UNDERSTANDING THE DUAL FORMULATION FOR THE HEDGING OF PATH-DEPENDENT OPTIONS WITH PRICE IMPACT

• 成果类型: Article
• 署名作者: Bouchard, Bruno; Tan, Xiaolu
• 署名单位: Universite PSL; Universite Paris-Dauphine; Centre National de la Recherche Scientifique (CNRS); Chinese University of Hong Kong
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1719
• 发表日期: 2022
• 页码: 1705-1733
• 关键词: integral-representation

摘要

We consider a general path-dependent version of the hedging problem with price impact of Bouchard et al. (SIAM J. Control Optim. 57 (2019) 4125-49), 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 Ito's lemma for path-dependent functionals that are only C-0,C-1 in the sense of Dupire.