Robust Pricing-Hedging Duality

• Robust Pricing-Hedging Duality is the equality between the minimal superhedging cost and the supremum of expected payoffs over calibrated martingale measures.
• It frames the hedging problem as a Monge–Kantorovich optimal transport problem with martingale constraints to capture worst-case market scenarios.
• Methodologies such as discretization, optimal transport techniques, and neural network approximations provide practical, explicit hedging strategies amidst market frictions and uncertainty.

Robust pricing-hedging duality refers to the equality between the minimal super-hedging cost of a financial claim—attainable using (semi-)static and dynamic trading strategies under model uncertainty—and the supremum of expected payoffs computed over a (typically non-dominated) family of martingale measures that incorporate observable market prices. This principle encapsulates a core connection: the robust (i.e., model-independent or “worst-case”) hedging cost is characterized as a solution to a Monge–Kantorovich type optimal transport problem with martingale constraints. The mathematical formulation, core techniques, and applications of this duality framework have been developed and refined across a range of settings, including continuous and discrete time, markets with frictions, American exercise features, incomplete market data, and information asymmetry.

1. Formulation of the Robust Hedging Problem and Its Dual

Robust pricing-hedging duality is formally motivated by the superhedging problem for a contingent claim $G$ on a market featuring a risky asset $S$, usually modeled as a continuous (or càdlàg) price process. The minimal super-hedging cost is defined by the infimum of initial capital necessary to construct a portfolio—composed of static positions in European options (with prices implying a terminal marginal law $\mu$) and a dynamic trading strategy $\gamma$—so that

$g(S_T) + \int_0^T \gamma_t(S) dS_t \geq G(S)$

on every price path $S$ (Dolinsky et al., 2012). The robust pricing side considers the supremum over all possible martingale measures $Q$ consistent with the observed market prices (i.e., with prescribed marginals):

$$V(G) = \sup_{Q \in \mathcal{M}_\mu} \mathbb{E}_Q [G(S)]$$

where $\mathcal{M}_\mu$ is the set of martingale measures with $\mu$ as the marginal at $T$.

This equivalence between the super-replication cost and the supremum over calibrated martingale measures constitutes the robust pricing-hedging duality. The dual formulation maps precisely to a Monge–Kantorovich optimal transport problem constrained by the martingale property, highlighting the interplay between mass transport and model-independent finance.

2. Connection to Martingale Optimal Transport and Monge–Kantorovich Duality

Classical optimal transport seeks a coupling $\pi$ between initial and terminal distributions minimizing a given cost function $c(x, y)$. In robust finance, the cost is the pay-off $G(S)$, the coupling is a martingale measure, and the observed option prices specify $\mu$ as the terminal marginal. Dual variables (analogous to Kantorovich potentials) correspond to static payoffs $g$ and dynamic trading $\gamma$, ensuring mass is transported from a Dirac at $S_0$ to $\mu$ via a martingale (Dolinsky et al., 2012).

For example, robust lower bounds for the forward-start straddle require minimizing $\int |y-x|\, \rho(dx,dy)$ over couplings $\rho$ in $\mathcal{M}(\mu,\nu)$, i.e., the set of all martingale couplings with prescribed one-dimensional marginals $\mu$ at $T_0$ and $\nu$ at $T_1$ (Hobson et al., 2013). The dual involves constructing explicit semi-static subhedges via solutions to a Lagrangian inequality

$|y-x| \geq \psi(y) - \psi(x) + \delta(x)(x-y)$

with tightness on the optimal transport support. Such dual representations connect directly with geometric ideas from optimal transport and reinforce the role of convex order, martingale couplings, and variational inequalities.

3. Methodology: Discretizations, Approximations, and Explicit Constructions

Establishing robust duality requires passing from continuous-time (and infinite-dimensional) problems to tractable approximations. The canonical approach involves discretizing the price path using random crossing times, yielding piecewise constant paths on countable sets (Dolinsky et al., 2012). The discrete duals are then constructed, and min–max theorems (Kantorovich duality) are employed to relate the superhedging cost to the dual optimal transport problem. Uniform continuity and tightness arguments ensure convergence:

$\lim_{N \to \infty} V_N(G) = V(G)$

where $V_N(G)$ is the value in the $N$-discretized model. This approach not only provides theoretical guarantees but also explicit, near-optimal hedging strategies. Similar discretization and approximation methods enable robust pricing-hedging duality in multi-marginal and high-dimensional settings (Eckstein et al., 2019).

