Optimal mean-variance portfolio selection

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• 作者：Jesper Lund Pedersen ; Goran Peskir
• 关键词：Nonlinear optimal control ; Static optimality ; Dynamic optimality ; Mean ; variance analysis ; The Hamilton–Jacobi–Bellman equation ; Martingale ; Geometric Brownian motion ; Markov process
• 刊名：Mathematics and Financial Economics
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：11
• 期：2
• 页码：137-160
• 全文大小：749KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Quantitative Finance; Finance, general; Macroeconomics/Monetary Economics//Financial Economics; Economic Theory/Quantitative Economics/Mathematical Methods; Applications of Mathematics; Statistics for
• 出版者：Springer Berlin Heidelberg
• ISSN：1862-9660
• 卷排序：11

文摘

Assuming that the wealth process \(X^u\) is generated self-financially from the given initial wealth by holding its fraction u in a risky stock (whose price follows a geometric Brownian motion with drift \(\mu \in \mathbb {R}\) and volatility \(\sigma >0\)) and its remaining fraction \(1 -u\) in a riskless bond (whose price compounds exponentially with interest rate \(r \in \mathbb {R}\)), and letting \(\mathsf{P}_{t,x}\) denote a probability measure under which \(X^u\) takes value x at time t, we study the dynamic version of the nonlinear mean-variance optimal control problem where t runs from 0 to the given terminal time \(T>0\), the supremum is taken over admissible controls u, and \(c>0\) is a given constant. By employing the method of Lagrange multipliers we show that the nonlinear problem can be reduced to a family of linear problems. Solving the latter using a classic Hamilton-Jacobi-Bellman approach we find that the optimal dynamic control is given by $$\begin{aligned} u_*(t,x) = \frac{\delta }{2\; c\; \sigma }\; \frac{1}{x}\, e^{(\delta ^2-r)(T-t)} \end{aligned}$$where \(\delta = (\mu -r)/\sigma \). The dynamic formulation of the problem and the method of solution are applied to the constrained problems of maximising/minimising the mean/variance subject to the upper/lower bound on the variance/mean from which the nonlinear problem above is obtained by optimising the Lagrangian itself.KeywordsNonlinear optimal controlStatic optimalityDynamic optimalityMean-variance analysisThe Hamilton–Jacobi–Bellman equationMartingaleGeometric Brownian motionMarkov processMathematics Subject ClassificationPrimary 60H3060J65Secondary 49L2091G80JEL Classification C61G111 IntroductionImagine an investor who has an initial wealth which he wishes to exchange between a risky stock and a riskless bond in a self-financing manner dynamically in time so as to maximise his return and minimise his risk at the given terminal time. In line with the mean-variance analysis of Markowitz [11] where the optimal portfolio selection problem of this kind was solved in a single period model (see e.g. Merton [12] and the references therein) we will identify the return with the expectation of the terminal wealth and the risk with the variance of the terminal wealth. The quadratic nonlinearity of the variance then moves the resulting optimal control problem outside the scope of the standard optimal control theory (see e.g. [5]) which may be viewed as dynamic programming in the sense of solving the Hamilton–Jacobi–Bellman (HJB) equation and obtaining an optimal control which remains optimal independently from the initial (and hence any subsequent) value of the wealth. Consequently the results and methods of the standard/linear optimal control theory are not directly applicable in this new/nonlinear setting. The purpose of the present paper is to develop a new methodology for solving nonlinear optimal control problems of this kind and demonstrate its use in the optimal mean-variance portfolio selection problem stated above. This is done in parallel to the novel methodology for solving nonlinear optimal stopping problems that was recently developed in [13] when tackling an optimal mean-variance selling problem.