Dispersion-Constrained Martingale Schrödinger Bridges and the Joint SPX/VIX Calibration Problem

Guyon's paper addresses the longstanding calibration puzzle of jointly fitting the implied volatility surface of the S&P 500 (SPX) and the VIX options market simultaneously under a single coherent model. The puzzle arises because the VIX — defined as VIX²_T = (1/Δ) E_Q[∫_{T}^{T+Δ} σ²_t dt | F_T] under the risk-neutral measure Q — constrains the instantaneous variance forward curve in a way that is difficult to reconcile with the smile shapes observed in SPX vanilla options, particularly for short maturities where the SPX smile is steep and the VIX smile is convex. Standard stochastic volatility models calibrated to one surface typically fail to match the other — rough Heston and SABR models reproduce SPX term structure and skew but systematically misprice VIX options, while mean-reverting variance models that fit VIX smiles produce SPX smiles with incorrect slope and term structure. The core contribution is to embed the joint calibration problem within a variational framework in which the risk-neutral measure Q* is identified as the solution to a dispersion-constrained martingale Schrödinger problem: minimize the relative entropy H(Q|Q_ref) subject to three classes of constraints — the static SPX marginal constraint (the risk-neutral distribution of S_T under Q equals the market-implied marginal μ_T for each maturity T), the VIX constraint (the conditional variance E_Q[VIX²_T | F_T] matches the market price of VIX futures and the distributional constraint from VIX option prices defines the marginal of VIX_T under Q), and the martingale constraint (the undiscounted price process (S_t)_{t≥0} is a local martingale under Q). Together these three constraints form the dispersion constraint because the VIX definition ties the conditional expected quadratic variation to the observable VIX futures level, restricting the set of admissible measures Q in a way that is more stringent than either the marginal or martingale constraint alone. The Schrödinger bridge formulation — minimize H(Q|Q_ref) over Q in M(μ, ν) subject to the dispersion constraint — has the critical property that the optimizer Q* is unique whenever it exists (by strict convexity of relative entropy) and is characterized by an exponential tilting of Q_ref: Q*(dω) = exp(φ(ω_{T_1}) + ψ(ω_{T_2}) + λ · g(ω)) Q_ref(dω) where φ, ψ are the Lagrange multipliers for the two marginal constraints, λ is the Lagrange multiplier for the dispersion constraint, and g(ω) is a functional encoding the VIX definition in terms of the path ω. The dual problem is: maximize inf_{Q} [H(Q|Q_ref) - E_Q[φ(X_{T_1}) + ψ(X_{T_2}) + λ · g]] over (φ, ψ, λ), which has a smooth concave dual function in the Lagrange multipliers. The exponential tilting structure means that Q* takes the form of a reference process Q_ref — typically a continuous-time diffusion or a local volatility model — tilted by an exponential density that incorporates the market constraints; this is the martingale analogue of the classical Sinkhorn–Knopp theorem for coupling probability measures, and it establishes that the joint SPX/VIX calibration problem admits a unique solution in the minimum-entropy class. The algorithmic implementation proceeds via a Sinkhorn-type fixed-point iteration on the Lagrange multipliers: initialize (φ_0, ψ_0, λ_0), then alternate between updating φ to match the SPX marginal constraint (a single-period Schrödinger equation solved by a path-integral over Q_ref), updating ψ to match the VIX marginal constraint, and updating λ to match the VIX futures level and dispersion target. Each iteration step requires a Monte Carlo or PDE solve over the reference process Q_ref, and convergence is established by the strict convexity of the dual function and the invertibility of the constraint Jacobian at the optimum. The rate of convergence is geometric in the number of Sinkhorn sweeps with contraction rate determined by the relative entropy distance between Q_ref and Q*, meaning that a well-chosen reference process Q_ref — one that is close in relative entropy to the market measure — accelerates convergence significantly.

The dispersion-constrained martingale Schrödinger problem is: Q* = argmin_{Q in M(μ_S, ν_S)} H(Q | Q_ref) subject to E_Q[VIX²_T] = v²_{mkt} and E_Q[Φ_{VIX}(VIX_T)] = P_{VIX,mkt} for each VIX option payoff Φ_{VIX}. Here H(Q|Q_ref) = ∫ log(dQ/dQ_ref) dQ is the relative entropy, M(μ_S, ν_S) is the set of martingale couplings of the SPX marginals at two maturities T_1 < T_2, and the VIX definition gives VIX²_T = (1/Δ) E_Q[∫_T^{T+Δ} σ²_s ds | F_T] = (1/Δ) E_Q[-2 log(S_{T+Δ}/S_T) + 2(S_{T+Δ}/S_T - 1) | F_T] under the log-expansion for small Δ. The Lagrangian dual is: L*(φ, ψ, λ) = log E_{Q_ref}[exp(φ(S_{T_1}) + ψ(S_{T_2}) + λ · g(S_{[T,T+Δ]}))] - ∫ φ dμ_S - ∫ ψ dν_S - λ v²_{mkt} where g(S_{[T,T+Δ]}) = (1/Δ) ∫_T^{T+Δ} σ²_s ds is the realized variance functional. The optimal Q* satisfies dQ*/dQ_ref = Z^{-1} exp(φ*(S_{T_1}) + ψ*(S_{T_2}) + λ* g), so Q* is an exponential tilt of Q_ref with Lagrange multipliers (φ*, ψ*, λ*) uniquely determined by the market constraints. The Sinkhorn update for φ: φ_{n+1}(x) = log(μ_S(x)) - log E_{Q_ref}[exp(ψ_n(S_{T_2}) + λ_n g) | S_{T_1} = x] for each x in the SPX marginal support.

