Optimal Mean-Variance Portfolio Selection

J. L. Pedersen, Goran Peskir

    Research output: Contribution to journalArticlepeer-review

    Abstract

    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 $\mu \in \R$ and volatility $\sigma>0$) and its remaining fraction $1 \m u$ in a riskless bond (whose price compounds exponentially with interest rate $r \in \R$), and letting $\PP_{t,x}$ 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 \begin{equation} \sup_u \Big[ \EE_{t,X_t^u}(X_T^u) - c \Var_{t,X_t^u}(X_T^u) \Big] \end{equation} 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{equation} u_*(t,x) = \frac{\delta}{2 c \sigma} \frac{1}{x} e^{(\delta^2-r)(T-t)} \end{equation} where $\delta = (\mu \m r)/\sigma$. 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.
    Original languageEnglish
    Pages (from-to)127-160
    Number of pages34
    JournalMathematics and Financial Economics
    Volume11
    Issue number2
    Early online date20 Jun 2016
    DOIs
    Publication statusPublished - Mar 2017

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