ and $\mathsf m_j^S(P):=\Law _P(e^{X_j})\in (\Pp _1(\R _+),W_1)$, and, for finite $N$ and $\mathcal L\subseteq \{2,\ldots ,J\}$, define

The full moving-marginal topology $\tau _{\rm mov}$ is generated jointly by all $\tau _{\rm mov}^{N,\mathcal L}$, so a basic neighbourhood constrains finitely many coordinates; target and approximant may have different mark spaces, since every generator is a model-level map. The topology $\tau _{\rm mov}^{N,\mathcal L}$ is generated by the pseudometric

so $P$ lies in the $\tau _{\rm mov}^{N,\mathcal L}$-closure of a class $\mathfrak A\subseteq \mathfrak M$ exactly when its finite-order calibration error $\inf _{Q\in \mathfrak A}D_{N,\mathcal L}(P,Q)$ vanishes; exact martingality constrains the candidates rather than entering the error. For an approximating class $\mathfrak A\subseteq \mathfrak M$ and a target $P\in \mathfrak M$, write

Thus $k_*=\infty$ means zero calibration error at every finite order and maturity set; both constructions below achieve it with a single sequence.

Density is proved along one constructed sequence, by showing that every generator converges and applying Lemma 2.3; adapted Wasserstein convergence (Section 5) is a step of that proof, not a finer topology.

2.6Moving marginals and exact martingality

For the discounted asset $S$ and $X=\log S$ (so the log forward is $u=X$), martingality is the exponential barycentre identity

for every state $(x,y)$ reached by the chain.

Both finite classes below have this form, with $\alpha _h$ the factor variances (Definition 5.1) or $\beta _{h,n}$ (Definition 9.6, Lemma 9.7). Under Assumption 3.1, $\Law _P(X_j)=\Law (x_0+S_j)*\gamma _{A_j^P}$ with $S_j:=\sum _{h<j}L_h$ and $A_j^P$ the sum of the target’s factor variances, and matching (12) is impossible if $A_j<A_j^P$, forces $S_j$ to be finitely supported if $A_j=A_j^P$, and forces it to be a finite Gaussian location mixture with a common variance if $A_j>A_j^P$. For generic targets the marginals can therefore be approximated — in $W_p$ by Theorems 7.3 and 9.13, and in total variation, since the densities converge uniformly (Lemmas 6.2 and 9.10) — but not attained. This is why the formulation lets the marginals move; holding them fixed would in addition cap the dynamics (Proposition 7.5). The approximants below satisfy (11) exactly, so their spot laws are in convex order. A nondegenerate time-zero law $\mu _0$ with $\int e^{q|x|}\mu _0(dx)<\infty$ can be included by quantizing $X_0-\log \int e^x\mu _0(dx)$ with Lemma 4.1, shifting back and adding an independent $\N (-h_n^2/2,h_n^2)$, $h_n\downarrow 0$, which keeps $\E [e^{X_0}]$ exact.

3Model class and Gaussian factorization

A variance floor with imperfect correlation leaves a Brownian direction unspanned by the volatility, so each transition splits into a residual return carrying the whole exponential barycentre and an independent mean-one Gaussian factor. This restriction on the target is checkable from the model’s coefficients (Proposition 3.3); the factor keeps martingality exact under quantization, supplies the smoothing of Section 6 and yields the rate of Section 8. Section 9 drops it and pays for it.

Fix dates $0=t_0<t_1<\cdots <t_J$ containing the observation step and all maturities, a compact convex $E\subset \R ^d$, and states $Z_j=(X_j,Y_j)\in \R \times E$ with known root $(x_0,y_0)$, where $Y$ is the latent volatility state. With $\mathcal F_j=\sigma (Z_h:0\le h\le j)$, homogeneity means that for kernels $K_j^{\rm ret}$,

For $\alpha >0$ put $\gamma _\alpha :=\N (-\alpha /2,\alpha )$, so that $\E [e^G]=1$ for $G\sim \gamma _\alpha$.

By (14) and $\E [e^{G_j}]=1$, (11) holds exactly. Fix $p>2$ and the metric $d((\ell ,y),(\bar \ell ,\bar y)):=|\ell -\bar \ell |+|y-\bar y|$ on $\R \times E$.