June 2022 Understanding the dual formulation for the hedging of path-dependent options with price impact
Bruno Bouchard, Xiaolu Tan
Author Affiliations +
Ann. Appl. Probab. 32(3): 1705-1733 (June 2022). DOI: 10.1214/21-AAP1719

Abstract

We consider a general path-dependent version of the hedging problem with price impact of Bouchard et al. (SIAM J. Control Optim. 57 (2019) 4125–4149), 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. (Comm. Pure Appl. Math. 60 (2007) 1081–1110). 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 C0,1 in the sense of Dupire.

Funding Statement

This work has benefited from the financial support of the Initiative de Recherche “Méthodes non-linéaires pour la gestion des risques financiers” sponsored by AXA Research Fund.
The research of Xiaolu Tan is supported by CUHK startup grant and Hong Kong RGC General Research Fund 14302921.

Acknowledgments

The authors are grateful to Pierre Cardaliaguet for helpful discussion and suggestions.

Citation

Download Citation

Bruno Bouchard. Xiaolu Tan. "Understanding the dual formulation for the hedging of path-dependent options with price impact." Ann. Appl. Probab. 32 (3) 1705 - 1733, June 2022. https://doi.org/10.1214/21-AAP1719

Information

Received: 1 December 2019; Revised: 1 January 2021; Published: June 2022
First available in Project Euclid: 29 May 2022

MathSciNet: MR4429999
zbMATH: 1498.91430
Digital Object Identifier: 10.1214/21-AAP1719

Subjects:
Primary: 49N15 , 91G20
Secondary: 49N60

Keywords: Dupire derivative , Market impact , second order BSDEs

Rights: Copyright © 2022 Institute of Mathematical Statistics

Vol.32 • No. 3 • June 2022
Back to Top