[Paper Review] Superreplication under Model Uncertainty in Discrete Time

Marcel Nutz|arXiv (Cornell University)|Jan 15, 2013

Stochastic processes and financial applications27 references4 citations

TL;DR

This paper establishes the existence of optimal superreplicating strategies and characterizes the minimal superreplication price under model uncertainty in discrete-time financial markets. Using medial limits and a closedness result for the set of superreplicable claims, it proves that the minimal price equals the supremum of continuous linear pricing functionals on a Banach space, generalizing classical duality to nondominated, Knightian uncertainty settings.

ABSTRACT

We study the superreplication of contingent claims under model uncertainty in discrete time. We show that optimal superreplicating strategies exist in a general measure-theoretic setting; moreover, we characterize the minimal superreplication price as the supremum over all continuous linear pricing functionals on a suitable Banach space. The main ingredient is a closedness result for the set of claims which can be superreplicated from zero capital; its proof relies on medial limits.

Motivation & Objective

• To establish the existence of optimal superreplicating strategies when the true probability model is unknown and multiple measures are considered.
• To characterize the minimal superreplication price as a supremum over continuous linear pricing functionals in a general measure-theoretic framework.
• To extend classical superreplication duality to nondominated models where no reference probability measure dominates all possible models.
• To address the computational and theoretical challenges arising from Knightian uncertainty in discrete-time financial markets.
• To provide a robust framework for pricing and hedging contingent claims under model risk.

Proposed method

• Uses medial limits to prove closedness of the set of claims superreplicable from zero capital in the L1 norm.
• Applies the Hahn-Banach theorem to separate the claim from the superreplication cone, leading to a continuous linear functional.
• Characterizes the minimal superreplication price as the supremum of continuous linear functionals on a Banach space of claims.
• Assumes each model in the collection 𝒫 is a martingale measure, allowing the replacement of each P ∈ 𝒫 with an equivalent martingale measure.
• Establishes duality between wealth processes and linear pricing functionals under nondominated, model-uncertain settings.
• Uses sequential convergence in L1 to analyze continuity properties of functionals, particularly in counterexamples involving Dirac measures.

Experimental results

Research questions

• RQ1 Does an optimal superreplicating strategy exist when model uncertainty is present and the set of models is nondominated?
• RQ2 Can the minimal superreplication price be characterized as a supremum over continuous linear functionals in a general measure-theoretic setting?
• RQ3 What conditions ensure that the supremum of pricing functionals coincides with the essential supremum over models in 𝒫?
• RQ4 How does the use of medial limits enable the proof of closedness for the superreplication cone in L1?
• RQ5 Under what conditions can a sequentially continuous functional on L1 fail to be topologically continuous, and what does this imply for pricing functionals?

Key findings

• An optimal superreplicating strategy exists for any contingent claim under model uncertainty in discrete time.
• The minimal superreplication price is given by the supremum of all continuous linear pricing functionals on a suitable Banach space.
• The set of claims superreplicable from zero capital is closed in the L1 norm, a key result proven via medial limits.
• The duality result generalizes classical superreplication duality to nondominated models, removing the need for a dominating probability measure.
• A counterexample shows that sequentially continuous linear functionals on L1 need not be topologically continuous, highlighting the subtlety of convergence in model-uncertain settings.
• The result holds under universal completeness of the σ-fields, ensuring technical robustness of the duality framework.