Explicit Efficient Frontier of a Continuous-Time Mean-Variance Portfolio Selection Problem

Abstract

In this work we consider a continuous-time mean-variance portfolio selection problem that is formulated as a hi-criteria optimization problem. The objective is to maximize the expected return and minimize the variance of the terminal wealth. By putting weights on the two criteria one obtains a single objective stochastic control problem which is however not in the standard form. We show that this non-standard problem can be "embedded" into a class of auxiliary stochastic linear-quadratic (LQ) problems. By solving the latter based on the recent development on stochastic LQ problems with indefinite control weighting matrices, we derive the efficient frontier in a closed form for the original mean-variance problem. 1 INTRODUCTION Portfolio selection is to seek a best allocation of wealth among a basket of securities. The mean-variance approach by Markowitz [9] provides a fundamental basis for portfolio construction in a single period. The most important contribution of this model is that it quantifies the risk by using the variance which enables investors to seek highest return after specifying their acceptable risk level. This approach becomes the foundation of modern finance theory and inspires literally hundreds of extensions and applications. In particular, in the case where the variance matrix is positive definite and short-selling is allowed, an analytic solution was obtained by Merton [10]. Perold [12] developed a more

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“…By resorting to the method of dynamic programming, the mean-variance model in a continuous time setting was developed a bit later (see [10,6,7]). By employing the embedding technique, Zhou and Li [26] turned the continuous-time mean-variance problem into a stochastic linear quadratic problem, which could be solved by using the results in Chen, Li and Zhou [5].…”

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“…For a single-period investment (see References 3,4), Markowitz considered the mean-variance model as a quadratic programming problem. Based on the result in Chen, Li and Zhou, 5 Zhou and Li 6 solved the continuous-time mean-variance problem by employing the embedding technique to deal with a stochastic LQ problem. By functional analysis approach and introducing a parameter in the quadratic optimization problem, Ji and Xue 7 first developed a novel stochastic maximum principle to tackle the classical stochastic linear quadratic problem with nonconvex control domain.…”