Duality for pathwise superhedging in continuous time

Bartl, Daniel; Kupper, Michael; Prömel, David J.; Tangpi, Ludovic

doi:10.1007/s00780-019-00395-2

Cited by 29 publications

(50 citation statements)

References 56 publications

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“…But this does not change the essential mathematical structure; see [19]. For related examples and discussions of the role of Ω as a prediction set, we refer to [6,5,38].…”

Section: Financial Applicationsmentioning

confidence: 99%

“…In particular, [18] provides a general representation result. [19] proves duality in discrete time and [4,6,5] for a σ-compact set Ω. Our approach is similar to that of [5] but without the assumption of σ-compactness.…”

Section: Introductionmentioning

confidence: 96%

“…[19] proves duality in discrete time and [4,6,5] for a σ-compact set Ω. Our approach is similar to that of [5] but without the assumption of σ-compactness. Instead, we use the S-topology introduced by Jakubowski [39] which provides an efficient characterization of compact sets via up-crossings.…”

Section: Introductionmentioning

confidence: 96%

“…This construction provides duality for upper semicontinuous functions. We then further enlarge the quotient set I by taking its Fatou-closure as defined in Subsection 2.3 and prove the duality for measurable functions by using Choquet's capacitability theorem as done earlier in [4,6,5,9]. These results are stated in Theorem 3.1.…”

Section: Introductionmentioning

confidence: 98%

“…The initial duality results [25,26,27,38] are proved by real-analytic techniques and only for uniformly continuous functions. Alternatively, [4,6,5,18,19] use functional analytical tools. In particular, [18] provides a general representation result.…”

Section: Introductionmentioning

confidence: 99%

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Martingale optimal transport duality

et al. 2020

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We derive a Kantorovich-type duality for convex functionals defined for functions on a subset of the Skorokhod space through a quotient set. Our dual representation takes the form of a Choquet capacity generated by martingale measures satisfying additional constraints to ensure compatibility with the quotient set. The quotient set contains pathwise stochastic integrals. We work with two alternative definitions of such integrals as limits of integrals of simple integrands. Another important ingredient of our analysis is a regularized version of Jakubowskis S-topology on the Skorokhod space.

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“…But this does not change the essential mathematical structure; see [19]. For related examples and discussions of the role of Ω as a prediction set, we refer to [6,5,38].…”

Section: Financial Applicationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Martingale optimal transport duality

et al. 2020

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Pathwise superhedging on prediction sets

Bartl

Kupper

Neufeld³

2019

Finance Stoch

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In this paper we provide a pricing-hedging duality for the model-independent superhedging price with respect to a prediction set Ξ ⊆ C[0, T ], where the superhedging property needs to hold pathwise, but only for paths lying in Ξ. For any Borel measurable claim ξ which is bounded from below, the superhedging price coincides with the supremum over all pricing functionals E Q [ξ] with respect to martingale measures Q concentrated on the prediction set Ξ. This allows to include beliefs in future paths of the price process expressed by the set Ξ, while eliminating all those which are seen as impossible. Moreover, we provide several examples to justify our setup. .sg. setting, superhedging is required to hold true for every possible future path in C[0, T ] of the price process. Such an approach has started with the seminal work [14] and has been recently lead to attention in various other works; we refer to [1,3,6,10], to name but a few.However, it turns out that the concept of superhedging is too robust leading to too high prices. In fact, for stochastic volatility or rough volatility models, it turns out that the classical superhedging price coincides with the model-independent one and is so high that for Markovian payoffs of the form Γ(S T ), like e.g. the European Call and Put option, the optimal superhedging strategy can be chosen to be of buy-and-hold type, see [9,23]. To reduce the model-independent superhedging price, inspired by the work of [22], [15] introduced the concept of prediction sets, where agents may allow to exclude paths which they consider to be impossible to model future price paths. Hence they require the superhedging property only to hold true on every path in Ξ ⊆ C[0, T ] of their prediction set.Whereas the pricing-hedging duality is well-understood for the pathwise superhedging with respect to all paths in C[0, T ], it turns out that the problem becomes considerably more difficult when requiring the superhedging property only to hold true on the prediction set Ξ ⊆ C[0, T ]. To illustrate the difficulty, consider the examples where the agent may believe in the Black-Scholes model, or is uncertain about the volatility like in the G-expectation (see [28]) and hence models his/her beliefs by requiring

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