Abstract
We provide a model-free pricing–hedging duality in continuous time. For a frictionless market consisting of d risky assets with continuous price trajectories, we show that the purely analytic problem of finding the minimal superhedging price of a path-dependent European option has the same value as the purely probabilistic problem of finding the supremum of the expectations of the option over all martingale measures. The superhedging problem is formulated with simple trading strategies, the claim is the limit inferior of continuous functions, which allows upper and lower semi-continuous claims, and superhedging is required in the pathwise sense on a sigma -compact sample space of price trajectories. If the sample space is stable under stopping, the probabilistic problem reduces to finding the supremum over all martingale measures with compact support. As an application of the general results, we deduce dualities for Vovk’s outer measure and semi-static superhedging with finitely many securities.
Highlights
The superhedging (i.e., inequality (1.1)) is assumed to hold P -almost surely and the set of absolutely continuous local martingale measures is non-empty, which is guaranteed by the exclusion of some form of arbitrage; see [19, Corollary 9.1.2] for the precise formulation
If X is the limit inferior of a sequence of continuous functions, under the assumptions that is σ -compact and contains all its stopped paths, we show that the superhedging price coincides with the supremum of EQ[X] over all martingale measures Q
Remark 2.6 While the pathwise pricing–hedging duality results in [21, 31] hold for sufficiently regular claims when trading is limited to simple predictable processes, the following example shows the necessity of “lim inf” for claims in Cδσ
Summary
Given the space C([0, T ], Rd ) of all continuous price trajectories, the superhedging problem of a contingent claim X : C([0, T ], Rd ) → R consists of finding the infimum over all λ ∈ R such that there exists a trading strategy H which satisfies λ + (H · S)T (ω) ≥ X(ω), ω ∈ C([0, T ], Rd ),. We focus on the pathwise/model-free approach and assume that the superhedging requirement (1.1) has to hold pointwise for all price trajectories in a given set ⊆ C([0, T ], Rd ). In this pathwise setting, finding the minimal superhedging price turns out to be a purely analytic problem whose formulation is independent of any probabilistic assumptions. In the continuous-time setting, already the definition of a pathwise “stochastic integral” is a non-trivial issue We circumvent this problem by working with simple strategies and consider as “stochastic” integral the pointwise limit inferior of pathwise integrals against simple strategies, an approach that was proposed by Perkowski and Prömel [40] to define an outer measure allowing to study stochastic integration under model ambiguity. A criterion for the sample path regularity of stochastic processes and the construction of a counterexample are given in the Appendix
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