Robust pricing–hedging dualities in continuous time

Hou, Zhaoxu; Obłój, Jan

doi:10.1007/s00780-018-0363-9

Cited by 71 publications

(75 citation statements)

References 54 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%

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%

Martingale optimal transport duality

et al. 2020

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“…To this end, we need an additional assumption, which rules out an arbitrage opportunity, that is, a case where

V_{h_{1}, \dots, h_{N}}^{double-struckP} false(H false) = V_{h_{1}, \dots, h_{N}} false(H false) = - \infty

. Thus, as in Hou and Obłój (; see assumption 3.7 and remark 3.8 there), we assume the following. Assumption There is

ε > 0

such that for any

0true (y_{1}, \dots, y_{N}) \in \prod_{i = 1}^{N} [{scriptP}_{i} - ε, {scriptP}_{i} + ε]

, we can find a probability measure

Q \in M

for which

{double-struckE}_{Q} false[h_{i} (S) false] = y_{i}

i = 1, \dots, N

.…”

Section: Semistatic Hedgingmentioning

confidence: 99%

Super‐replication in fully incomplete markets

Dolinsky

Neufeld

2017

Mathematical Finance

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In this work, we introduce the notion of fully incomplete markets. We prove that for these markets, the super‐replication price coincides with the model‐free super‐replication price. Namely, the knowledge of the model does not reduce the super‐replication price. We provide two families of fully incomplete models: stochastic volatility models and rough volatility models. Moreover, we give several computational examples. Our approach is purely probabilistic.

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“…Naturally, such additional assets constrain the set of martingale measures, which may be used for pricing. General pricing-hedging duality results, in different variations of this setting, both in continuous and in discrete time, can be found in, for example, Acciaio, Beiglböck, Penkner, and Schachermayer (2016), Burzoni, Frittelli, Hou, Maggis, and Obłój (2018), Beiglböck et al (2013), Dolinsky and Soner (2014), Hou and Obłój (2018), Guo, Tan, and Touzi (2017), Tan and Touzi (2013), and we refer to the survey papers Hobson (2011) and Obłój (2004) for more details.…”

Section: Introductionmentioning

confidence: 99%

The robust pricing–hedging duality for American options in discrete time financial markets

Aksamit

Deng

Obłój

et al. 2018

Mathematical Finance

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We investigate the pricing–hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, for example, a family of European options, only statically. In the first part of the paper, we consider an abstract setting, which includes the classical case with a fixed reference probability measure as well as the robust framework with a nondominated family of probability measures. Our first insight is that, by considering an enlargement of the space, we can see American options as European options and recover the pricing–hedging duality, which may fail in the original formulation. This can be seen as a weak formulation of the original problem. Our second insight is that a duality gap arises from the lack of dynamic consistency, and hence that a different enlargement, which reintroduces dynamic consistency is sufficient to recover the pricing–hedging duality: It is enough to consider fictitious extensions of the market in which all the assets are traded dynamically. In the second part of the paper, we study two important examples of the robust framework: the setup of Bouchard and Nutz and the martingale optimal transport setup of Beiglböck, Henry‐Labordère, and Penkner, and show that our general results apply in both cases and enable us to obtain the pricing–hedging duality for American options.

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Robust pricing–hedging dualities in continuous time

Cited by 71 publications

References 54 publications

Martingale optimal transport duality

Martingale optimal transport duality

Super‐replication in fully incomplete markets

The robust pricing–hedging duality for American options in discrete time financial markets

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