Robust Markowitz mean-variance portfolio selection under ambiguous volatility and correlation *

Amine Ismail (1, 2) , Huyên Pham (1, 3)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 Natixis Asset Management
3 CREST - Centre de Recherche en Économie et Statistique

This paper studies a robust continuous-time Markowitz portfolio selection problem where the model uncertainty carries on the variance-covariance matrix of the risky assets. This problem is formulated into a min-max mean-variance problem over a set of non-dominated probability measures that is solved by a McKean-Vlasov dynamic programming approach, which allows us to characterize the solution in terms of a Bellman-Isaacs equation in the Wasserstein space of probability measures. We provide explicit solutions for the optimal robust portfolio strategies in the case of uncertain volatilities and ambiguous correlation between two risky assets, and then derive the robust efficient frontier in closed-form. We obtain a lower bound for the Sharpe ratio of any robust efficient portfolio strategy, and compare the performance of Sharpe ratios for a robust investor and for an investor with a misspecified model. MSC Classification: 91G10, 91G80, 60H30

Mots clés: Wasserstein space; volatility uncertainty; Continuous-time Markowitz problem; ambiguous corre- lation; McKean-Vlasov; dynamic programming

Domaines: Probabilités [math.PR]; Optimisation et contrôle [math.OC]

Amine Ismail, Huyên Pham. Robust Markowitz mean-variance portfolio selection under ambiguous volatility and correlation *. 2016. ⟨hal-01385585v1⟩