Pathwise superreplication via Vovk’s outer measure

Beiglböck, Mathias; Cox, Alexander M. G.; Huesmann, Martin; Perkowski, Nicolas; Prömel, David J.

doi:10.1007/s00780-017-0338-2

Cited by 28 publications

(29 citation statements)

References 38 publications

(66 reference statements)

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“…Next we introduce a particular class of prediction sets. Our definition is closely related to time-invariant sets in Vovk [51], also recently used in Beiglböck et al [4], but slightly different as we work with all continuous functions and also require that maturities T i are preserved.…”

Section: Remark 310mentioning

confidence: 99%

Robust pricing–hedging dualities in continuous time

Hou

Obłój

2018

Finance Stoch

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We pursue a robust approach to pricing and hedging in mathematical finance. We consider a continuous-time setting in which some underlying assets and options, with continuous price paths, are available for dynamic trading and a further set of European options, possibly with varying maturities, is available for static trading. Motivated by the notion of prediction set in Mykland (Ann. Stat. 31:1413Stat. 31: -1438Stat. 31: , 2003, we include in our setup modelling beliefs by allowing to specify a set of paths to be considered, e.g. superreplication of a contingent claim is required only for paths falling in the given set. Our framework thus interpolates between model-independent and model-specific settings and allows us to quantify the impact of making assumptions or gaining information. We obtain a general pricing-hedging duality result: the infimum over superhedging prices of an exotic option with payoff G is equal to the supremum of expectations of G under calibrated martingale measures. Our results include in particular the martingale optimal transport duality of Dolinsky and Soner (Probab. Theory Relat. Fields 160:391-427, 2014) and extend it to multiple dimensions, multiple maturities and beliefs which are invariant under time-changes. In a general setting with arbitrary beliefs and for a uniformly continuous G, the asserted duality holds between limiting values of perturbed problems.

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Section: Remark 310mentioning

confidence: 99%

Robust pricing–hedging dualities in continuous time

Hou

Obłój

2018

Finance Stoch

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“…While in the classical setup no-arbitrage theory, including dynamic understanding of the superhedging price, is well developed in continuous time, see Föllmer and Kramkov [1997], Delbaen and Schachermayer [2006], in the robust setting an extension of abstract no-arbitrage theory, as developed in Bouchard and Nutz [2015] or Burzoni et al, to the continuous time is still open. This is despite a body of works which have achieved either particular or generic steps towards such a goal, large enough so that we can not do it justice in this introduction but refer to Avellaneda et al [1996], Lyons [1995], Denis and Martini [2006], Cox and Obłój [2011], Denis and Kervarec [2013], Epstein and Ji [2014], Biagini et al [2017], Hou and Obłój [2018], Beiglböck et al [2017], Bartl et al [2017] and the references therein. We note that d may be large and our assets may include both primary and derivate assets.…”

Section: Introductionmentioning

confidence: 99%

The Robust Superreplication Problem: A Dynamic Approach

Carassus¹,

Obłój²,

Wiesel³

2019

SIAM J. Finan. Math.

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In the frictionless discrete time financial market of Bouchard et al.(2015) we consider a trader who, due to regulatory requirements or internal risk management reasons, is required to hedge a claim ξ in a risk-conservative way relative to a family of probability measures P. We first describe the evolution of π t (ξ)the superhedging price at time t of the liability ξ at maturity T -via a dynamic programming principle and show that π t (ξ) can be seen as a concave envelope of π t+1 (ξ) evaluated at today's prices. Then we consider an optimal investment problem for the trader who is rolling over her robust superhedge and phrase this as a robust maximisation problem, where the expected utility of inter-temporal consumption is optimised subject to a robust superhedging constraint. This utility maximisation is carrried out under a new family of measures P u , which no longer have to capture regulatory or institutional risk views but rather represent trader's subjective views on market dynamics. Under suitable assumptions on the trader's utility functions, we show that optimal investment and consumption strategies exist and further specify when, and in what sense, these may be unique.

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“…Moreover, let I(G) ⊂ B(Ω) be the Fatou-closure of I(G), i.e., the smallest set of extended real-valued Borel measurable functions containing I(G) with the property that for every sequence {ℓ n } n∈N ⊂ I(G) satisfying a uniform lower bound ℓ n ≥ −λ for some λ ∈ B + q , lim inf n ℓ n ∈ I(G). In the context of financial applications, similar integrals were first constructed in [58] and later used in [7,49,59]. Their properties have recently been studied in [48].…”

Section: Integrals and Quotient Setsmentioning

confidence: 99%

“…Since there is no a priori given probabilistic structure, the definition of the integral must be pathwise and is a delicate aspect of the problem. Starting from simple integrands, we first extend the integrands using the theory developed by Vovk [58,59], later by [49] and used in [7] to prove duality. This construction provides duality for upper semicontinuous functions.…”

Section: Introductionmentioning

confidence: 99%

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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Pathwise superreplication via Vovk’s outer measure

Cited by 28 publications

References 38 publications

Robust pricing–hedging dualities in continuous time

Robust pricing–hedging dualities in continuous time

The Robust Superreplication Problem: A Dynamic Approach

Martingale optimal transport duality

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