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Model-independent Bounds for Option Prices: A Mass Transport Approach
Mathias Beiglböck, Pierre Henry-Labordère, Friedrich Penkner
TL;DR
Exotic options may have widely varying prices across models calibrated to the same market data, motivating model-independent bounds. The paper formulates these bounds through Monge-Kantorovich optimal transport and a semi-static hedging dual, proving no duality gap and primal attainment under stated conditions. The dual supremum, however, is not generally attained.
Problem
Calibrated models need not determine unique forward-price dynamics, so the paper studies bounds for exotic options across martingale models sharing prescribed marginals.
Method
The paper casts pricing as an infinite-dimensional martingale-transport problem and constructs a dual using static vanilla portfolios plus dynamic delta trading.
Results
Under mild regularity and linear-growth assumptions, the primal and dual values coincide, with the primal value attained by a calibrated martingale measure.
Takeaways & Limitations
The optimal-transport formulation provides robust model-independent option bounds with a direct semi-static hedging interpretation.
Takeaways & Limitations
The dual supremum is generally not attained, and a generalization of the two-period c-convex result to multiple periods is unavailable.
Abstract
from arXiv · showhide
In this paper we investigate model-independent bounds for exotic options written on a risky asset. Based on arguments from the theory of Monge-Kantorovich mass-transport we establish a dual version of the problem that has a natural financial interpretation in terms of semi-static hedging. In particular we prove that there is no duality gap.
1. Introduction
The paper frames exotic-option pricing under calibrated but nonunique martingale models as an infinite-dimensional optimization problem, then derives a semi-static hedging dual with no duality gap under mild assumptions. Its optimal-transport approach yields robust model-independent bounds whose primal value is attained, while the dual supremum need not be.
- Motivation: Different models calibrated to the same liquid-option data can produce a wide range of exotic-option prices because forward-price dynamics are not uniquely determined.The framework therefore seeks lower and upper bounds across models sharing prescribed marginals.
- Model-independent pricing: The admissible models are discrete-time martingale measures with prescribed one-dimensional marginals inferred from continuum call prices.The primal lower bound minimizes the exotic payoff expectation over this calibrated martingale class.
- Semi-static hedging: The dual problem consists of static vanilla positions and a self-financing delta strategy that subhedges the exotic payoff.The strategy uses integrable vanilla portfolios and bounded measurable trading functions, with the forward-position term absorbed into the first static payoff.
- Main result: Under mild regularity and linear-growth conditions, the primal and dual values coincide, and the primal value is attained by a calibrated martingale measure.The result applies to lower bounds and, by applying the theorem to the negative payoff, gives sharp upper bounds through semi-static superhedging.
- Main result: The dual value is unchanged when strategies are restricted to finite call-option combinations with continuous trading functions, preserving the financial interpretation.The paper identifies this restriction as sufficient for the dual problem under its theorem.
- Novelty and scope: The approach applies optimal-transport duality to mathematical finance to establish a robust, model-free version of extremal-price replication results.The paper notes that dual attainment is generally absent, even though the primal value is attained.
2. Optimal Transport
The section formulates optimal transport over probability measures with fixed marginals, then restricts transport plans to martingales and develops the associated dual problem. Compactness and continuity results support the duality framework used for model-independent option bounds.
- A transport plan is a probability measure on Rn whose coordinate marginals are the prescribed measures µ1, . . . , µn.
- The primal Monge-Kantorovich problem minimizes the cost functional Iπ(Φ) over all transport plans with the specified marginals.
- The dual problem maximizes the corresponding marginal-function objective over functions satisfying the pointwise domination constraint.
- The dual optimal-transport bound has a financial interpretation as a static portfolio of European options with maturities ti and payoffs ui.
- Martingale transport plans are characterized by vanishing integrals of bounded continuous trading functions multiplied by successive increments xj+1 − xj.
- The martingale transport set M(µ1, . . . , µn) is compact in the weak topology, using compactness of all transport plans and closedness from the martingale characterization.
3. Proof of Theorem 1
The proof combines Monge–Kantorovich duality with a Min–Max theorem to establish equality between martingale transport bounds and semi-static hedging values. It also proves primal attainment, while showing that dual attainment can fail and illustrates the framework numerically and theoretically.
- The argument combines a Monge–Kantorovich duality theorem with a decision-theoretic Min–Max theorem.
- Martingale constraints are enforced through dynamic trading terms, whose scaling makes any non-martingale transport plan yield an arbitrarily large dual value.
- Primal attainment follows from compactness of martingale transport plans and lower semi-continuity, whereas the dual supremum need not be attained.
- When a dual maximizer exists, its semi-static subhedge perfectly replicates the payoff almost surely under a primal optimizer.
- 4.1. A numerical example: forward-start options: For forward-start options, numerical linear programming reduces the dual constraints to m+3 constraints per s1 and shows that vanilla smiles poorly constrain these options.
- Under the stated regularity and integrability conditions, the primal value is attained and there is no duality gap.
Appendix
The appendix develops the duality proof for lower semi-continuous transport costs by restricting and approximating admissible dual functions. It extends the result from compactly supported costs to general lower semi-continuous costs through iterative replacement arguments.
- The appendix invokes the classical optimal-transport duality equation as the starting point for Proposition 2.1.
- For lower semi-continuous costs, the proof shows that the duality equation remains valid when the dual functions are restricted to the class S.
- A bounded continuous function can be approximated from below by a function in S while preserving an arbitrarily small integral error, allowing the dual class to be changed from S to Cb(R).
- For compactly supported costs, replacing each dual component by a bounded continuous function preserves feasibility and yields uniformly continuous replacements through an infimum construction.
- Applying the same replacement argument iteratively establishes the duality relation for general lower semi-continuous costs Φ: R^n → [0, ∞].