Continuous-time mean-variance portfolios: A comparison

In this article, we present and compare three mean-variance optimal portfolio approaches in a continuous-time market setting. These methods are the L 2-projection as presented in Schweizer [M. Schweizer, Approximation of random variables by stochastic integrals, Ann. Prob. 22 (1995), pp. 1536-1575], the Lagrangian function approach of Korn and Trautmann [R. Korn and S. Trautmann, Continuous-time portfolio optimization under terminal wealth constraints, ZOR-Math. Methods Oper. Res. 42 (1995), pp. 69-92] and the direct deterministic approach of Lindberg [C. Lindberg, Portfolio optimization when expected stock returns are determined by exposure to risk, Bernoulli 15 (2009), pp. 464-474]. As the underlying model, we choose the recent innovative market parameterization introduced by Lindberg (2009) that has the particular aim to overcome the estimation problems of the stock price drift parameters. We derive some new results for the Lagrangian function approach, in particular explicit representations for the optimal portfolio process. Further, we compare the different optimization frameworks in detail and highlight their attractive and not so attractive features by numerical examples.