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Understanding the dual formulation for the hedging of path-dependent options with price impact

Bouchard, Bruno; Tan, Xiaolu (2022), Understanding the dual formulation for the hedging of path-dependent options with price impact, Annals of Applied Probability, 32, 3, p. 1705-1733. 10.1214/21-AAP1719

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Type
Article accepté pour publication ou publié
Date
2022
Journal name
Annals of Applied Probability
Volume
32
Number
3
Publisher
Institute of Mathematical Statistics
Published in
Paris
Pages
1705-1733
Publication identifier
10.1214/21-AAP1719
Metadata
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Author(s)
Bouchard, Bruno
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Tan, Xiaolu
The Chinese University of Hong Kong [Hong Kong]
Abstract (EN)
We consider a general path-dependent version of the hedging problem with price impact of Bouchard et al. (2019), in which a dual formulation for the super-hedging price is obtained by means of PDE arguments, in a Markovian setting and under strong regularity conditions. Using only probabilistic arguments, we prove, in a path-dependent setting and under weak regularity conditions, that any solution to this dual problem actually allows one to construct explicitly a perfect hedging portfolio. From a pure probabilistic point of view, our approach also allows one to exhibit solutions to a specific class of second order forward backward stochastic differential equations, in the sense of Cheridito et al. (2007). Existence of a solution to the dual optimal control problem is also addressed in particular settings. As a by-product of our arguments, we prove a version of Itô's Lemma for path-dependent functionals that are only C^{0,1} in the sense of Dupire.
Subjects / Keywords
Dupire derivative . Market impact . second order BSDEs

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