Journal of Mathematics

Volume 2013 (2013), Article ID 726297, 9 pages

http://dx.doi.org/10.1155/2013/726297

Research Article

Mean-Variance Portfolio Selection with Margin Requirements

Yuan Zhou1,2 and Zhe Wu¹

¹School of Mathematics, Fudan University, Shanghai 200433, China

²Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA

Received 18 November 2012; Revised 23 February 2013; Accepted 24 February 2013

Academic Editor: Cedric Yiu

Copyright © 2013 Yuan Zhou and Zhe Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. 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 [4–6]. 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 , 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 [7–9] 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 control variables then the new volatility matrix is singular and moreover are not independent therefore the key lemma for optimization is not applied in our case. We carefully decompose the problem into several subproblems and figure out the solutions of HJB equation in two regions and then prove it is the viscosity solution of HJB equation. Finally we formulate solution of optimal portfolio and the efficient frontier.

2. Problem Formulation

First, we prescribe a few notations. Let be a complete probability space and a standard d-dimensional Brownian motion defined on the probability space the natural filtration generated by . denotes the set composed of all the P null set and its subsets; define the augmented filtration as , for all, where , then is complete and continuous filtration.

We define , for the positive part and negative part of a real number . represents n-dimensional real Euclidean space. is nonnegative subset of n-dimensional real Euclidean space. is measurable for all , and .

We consider the Merton's market model. There are securities, trading continuously in finite time horizon . Assuming the interest rate is , the time price of the risk-free asset satisfies

with the initial price . The remaining risky assets prices at time are, respectively, , which satisfy the following stochastic differential equations:

where is the appreciation rate of the risky assets and is the volatility matrix.

Throughout the paper, we assume the following.

(A1) Function are deterministic and piecewise continuous on . (A2) Non-degenerate Condition: there exists a constant , s.t.

We also define the relative risk coefficient

where .

Let be initial wealth of the investor. Suppose is time total wealth of the investor on th asset, is time wealth vector of risky assets. Then time total wealth of the investor is . Suppose the investor is self-financing, then

where is the margin rate.

Note. When , 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

where the expected return is chosen as . If the investor puts all the money in the risk-free asset, his return will be with zero risk. Since the investor will always choose high return with low risk, therefore the lowest expected return must be .

For fixed , the optimal solution is called efficient strategy. Corresponding minimal variance is and is called efficient point. When runs all over , the set constructed by all the efficient points in variance-mean plane is called the efficient frontier.

3. 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.

Considering one dimensional system

where , , , , , we assume matrix is non-degenerate. These coefficients are all deterministic and piecewise continuous on , therefore they are all bounded on .

The class of admissible controls is the set . For any given , by the result of stochastic differential equations theory there is a unique solution of (8), and is called feasible pair. Our goal is to find admissible to minimize the following cost function

We define the value function of (9)

Obviously, it is nonnegative.

3.1. 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 . 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.

In order to find solution of the HJB equation, it is clear that for fixed if (that will be proved later) we must solve

where , . We need the following lemma [10].

Lemma 1. If and then there is a unique , s.t.

For nonnegative constant , the unique minimizer of

over is , , here . Furthermore, and

To solve the HJB equation (11), above lemma is not directly applicable. Now we introduce some transformation.

Note for any , there is (may be not unique) a matrix with or for , such that , . Then

Define space with or for .

We have following important lemma.

Lemma 2. If is negative, there is at least one and a minimizer of

over and for all .

If is positive, there is at least one and a minimizer of over and for all .

Proof. In order to solve (12), we fixed a diagonal matrix with diagonals either or for . Let

If , take and . Since is nonsingular, by Lemma 1, there exist vectors , , such that is minimizer of and .

Now because

Note is a finite set, first part of lemma holds.

If , take and . because is nonsingular, by Lemma 1 again, there exist vectors , such that is minimizer of and .

Because (19) is true for this case, we have the second part of lemma.

Note. The minimizer may be not unique. In lemma is dependent only on the sign of .

For , it is obvious that there is unique minimizer and .

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 ):

Here , are determined by Lemmas 1 and 2. If there are two solutions in some interval of , we can always take same on that interval for it is piecewise constant. That makes our solution piecewise continuous.

Since , the in Lemma 2 equals having same sign as . By the lemmas, we know and are independent of and except the sign of .

