Robust Continuous-Time Hedging

• Robust hedging in continuous time is a framework that minimizes reliance on a single probabilistic model by accommodating diverse market dynamics.
• It utilizes dual representations via martingale measures and optimal transport methods to derive minimal superhedging prices.
• The approach integrates pathwise and functional techniques to address trading frictions, jumps, and risk associated with discrete hedging.

Robust hedging in continuous time encompasses methodologies for constructing hedging strategies and pricing contingent claims in the presence of model uncertainty, trading frictions, pathwise regularity constraints, or structural market features. The focus is on minimizing reliance on a unique probabilistic model, ensuring that hedging performance controls losses across a broad family of possible price dynamics, including martingale measures, jump-diffusion processes, or even non-semimartingale paths. Mathematical frameworks span sublinear expectations, nonlinear optimal transport, pathwise functional calculus, and viscosity solution theory for nonlinear PDEs.

1. Market and Model Frameworks

Robust hedging frameworks are typically formulated on a path space such as $\Omega = C([0,T];\mathbb{R}_+^d)$ (continuous paths) or $D([0,T];\mathbb{R}^d)$ (càdlàg paths) endowed with the canonical filtration $(\mathcal{F}_t)_{0 \le t \le T}$. The asset universe generally consists of dynamically tradeable underlyings and possibly other options, with availability of a finite or countable set of European options for static positions (Hou et al., 2015). Model uncertainty is encoded via a nondominated family of probability measures $\mathcal{P} \subset \mathcal{P}(\Omega)$ or through volatility constraints/interpolation sets, e.g., prescribing sets of attainable quadratic variation densities or semimartingale characteristics (Biagini et al., 2014, Nutz, 2014, Guo et al., 6 Oct 2025).

A central technical requirement in the jump-diffusion context is the "dominating diffusion property," ensuring that diffusion dominates jump activity so that optional decompositions and supermartingale arguments apply (Nutz, 2014, Rodrigues, 17 Jun 2025). Saturation, i.e., closure under equivalent local martingale changes-of-measure, is typically assumed for mathematical completeness of duality theory.

2. Primal Superhedging Problem

The primal robust superhedging problem is stated as the minimal initial capital required to cover the claim $f$ "quasi-surely" (i.e., for all $P \in \mathcal{P}$), possibly with trading only required on a given prediction set $\Xi \subset \Omega$ representing market beliefs (Hou et al., 2015). The admissible strategies include dynamic trading in underlyings and static positions in available options, with integrands subject to pathwise or martingale-based admissibility constraints:

$$\pi(f) := \inf \left\{ x \in \mathbb{R} : \exists\, H \text{ admissible},\ x + \int_0^T H_s \cdot dS_s \geq f \quad \mathcal{P}\text{–q.s.} \right\}$$

or, in the presence of static options $g_i$,

$$\sum_{i=1}^n h_i g_i(S) + (\phi \cdot S)_T \ge G(S)\quad \forall S \in \Xi$$

(Hou et al., 2015, Nutz, 2014, Bouchard et al., 2020, Biagini et al., 2014).

In American claims, the robust hedging price requires the inequality to hold at all exercise times (Rodrigues, 17 Jun 2025, Guo et al., 6 Oct 2025).

3. Dual Representation: Martingale Measures and Optimal Transport

A profound consequence of robust superhedging duality is the representation of $\pi(f)$ as the supremum of expectations over a suitably defined class of local martingale measures calibrated to market observables and compatibility constraints:

$\pi(f) = \sup_{Q \in \mathcal{Q}} E^Q[f]$

where $\mathcal{Q}$ is the set of (local) martingale measures such that $Q \sim P$ on each $\mathcal{F}_t$ (prior to $\tau$ in general lifetime models), and under which statically traded options calibrate to the observed prices (Hou et al., 2015, Nutz, 2014, Biagini et al., 2014, Dolinsky et al., 2012).

In the model-independent setting, where no prediction set is specified beyond continuity, the dual reduces to a continuous-time Kantorovich-type martingale optimal transport (MOT) problem: maximizing the expectation of the payoff over all martingale measures matching the observed terminal distribution $\mu$ (Dolinsky et al., 2012, Hou et al., 2015).

Key duality theorems extend to multi-dimensional, multi-maturity, and jump-diffusion models, so long as admissibility, interiority, and model richness conditions are met.

4. Pathwise and Functional Approaches

Pathwise robust hedging approaches eschew probabilistic modeling entirely, working instead with integration by parts, deterministic regularity, and functional calculus. The Bender–Sottinen–Valkeila (BSV) and Cont–Fournié (CF) frameworks yield hedges as functionals of the realized trajectory and its "hindsight factors," such as running maxima/minima or quadratic variation (Tikanmäki, 2011). The functional Itô–Dupire calculus yields a pathwise (Föllmer-type) change-of-variables formula:

$$F_t(x_t) = F_0(x_0) + \int_0^t \mathcal{D}_u F(x_u)\, du + \frac12 \int_0^t \nabla^2_x F_u(x_u)\, d[x]_u + \int_0^t \nabla_x F_u(x_u)\, dx(u)$$

and, crucially, pathwise hedges in the form of the vertical derivative $\nabla_x F$. Whenever both functional and BSV hedges exist, they coincide across all models sharing the same quadratic variation structure (Tikanmäki, 2011).

