Convex duality in constrained mean-variance portfolio optimization

Abstract

We apply conjugate duality to establish the existence of optimal portfolios in an asset-allocation problem, with the goal of minimizing the variance of the final wealth which results from trading over a fixed, finite horizon in a continuous-time, complete market, subject to the constraints that the expected final wealth equal a specified target value and the portfolio of the investor (defined by the dollar amount invested in each stock) take values in a given closed, convex set. The asset prices are modelled by Itô processes, for which the market parameters are random processes adapted to the information filtration available to the investor. We synthesize a dual optimization problem and establish a set of optimality relations, similar to the Euler-Lagrange and transversality relations of calculus of variations, giving necessary and sufficient conditions for the given optimization problem and its dual to each have a solution, with zero duality gap. We then solve these relations, to establish the existence of an optimal portfolio.

Convex Duality in Mean-Variance Hedging Under Convex Trading Constraints

“…However, the mathematics are somewhat different since our 'quadratic random utility' U(x, ω) = − 1 2 |x − H (ω)| 2 is not increasing in x and the duality is taken in a different space. A fairly close precursor of our work is due to Labbé and Heunis [19], who studied the same problem when S is given by a complete Itô process model and the constraints do not depend on ω and t. Their duality is very similar to parts of our Theorem 5.2, but their formulations and in particular their proofs strongly depend on the availability and use of Itô's representation theorem. We do not need that at all, since S and the underlying filtration F are general in our setting.…”

“…For a better understanding of our assumptions, we now spell them out in a multidimensional Itô process model. This is one standard example of a financial market, and it illustrates that our assumptions are weaker than those in [13], [14], and [19]. Example 3.1.…”

“…of undiscounted payoffs attainable with constrained trading strategies considered in [13], [14], and [19] are closed in L 2 (P).…”

“…Our assumptions are clearly far less restrictive than completeness of the (unconstrained) financial market. The latter is imposed in [13] and [19] by the conditions that µ, r, and σ are uniformly bounded in t and ω, that σ −1 exists and is uniformly bounded in t and ω as well, and that F is the P-augmentation of the filtration generated by W . The last two assumptions allow us to use Itô's representation theorem and then rewrite integrals of W as integrals of S.…”

“…In [13] and [19], the constraints are not formulated in terms of the number of shares ϑ i , but in terms of the cash amounts π i := ϑ iSi invested in the different assets. To see that this can also be handled in our formulation, let C π be a closed-valued predictable correspondence which describes constraints on the cash amounts.…”