Superhedging problem under ratio constraint: BSDE approaches with Malliavin calculus

Abstract

The superhedging problem with portfolio constraints has an extensive literature, e.g., [5], [6], [8], [10], [1] [4], etc. In particular, Broadie et al. (1998) show that the smallest superhedging cost of a contingent claim with certain constraints is equal to the price of a dominating claim without constraints, which is called trivial case in our paper. In this paper, we study superhedging problem for the proportion constraint, i.e. ratio constraint, via BSDEs. The BSDEs with constraint is first considered in [7], then [12] provided some deep results in this topic, following this paper, there are [13], [14], [15]. From non-trivial examples discovered in previous paper [2], we apply Malliavin calculus combining preparations about BSDE theory to give sufficient and necessary conditions for the existence of non-trivial options from probabilistic point of view.