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.

Mean‐variance Policy for Discrete‐time Cone‐constrained Markets: Time Consistency in Efficiency and the Minimum‐variance Signed Supermartingale Measure

“…, T − 1. Constrained dynamic mean-variance portfolio selection problems with various constraints have been attracting increasing attention in the last decade, e.g., Li, Zhou, and Lim (2002), Zhu, Li, and Wang (2004), Bielecki et al (2005), Sun and Wang (2006), Labbé and Heunis (2007), and Czichowsky and Schweizer (2010). Recently, Czichowsky and Schweizer (2013) further considered cone-constrained continuous-time mean-variance portfolio selection with semimartingale price processes.…”

Convex Duality in Mean-Variance Hedging Under Convex Trading Constraints

“…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. Extending [13] and [19], this need not be deterministic. SinceS i > 0, we can define the correspondence C ϑ by C ϑ (ω, t) := diag(S i (ω, t)) −1 C π (ω, t), which is, by Proposition 2.3 again, a closed-valued predictable correspondence and describes by definition the same constraints as C π , but in the number of shares.…”

“…whereā > 0 and 1/ā are in L ∞ (P),c ∈ L 2 (P), q ∈ R, and C π ≡ K ⊆ R d is a fixed closed and convex set; see Equation (5.2) of [19]. To obtain a solution to this problem, we observe thatāS 0 T (x + G T ( X (C π ))) is convex and closed in L 2 (P), sinceā and 1/ā are in L ∞ (P), and that we can write…”

“…In the Itô process framework of Example 3.1 and if F is the P-augmentation of the filtration generated by W , the above dual problem (5.20) for stochastic processes specialises to the dual problem considered in [19,Equation 5.37]; this only requires some adjustments to the notation, along the lines of Remark 4.1(b). Now set G c := G T ( S (C)) so that Lemma 5.4 gives an explicit representation of the support function δ(·|G c ).…”

Optimal Multi-Period Mean-Variance Policy with Management Costs

“…Czichowsky and Schweizer [5] studied dynamic mean-variance hedging problem with general convex constraints. Labbe and Heunis [11], Sun and Wang [18] considered dynamic mean-variance models with general convex constraints. Gülpinar et al [9] solves multiperiod mean-variance model with presence of transaction costs.…”