The pathwise gamma-hedging approach demonstrates that, even without stochastic integrals, discrete-time hedging with cancellation conditions on derivatives (delta/gamma neutrality) ensures vanishing hedging error along refining partitions for paths of vanishing $p$-variation ($p < 3$) (Armstrong et al., 13 Jan 2026). This methodology extends to path-dependent payoffs and barrier options by careful partition design.

5. Robust Hedging with Jumps, American Options, and Transaction Costs

The robust superhedging problem generalizes to the Skorokhod space with jumps, using quasi-sure versions of the optional decomposition theorem and accommodating non-smooth path-dependent functionals that are Dupire-concave and non-increasing in time (Bouchard et al., 2020, Nutz, 2014). The duality holds without requiring domination of jumps by diffusion in Bouchard–Tan’s setting (Bouchard et al., 2020).

For American options, robust duality is established via aggregation of Snell envelopes over all prior models, and the construction of a universal minimal hedging strategy through the optional decomposition (Rodrigues, 17 Jun 2025, Guo et al., 6 Oct 2025). The robust American price admits a dual representation as the worst-case value over all martingale measures and stopping times:

$$\pi(\xi) = \sup_{P \in \mathcal{P}} \sup_{\tau} E^P[\xi_\tau]$$

Even in the presence of jumps and volatility constraints, superhedging prices and optimal strategies exist and can be characterized via robust (aggregated) Snell envelopes and the corresponding minimal supermartingale decomposition (Rodrigues, 17 Jun 2025).

Under transaction costs, robust (conservative) delta hedging is achieved using "enlarged volatility" schemes and optimized rebalancing (hitting times of the delta), combined with central limit theorems for hedging error and mean squared error analysis (Fukasawa, 2011). This scheme guarantees superhedging for convex (resp. concave) payoffs when remaining quadratic variation is at its upper (lower) bound, even in the presence of asymptotically small transaction costs.

6. Extensions: Relaxed Hedging, Market Convergence, and Structural Generalizations

Relaxed hedging in robust frameworks considers shortfall-acceptance criteria, replacing almost-sure coverage with controlled risk tolerance and yielding duality with a penalty term accounting for acceptable risk (Tangpi, 2018). This approach connects to convex risk measures and optimized certainty equivalents, providing a spectrum from superhedging to efficient "acceptable" risk pricing.

The convergence of discrete-time robust hedging to continuous time is established via Markovian uncertainty sets and discrete-time martingale approximation arguments. Under generalized Markov chain approximations, the uncertainty sets and their associated discrete superhedging prices converge (in Hausdorff metric and weak expectation) to their continuous-time analogs, ensuring structural preservation (Criens, 2024).

Permanent market impact, pathwise liquidity constraints, and feedback-based robust pricing can be addressed using quasi-linear viscosity solution theory for appropriate nonlinear PDEs (stochastic target problems), delivering model-free, worst-case hedging prices and strategies (Bouchard et al., 2015).

The martingale optimal transport duality framework further links robust continuous-time hedging to fully model-independent pricing, where dual solutions (worst-case martingale measures) are constructed purely from observable prices of vanilla options, and explicit near-optimal strategies are obtained by discretization and lifting (Dolinsky et al., 2012).

7. Connections, Limitations, and Practical Implementation

Robust hedging in continuous time unifies model-independent, model-specific, and pathwise approaches. Key implications include the existence of optimal strategies, clean superhedging/duality formulas, algorithmic (e.g., recursive) implementation, and strong connections to transport/aggregation theory. Limiting assumptions may include the need for pathwise regularity, dominating diffusion for optional decomposition, and interiority of market observables (e.g., option prices inside no-arbitrage region) (Hou et al., 2015, Nutz, 2014, Rodrigues, 17 Jun 2025).

Robust hedging methods are effective under various market uncertainties, including volatility ambiguity, stochastic volatility with rough/Markovian structure (Garnier et al., 2018), jump risk, and liquidity constraints. Implementation guidelines involve calibration to realized and implied volatilities, efficient parameter estimation, pathwise hedging using functional calculus, and discretization schemes adapted to observed data (Fukasawa, 2011, Garnier et al., 2018, Armstrong et al., 13 Jan 2026).

The literature provides a rigorous, unified theory for robust hedging and pricing, encompassing superhedging duality, aggregation of martingale measures, structural extensions, and practical computation—ensuring relevance for financial markets facing model risk and structural uncertainty (Hou et al., 2015, Nutz, 2014, Dolinsky et al., 2012, Biagini et al., 2014, Rodrigues, 17 Jun 2025, Guo et al., 6 Oct 2025, Bouchard et al., 2020, Tangpi, 2018, Fukasawa, 2011, Tikanmäki, 2011, Armstrong et al., 13 Jan 2026, Criens, 2024, Bouchard et al., 2015).