Semi-Static Hedging and the Duality of Model-Free Option Bounds

Beiglböck, Henry-Labordère, and Penkner recast the problem of model-independent option bounds as a martingale optimal transport problem: given the terminal marginal distributions μ and ν of a price process at two dates T_1 < T_2, implied by the vanilla option prices at those maturities, what is the supremum (respectively infimum) of the price of an exotic payoff Φ(S_{T_1}, S_{T_2}) over all martingale measures consistent with those marginals? The primal problem is a linear programme over the set M(μ, ν) of martingale couplings — joint distributions of (S_{T_1}, S_{T_2}) that have μ and ν as marginals and satisfy the martingale constraint E_Q[S_{T_2} | S_{T_1}] = S_{T_1} — and the dual problem is a static superreplication strategy: a combination of vanilla options at T_1 and T_2 and a dynamic hedge in the underlying between T_1 and T_2. The principal result of the paper is the no-duality-gap theorem: under mild regularity conditions on Φ, the primal supremum equals the dual infimum, and the dual minimizer is an explicit semi-static hedge that achieves the bound. The Kantorovich duality structure of the problem mirrors classical optimal transport, with the key modification that the coupling must be a martingale. The role of the convex order condition — μ ≤_c ν, meaning ∫φ dμ ≤ ∫φ dν for all convex functions φ — as the necessary and sufficient condition for the existence of any martingale coupling is established via Strassen's theorem, which states that the convex order is equivalent to the existence of a measure-preserving martingale from μ to ν. The convex order condition is therefore the no-arbitrage condition on the vanilla marginals: if it is violated, the market prices of vanillas at T_1 and T_2 are mutually inconsistent. When the condition holds, the set M(μ, ν) is non-empty and the transport problem is well-posed, and the bound gives the sharpest possible constraint on the exotic price consistent with the observed vanilla data. The desk reading of this paper is motivated by the need for model-free guardrails on exotic exposures during events, where the vanilla surface provides reliable information about marginal distributions but the model-specific coupling assumption is the source of greatest uncertainty. The martingale transport bound replaces the model-specific price with the worst-case price over all consistent couplings, and the gap between this bound and the model price quantifies the model-assumption dependence of the exotic valuation.

The primal martingale transport problem is: UB = sup_{Q ∈ M(μ,ν)} E_Q[Φ(X, Y)] where X = S_{T_1}, Y = S_{T_2}, M(μ, ν) = {Q : Q ∘ π_X^{−1} = μ, Q ∘ π_Y^{−1} = ν, E_Q[Y | X] = X} is the set of martingale couplings, and Φ(x, y) is the exotic payoff. The dual problem is: LB = inf ∫f dμ + ∫g dν subject to f(x) + g(y) + h(x)(y − x) ≥ Φ(x, y) for all (x, y), where f and g are the Kantorovich potentials (vanilla payoff profiles at T_1 and T_2 respectively) and h(x) is the dynamic hedge ratio (the number of shares held between T_1 and T_2, evaluated as a function of X = x). The no-duality-gap result states UB = LB under regularity of Φ. The convex order condition μ ≤_c ν is ∫φ dμ ≤ ∫φ dν for all convex φ ∈ C^0(R), equivalent by Strassen (1965) to ∫(x − K)_+ dμ(x) ≤ ∫(x − K)_+ dν(x) for all K (call price monotonicity in maturity). A measure Q ∈ M(μ, ν) exists if and only if μ ≤_c ν.

The no-duality-gap result is the technical core of the paper and its proof is non-trivial, drawing on the theory of abstract linear programming in infinite-dimensional spaces and the minimax theorem for convex-concave functions. The key difficulty relative to classical transport is the martingale constraint, which makes the feasible set M(μ, ν) a proper subset of all couplings Π(μ, ν), and the dual feasibility constraint acquires the additional h(x)(y − x) term encoding the dynamic hedge; showing that no duality gap arises requires establishing that the primal is attained and that the dual infimum is approached closely enough for the minimax argument to close. The connection to Strassen (1965) and classical transport theory is both a strength and a limitation of the approach. Strassen's theorem establishes the existence of martingale couplings, and the Kantorovich duality theory for classical transport provides the structural template; the contribution of Beiglböck–Henry-Labordère–Penkner is to show that this template extends to the martingale case with the additional dynamic hedge term. The dual hedge is constructive in the sense that it is computable from the optimal dual potentials (f*, g*, h*), providing a strategy that not only achieves the price bound but is also a valid super-replication: holding the vanilla portfolio (f*(X), g*(Y)) and the dynamic hedge h*(X) units of the underlying guarantees a payoff of at least Φ(X, Y) under any consistent measure. Comparison with model-specific bounds — which typically assume a particular dynamics for S and then compute the bound within that model — reveals that the martingale transport bounds are typically wider (less informative) than model-specific bounds but are guaranteed to be valid regardless of which consistent model the true dynamics follows. The tightness of the bound as a function of payoff regularity and marginal spread is important: for smooth payoffs Φ and well-separated marginals with thin tails, the bounds can be quite informative; for discontinuous payoffs like barriers or digitals, or for marginals that are close together (short time between T_1 and T_2), the bounds are typically loose.

The primary desk application of martingale transport bounds is as a guardrail for exotic exposures during events, where the usual model-dependent pricing is subject to elevated uncertainty because the normal calibration procedure may fail or produce unreliable parameters in the event window. By computing the transport upper and lower bounds for each exotic in the book using the prevailing vanilla surface at each date, the desk obtains a model-free interval for the exotic's fair value that is guaranteed to contain the true fair value under any consistent dynamics, and the width of this interval is a direct measure of the model-assumption dependence of the position. The gap between the martingale transport bound and the model-computed price is an interpretable risk metric: a small gap indicates that the exotic's value is approximately pinned by the marginals (marginally-pinned) and the model assumption matters little, while a large gap indicates that the exotic's value is highly sensitive to the coupling assumption — the joint distribution of price at T_1 and T_2 — and model risk is therefore a primary valuation concern. Exotics with large gaps should receive wider bid-offer spreads and larger reserve adjustments during event windows when model calibration is unreliable, and the desk should maintain a real-time tracker of gap magnitudes for the principal positions.

The primary practical limitation of the martingale transport bound is numerical: computing the bound requires solving an infinite-dimensional linear programme discretized on the marginals μ and ν, and the resolution of the discretization directly determines the tightness of the computed bound. Coarse discretization of the marginals — using a small number of strikes per maturity slice — produces looser bounds with larger numerical error, while fine discretization increases computational cost quadratically. For maturities where the option market has limited strike coverage, the implied marginals are not well-identified at the tails, and the computed bounds are sensitive to tail extrapolation assumptions, introducing a source of model dependence that partially defeats the purpose of the model-free approach. The coarseness of bounds for path-dependent payoffs with multiple observation dates beyond T_1 and T_2 is a fundamental limitation: the paper treats only the case of two marginal dates, and extending the framework to n observation dates requires specifying n − 1 intermediate marginals and solving a much larger transport problem on the n-fold product space. In practice, bounds for barrier options, Asian options, or lookback options computed via the two-date framework are too wide to discriminate between different models, and tightening them requires either additional moment constraints on the intermediate path or a restriction to a smaller class of consistent measures, both of which introduce auxiliary modeling assumptions.