The use of relative entropy as the calibration objective — rather than a parametric model distance or an L² misfit on option prices — has a deep information-theoretic justification: among all martingale measures consistent with the observed option prices, the minimum-entropy measure Q* is the unique measure that makes the fewest additional assumptions beyond what the market prices reveal. In the Bayesian interpretation, Q_ref represents the prior belief about the risk-neutral dynamics (encoded in a reference model such as a local volatility surface), and Q* is the posterior belief after updating on the market constraints (SPX marginals, VIX marginals, VIX futures). The relative entropy H(Q|Q_ref) measures the information content of the market constraints relative to the prior Q_ref, and minimizing it selects the measure that is closest to the prior while satisfying all constraints — the Bayesian posterior in the information-geometric sense. The dispersion constraint — the requirement that the conditional quadratic variation matches the VIX definition — is what distinguishes this formulation from standard martingale transport or from entropic optimal transport: standard MOT enforces only the marginal and martingale constraints, while the dispersion constraint additionally requires that the path-integral functional g(ω) = (1/Δ) ∫_T^{T+Δ} σ²_s(ω) ds has a fixed conditional expectation under Q*. This path-dependence of the dispersion constraint means that Q* is not a coupling of marginals alone but a genuine measure on path space, and the Sinkhorn iteration must solve a path-space Schrödinger equation at each step rather than a static coupling problem. The technical machinery is therefore substantially harder than finite-dimensional MOT or even bi-marginal Schrödinger bridges, requiring Monte Carlo estimation of the exponential martingale and the realized variance functional under Q_ref. For the desk, the Guyon framework resolves a persistent practical problem: parametric rough volatility and stochastic volatility models calibrated to SPX smiles typically misprice VIX futures by 5–15% and VIX option skew by 20–40%, creating model inconsistency that obscures the fair value of VIX-correlated structured products. The minimum-entropy calibration produces a unique model-consistent measure Q* that exactly fits both markets simultaneously, enabling more reliable relative-value analysis between SPX variance swaps and VIX derivatives.

The dispersion-constrained Schrödinger bridge provides the desk with a calibration methodology that is simultaneously exact in fitting market prices (both SPX and VIX surfaces are matched to full precision rather than approximately), model-light (the reference process Q_ref is used only as a prior, not as a pricing model; the pricing model is Q* itself), and information-theoretically principled (the minimum-entropy principle provides a unique selection criterion when multiple calibrated models exist). For desks running structured products with joint SPX/VIX dependence — such as variance spread options, barrier products with VIX knockout triggers, or volatility target products — this is the appropriate calibration baseline. The Sinkhorn iteration converges in practice with 10–30 sweeps for typical SPX/VIX market configurations, each sweep requiring one Monte Carlo simulation of the reference process Q_ref to estimate the exponential tilting weights. With a reference process implemented as a local volatility model (which can be simulated in O(n) time for a path of n steps), the total calibration cost is O(30 × N × n) path evaluations where N is the number of Monte Carlo paths — comparable in cost to a single calibration of a parametric stochastic volatility model, but producing a calibration that is provably exact and unique. The desk should set Q_ref to the SPX-calibrated local volatility surface, which is a near-exact fit to the SPX marginals and therefore a low-entropy reference for Q*.

The Sinkhorn convergence rate — geometric with contraction factor determined by the relative entropy H(Q*|Q_ref) — depends critically on the quality of the reference process Q_ref. For a reference process that is a poor fit to the market (e.g., Black–Scholes with flat volatility), H(Q*|Q_ref) is large, the exponential tilting weights exp(φ* + ψ* + λ*g) are highly variable across Monte Carlo paths (large variance of the importance weights), and the Sinkhorn iteration converges slowly with high Monte Carlo estimation error. Practically, the reference process Q_ref must be pre-calibrated to approximate the SPX and VIX surfaces reasonably well before starting the Sinkhorn iteration, which requires solving a preliminary calibration problem and adds to the total computation cost. The dispersion constraint encodes the VIX definition under Q via the realized-variance functional g(ω) = (1/Δ) ∫_T^{T+Δ} σ²_s(ω) ds, which is a path integral over the instantaneous variance process. In a continuous-time diffusion Q_ref, the instantaneous variance σ²_s is the diffusion coefficient evaluated at (s, S_s), and g can be computed along each simulated path. However, if the reference process Q_ref includes jumps (e.g., a jump-diffusion local volatility model), the VIX definition must account for the jump component of quadratic variation: VIX²_T = E_Q[(1/Δ)〈S/S〉_{[T,T+Δ]} | F_T] where 〈S/S〉 is the quadratic variation including jumps, and the discrete-time VIX approximation used in practice may not accurately capture the jump component, introducing a definition mismatch between the model VIX and the market-observed VIX that biases the calibration.