Quadratic Hedging with Margin Requirements and Portfolio Constraints

Abstract

We consider a mean-variance portfolio optimization problem, namely, a problem of minimizing the variance of the final wealth that results from trading over a fixed finite horizon in a continuous-time complete market in the presence of convex portfolio constraints, taking into account the cost imposed by margin requirements on trades and subject to the further constraint that the expected final wealth equal a specified target value. Market parameters are chosen to be random processes adapted to the information filtration available to the investor and asset prices are modeled by Itô processes. To solve this problem we use an approach based on conjugate duality: we start by synthesizing a dual optimization problem, establish a set of optimality relations that describe an optimal solution in terms of solutions of the dual problem, thus giving necessary and sufficient conditions for the given optimization problem and its dual to each have a solution. Finally, we prove existence of a solution of the dual problem, and for a particular class of dual solutions, establish existence of an optimal portfolio and also describe it explicitly. The method elegantly and rather straightforwardly constructs a dual problem and its solution, as well as provides intuition for construction of the actual optimal portfolio.