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

Bruno Bouchard (1) , Xiaolu Tan (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 CUHK - The Chinese University of Hong Kong [Hong Kong]

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.

Domaines: Probabilités [math.PR]; Finance quantitative [q-fin.CP]

Bruno Bouchard, Xiaolu Tan. Understanding the dual formulation for the hedging of path-dependent options with price impact. The Annals of Applied Probability, 2022, 32 (3), pp.1705-1733. ⟨10.1214/21-AAP1719⟩. ⟨hal-02398881v2⟩