Defining , , from (21) and (22) we can get

By the terminal constraint , noting the equations are same for , we obtain

Therefore, for ,

Substituting these expressions into (23), we obtain for

Because

We have for

3.2. Value Function and Optimal Control

Firstly, we will construct a viscosity solution for HJB equation. Let

Define the spaces

We denote .

Now let us check that satisfies HJB equation on , for . Considering (20) for , we have

Substitute into (11) for

Since in , substituting the minimizer by Lemma 2 we can see satisfies HJB equation in .

By the same way, we can show also satisfies HJB equation in . For , we have

Substitute into (11) for

Since , substituting the minimizer we can see satisfies HJB equation in .

On , we have , then we have

by (29), for . So is continuous at .

Therefore,

that is to say, is differentiable on . However, may be not smooth on . It is clear does not exist on , since . This means that the classical solution of HJB equation (11) does not exist. Therefore, we have to work within the framework of viscosity solution (see the appendix of [6]).

By the definition of viscosity solutions, we have for all ,

Now let

For all , , we have

Therefore, is the subsolution of HJB equation (11). On the other hand, for all , , we have

Therefore, by the definition we know is the supersolution of HJB equation (11). Finally, it is easy to see that the terminal condition is satisfied. Therefore, by definition, is the viscosity solution of HJB equation (11). Moreover, for all , let then

So it satisfies the conditions in verification theorem [12, Theorem 3.1].

The main theorem in this section follows.

Theorem 3. Let for be the solution of (21)–(23). The regions are defined by (31). Then the function defined by

is the viscosity solution of HJB equation (11). Control function

is the optimal feedback control of system (8) for objective functional (9) with , the optimal value is . Here are constructed by Lemmas 1 and 2.

4. Optimal Strategy and Efficient Frontier

4.1. Optimal Strategy

Back to the portfolio problem. From [6] problem (7) is equivalent to for some :

In order to make use of the result in Section 3, let . We study optimal control problem

If we denote , , , , , (46) becomes

Before analyzing the efficient frontier of the original problem (7), we firstly give the optimal strategy of . The translated problem is a special case studied in the previous section, the optimal strategy is given in (44). In this problem,

The optimal strategy is

Substituting with , we get the following.

Theorem 4. For problem , the optimal portfolios are

Here are constructed by Lemmas 1 and 2.

4.2. Efficient Frontier

In this part, we will compute the efficient frontier of problem . Because

For fixed , we have

where , , are either , , or , , , respectively, on and . Now, if , we have a concave quadratic function in

If , we obtain

The minimum variance (optimal value functional) is

Since is Lagrange multiplier. We must maximize it. Note that . If , then . Maximize

we get

with maximum value

If , then . Maximize

We get with maximum value . Therefore the maximum of (55) equals (58).

Now we have the theorem to conclude the section.

Theorem 5. The efficient strategy of mean-variance problem can be formulated as

where is defined in (53), , , come from Lemmas 1 and 2. The minimum variance is

Here is determined by Lemmas 1 and 2.

Note. When coefficients are all constant then , and are all constant vectors.

5. Two Examples

Example 6. Suppose , , , , . Then the wealth equation is

The problem can be formulated as

The correspondence HJB equation is

Here . Letting , we need to minimize

We assume , then when ,

When ,

By (53) we have the optimal (note and ) and the optimal value ; they are all independent of margin rate .

Because , we have . By (64) and (65) we know . By Theorem 5, the efficient strategy of mean-variance problem can be formulated as

It is interesting to see that although the optimal strategies are different when changes, (which is the boundary of two regions and ) and the optimal values remain the same and therefore two regions and the efficient frontiers keep unchanged. But it is not always the same case, next example we will show that the optimal values and the efficient frontiers vary with margin rate .

Example 7. Let , , , , . In order to solve the HJB equation, we must minimize

When , we have .

When , short-selling is not allowed. We have

.

When , we consider four cases: , ; ; , ; , , then take the minimum. We have

.

From the example we know depends on . By Theorem 5, we know the optimal parameter , the boundary of two regions and , the optimal value are determined by therefore they all vary with . For the same return , the risk is lowest when (i.e., no margin required) and the risk is highest when short-selling is prohibited . The results of our paper cover these two extreme cases.

6. Conclusion

In this paper, we study the mean-variance portfolio selection problems for short selling with margin required. The problem is more complicated than free short-selling () and short-selling prohibiting (). We found the optimal strategies in the feedback form and the optimal value function which is the viscosity solution of the HJB equation. We have also given examples to show how the margin rate affects the optimal portfolio, optimal variance, and efficient frontier.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (10971127).

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