[Paper Review] Pathwise super-replication via Vovk's outer measure

Mathias Beiglböck, Alexander M. G. Cox+3 more|arXiv (Cornell University)|Jan 1, 2015

Stochastic processes and financial applications35 references9 citations

TL;DR

This paper establishes a model-independent super-replication theorem for exotic derivatives using Vovk's pathwise approach and Skorokhod embedding duality. It proves that the robust super-hedging price equals the primal expectation over martingale measures when information on finitely many marginals is given, covering options on realized variance, lookback, and discretely monitored Asian options.

ABSTRACT

Since Hobson's seminal paper [D. Hobson: Robust hedging of the lookback option. In: Finance Stoch. (1998)] the connection between model-independent pricing and the Skorokhod embedding problem has been a driving force in robust finance. We establish a general pricing-hedging duality for financial derivatives which are susceptible to the Skorokhod approach. Using Vovk's approach to mathematical finance we derive a model-independent super-replication theorem in continuous time, given information on finitely many marginals. Our result covers a broad range of exotic derivatives, including lookback options, discretely monitored Asian options, and options on realized variance.

Motivation & Objective

• To establish a robust pricing-hedging duality for exotic derivatives in continuous-time model-independent finance.
• To extend the Dolinsky-Soner super-replication theorem to include derivatives dependent on quadratic variation and volatility, such as options on realized variance.
• To develop a pathwise super-replication framework using Vovk's outer measure and quadratic variation-based path regularization.
• To prove that the primal super-replication price equals the dual hedging cost under finitely many marginal constraints.
• To cover time-invariant payoffs under the Skorokhod embedding framework, including lookback and discretely monitored options.

Proposed method

• Uses Vovk's pathwise approach to mathematical finance, defining an outer measure on continuous paths with quadratic variation.
• Introduces a pathwise stochastic integral (H·S) via simple strategies adapted to stopping times, ensuring well-defined capital processes.
• Applies Vovk's pathwise Dambis-Dubins-Schwarz theorem to time-rescale paths so that quadratic variation matches time.
• Employs Skorokhod embedding duality via the dual formulation of optimal transport, using convex order and martingale coupling.
• Imposes regularity via Υ-continuity and boundedness of functions f, ensuring convergence of hedging strategies.
• Uses Lagrangian relaxation and compactness arguments (via convex penalty functions) to handle marginal constraints and prove duality.

Experimental results

Research questions

• RQ1 Can a model-independent super-replication theorem be established for derivatives that depend on quadratic variation and are invariant under time-changed paths?
• RQ2 Does the duality between super-hedging cost and maximal expected payoff hold under finitely many marginal constraints in continuous time?
• RQ3 Can the framework of Vovk's outer measure and pathwise stochastic integration be used to extend the Dolinsky-Soner theorem to volatility-dependent payoffs?
• RQ4 How can the Skorokhod embedding problem be leveraged to derive robust hedging strategies for exotic options like lookback and discretely monitored Asian options?
• RQ5 Is it possible to construct a duality framework that includes both European-style options and path-dependent derivatives under a unified pathwise stochastic calculus?

Key findings

• The primal super-replication price Pn, defined as the supremum of EP[G] over all martingale measures with specified marginals, equals the dual hedging cost Dn.
• The dual cost Dn is given by the infimum over a constant a and hedging strategies (Hm) such that a + ∑j∈I ψj(Sj) + lim infm→∞(Hm·S)n ≥ G(ω) for all paths ω.
• The result holds for payoff functions G(ω) = γ(t(ω)↾[0,⟨ω⟩n], ⟨ω⟩1, ..., ⟨ω⟩n), including options on realized variance and lookback options.
• The framework excludes continuously monitored Asian options, which require time discretization, but covers their discretely monitored counterparts.
• The proof relies on compactness via convex penalty functions and the use of Υ-continuous martingales to control the dual representation.
• The duality is robust to marginal constraints and extends to include additional options of the same invariant form via further market information.