Abstract
We study the continuous-time mean-variance portfolio selection problem in the situation when investors must pay margin for short selling. The problem is essentially a nonlinear stochastic optimal control problem because the coefficients of positive and negative parts of control variables are different. We can not apply the results of stochastic linearquadratic (LQ) problem. Also the solution of corresponding Hamilton-Jacobi-Bellman (HJB) equation is not smooth. Li et al. (2002) studied the case when short selling is prohibited; therefore they only need to consider the positive part of control variables, whereas we need to handle both the positive part and the negative part of control variables. The main difficulty is that the positive part and the negative part are not independent. The previous results are not directly applicable. By decomposing the problem into several subproblems we figure out the solutions of HJB equation in two disjoint regions and then prove it is the viscosity solution of HJB equation. Finally we formulate solution of optimal portfolio and the efficient frontier. We also present two examples showing how different margin rates affect the optimal solutions and the efficient frontier.
Highlights
Modern portfolio theory was introduced by Markowitz in 1952 [1, 2]
When μ runs all over [μ0, ∞), the set constructed by all the efficient points in variance-mean plane is called the efficient frontier
It is clear Vxx does not exist on Γ3, since P1(t) ≠ P2(t). This means that the classical solution of HJB equation (11) does not exist
Summary
Modern portfolio theory was introduced by Markowitz in 1952 [1, 2]. It is a theory of finance which attempts to minimize risk for a given level of expected return, by carefully choosing the proportions of various assets. Markowitz established concept of the efficient frontier of the optimal portfolio: it is a curve showing the relation of the best possible expected level of return with respect to its level of risk (the standard deviation of the portfolio’s return). This theory has been widely accepted both in financial industry and academy. Cuoco and Liu [7] examined the optimal consumption and investment choices and the cost of hedging contingent claims in the presence of margin requirements They established existence of optimal policies by using martingale and duality techniques [10] under general assumptions on the securities price process and the investors preferences. We formulate solution of optimal portfolio and the efficient frontier
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