Mean-Variance Portfolio Selection with Margin Requirements

We study the continuous-timemean-variance portfolio selection problem in the situationwhen investorsmust paymargin 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.Themain 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.


Introduction
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.In the theory variance of portfolio return was chosen as measure of the risk by Markowitz.Markowitz also 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.There are lots of extensions and applications in the discrete time models [3].Relatively there are less discussions of the mean-variance portfolio problem about continuous-time model [4][5][6].In 2000, Zhou and Li [5] solved the continuous time mean-variance problem without short selling constraint by using the stochastic linear quadratic optimal control theory.Li et al. [6] in 2002 considered mean-variance portfolio selection with short selling prohibiting condition by solving HJB equation.
In financial market, the potential loss on a short sale can be huge when the price of the security goes up, therefore in practice the short seller will be required to post margin or collateral to cover possible losses.In this paper, we consider the situation that short selling is allowed and deposit of certain percentage margin according to the shorting is needed to avoid loss by default of short seller.We also want to investigate how margin rate  affect the optimal portfolio.In the extreme cases, when  = 0, it is the case short selling is free of margin and in another case  = ∞ means that short selling is not allowed.
There is some literature [7][8][9] that discussed the margin requirements for portfolio selections.For example, 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.However, the conditions for utility functions in [7] fail for mean-variance problem.
We use the model from [7], formulate a closed form of the optimal policies.The main difficulty lies when we use the approach of [6]: the positive part and the negative part of control variables  have different coefficients; if we define new

Problem Formulation
First, we prescribe a few notations.Let (Ω, F, ) be a complete probability space and {(), 0 ≤  ≤ } a standard d-dimensional Brownian motion defined on the probability space {F   } ≥0 the natural filtration generated by (⋅).N denotes the set composed of all the P null set and its subsets; define the augmented filtration as F  := (F   ⋃ N), for all  ∈ [0, ], where F   = (  |  ∈ [0, ]), then {F  } ≥0 is complete and continuous filtration.
Note.When  = 0, there is no margin requirement for short selling and it has been discussed in [5].
In general, the terminal wealth () of the investor is a random variable.In the mean-variance problem, people consider  = [()] as his return whereas variance of () as his risk.Therefore mean-variance problem is to minimize the risk of the investment for fixed expected return  = [()]: The portfolio selection problem can be formulated as min  (,  (⋅)) = var [ ()] where the expected return is chosen as  ≥  0 := V ∫  0 () .If the investor puts all the money in the risk-free asset, his return will be  0 with zero risk.Since the investor will always choose high return with low risk, therefore the lowest expected return must be  0 .
For fixed  ≥  0 , the optimal solution  * () is called efficient strategy.Corresponding minimal variance is () = min (, (⋅)) and ((), ) is called efficient point.When  runs all over [ 0 , ∞), the set constructed by all the efficient points in variance-mean plane is called the efficient frontier.

A Class of Stochastic Control Problem
The problem mentioned in the previous section is a special class of stochastic nonlinear quadratic control problem.As the state equation is nonlinear on control variables , the results of LQ problem can not apply.In this part, we will solve this control problem.
The class of admissible controls is the set u  =  2 F (, , R  ).For any given (⋅) ∈ u  , by the result of stochastic differential equations theory there is a unique solution (⋅; , (⋅)) ∈  2 F (, , R) of ( 8), and ((⋅), (⋅)) is called feasible pair.Our goal is to find admissible (⋅) ∈ u  to minimize the following cost function We define the value function of ( 9) Obviously, it is nonnegative.

HJB Equation.
We will try to find the optimal value function (, ).According to stochastic optimal control theory [11], the value function is satisfying the following HJB equation: where (, , ) = ()() +  1 () −  0 () − + ().Our idea is to find local solution firstly, then prove that it is the viscosity solution of HJB equation and finally construct the optimal control by using verification theorem.
For nonnegative constant , the unique minimizer of To solve the HJB equation ( 11), above lemma is not directly applicable.Now we introduce some transformation.
We have following important lemma.

Note. (1)
The minimizer may be not unique.In lemma λ1 , λ2 is dependent only on the sign of .
Following the approach of LQ problem, we assume the value function is quadratic: Note the coefficients (), (), () may be different for  positive and negative in Lemma 2.
Substituting (20) into HJB equation we have following ordinary differential equations (for  = 1, 2): Here  1 (),  2 () are determined by Lemmas 1 and 2. If there are two solutions in some interval of [0, ], we can always take same  on that interval for it is piecewise constant.That makes our solution piecewise continuous.