Optimal Superhedging under Non-Convex Constraints - A BSDE Approach

We apply theoretical results by Peng on supersolutions for Backward SDEs (BSDEs) to the problem of finding optimal superhedging strategies in a generalized Black-Scholes market under constraints. Constraints may be imposed simultaneously on wealth process and portfolio. They may be non-convex, time-dependent, and random. The BSDE method turns out to be an extremely useful tool for modeling realistic markets: in this paper, it is shown how more realistic constraints on the portfolio may be formulated via BSDE theory in terms of the amount of money invested, the portfolio proportion, or the number of shares held. Based on recent advances on numerical methods for BSDEs (in particular, the forward scheme by Bender and Denk [1]), a Monte Carlo method for approximating the superhedging price is given, which demonstrates the practical applicability of the BSDE method. Some numerical examples concerning European and American options under non-convex borrowing constraints are presented