  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 It{\^o}'s Lemma for
path-dependent functionals that are only C^{0,1} in the sense of Dupire.