Numerical methods include finite-dimensional linear programs for the primal MMOT problem (with convergence guarantees tied to Wasserstein distances and accuracy bounds) and penalization/deep neural network approaches for the dual superhedging formulation, which offer scalability by parameterizing the hedging strategy class.

4. Extensions: Transaction Costs, Market Frictions, Constraints, and Information

Robust duality remains valid under a range of market imperfections. With proportional transaction costs, the admissible set of dual measures relaxes from true martingale laws to approximate martingale laws, adapting conditional expectations to a cost "envelope":

$$(1-K) S_k \leq \mathbb{E}_Q [S_N | \mathcal{F}_k] \leq (1+K) S_k$$

The dual problem is optimized over these consistent price systems, reflecting the increased range of admissible price dynamics due to transaction costs (Dolinsky et al., 2013). With convex transaction costs and other trading constraints, the duality adjusts to include penalization for deviation from strict martingality or static option price constraints (Cheridito et al., 2016).

Advanced frameworks incorporate beliefs ("prediction sets") and information asymmetry by specifying the admissible path-space, possibly conditioned on additional observed information or signals (Hou et al., 2015, Aksamit et al., 2016). Informed agents restrict the superhedging problem to the paths compatible with their information, and the dynamic programming principle extends duality recursively over time intervals partitioned by arrival of new information.

Robust duality also generalizes to relaxed (risk-based) hedging, incorporating acceptance sets defined via, e.g., average value at risk or entropic risk measures. This yields a controlled trade-off between risk and hedging cost, narrowing superhedging/subhedging price bounds in incomplete or illiquid markets.

5. American Options, Path-Dependent Claims, and Model Uncertainty

For American options, robust pricing-hedging duality is achieved by reformulating exercise decisions through randomization or by "enlarging" the state space so that the American option becomes a European option with additional randomization variables (stopping policies) (Aksamit et al., 2016, Guo et al., 6 Oct 2025). In non-dominated models, the dynamic Snell envelope (associated with optimal stopping) is aggregated across the uncertainty set via reflected BSDEs or measurable selection arguments, providing both the duality and existence of minimal superhedging strategies (Rodrigues, 17 Jun 2025).

The duality extends to multi-asset, multi-marginal, and path-dependent claims, including digital, Asian, Bermudan, and forward-start options, with structural results on optimal transport plans reflecting covariance or linear increment properties of the marginals (Eckstein et al., 2019).

For "nonlinear" extensions, entropy regularization and Wasserstein penalizations control relaxation of marginal constraints, yielding nonlinear pricing-hedging duality. In the limit of infinite penalty, such frameworks recover classical MOT duality, providing a robust bridge between empirical uncertainty in option markets and worst-case scenario analysis (Doldi et al., 2020).

6. Fundamental Theorem, Arbitrage, and Market Completeness

The absence of arbitrage is equivalent to the non-emptiness of the set of admissible (generalized) martingale measures and the attainability of claims through superhedging strategies (Burzoni et al., 2016). This holds in both pathwise ("scenario-based") and quasi-sure (probability family) formulations, with equivalence of their FTAP and superhedging theorems (Obloj et al., 2018). Convex duality and representation results for increasing convex functionals underpin these equivalences. In markets with incomplete or partial data, duality yields robust bounds with stability under perturbations and explicit characterization of sensitivity to changes in market inputs or beliefs (Badikov et al., 2018).

Robust pricing-hedging duality thereby unifies classical probabilistic approaches with recent advances in model-independent finance, optimal transport, and risk measurement. It provides the theoretical foundation for hedging and pricing under model uncertainty, delivering both explicit strategies and practical algorithms that are stable and well-calibrated to observable market data.