#### Abstract

This paper studies a continuous-time dynamic mean-variance portfolio selection problem with the constraint of a higher borrowing rate, in which stock price is governed by a constant elasticity of variance (CEV) process. Firstly, we apply Lagrange duality theorem to change an original mean-variance problem into an equivalent optimization one. Secondly, we use dynamic programming principle to get the Hamilton-Jacobi-Bellman (HJB) equation for the value function, which is a more sophisticated nonlinear second-order partial differential equation. Furthermore, we use Legendre transform and dual theory to transform the HJB equation into its dual one. Finally, the closed-form solutions to the optimal investment strategy and efficient frontier are derived by applying variable change technique.

#### 1. Introduction

The main purpose of this paper is to focus on the portfolio selection problem under the constant elasticity of variance (CEV) model. The CEV model was originally proposed by Cox and Ross [1] as an alternative diffusion process for European option pricing. It is a natural extension of the geometric Brownian motion (GBM). The advantage of the CEV model is that it can explain the empirical bias exhibited by Black and Scholes [2] model, such as volatility smile. Therefore, the CEV model was often applied to analyze the option pricing formulas, for example, Schroder [3], Phelim and Yisong [4], Davydov and Linetsky [5], and so forth. Recently, the CEV model has been introduced into annuity contracts to study the optimal investment strategy in the defined contribution and defined benefit pension plan (referring to Xiao et al. [6], Gao [7, 8]), but those models were all considered in the utility function framework. In the existing literatures, as far as our knowledge, the CEV model in the mean-variance framework has not been reported.

In most of the real-world situations, different interest rates for borrowing and lending are often faced by the investors. It is clear that portfolio selection models with borrowing constraint will make it more practical. This attracts the attention of many authors, referring to Paxson [9], Fleming and Zariphopoulou [10], Vila and Zariphopoulou [11], TeplĂˇ [12], and Zariphopoulou [13]. However, those models were usually dealt with under expect utility criterion, and the risky asset price was usually supposed to be driven by a GBM. In addition, the risk and return relationship is implicit in the utility function approach and cannot be disentangled at the different level of the optimal strategy. As a matter of fact, the optimal investment strategy under the utility maximizing criterion is not necessarily mean-variance efficient.

This paper introduces a CEV model and borrowing constraint into the classical portfolio selection problem in a continuous-time mean-variance framework. For the mean-variance portfolio selection problem, stochastic linear-quadratic (LQ) control technique is an effective method (e.g., Zhou and Li [14], Li et al. [15], and Chiu and Li [16], Xie et al. [17]). But borrowing constraint forces this problem to become piecewise linear-quadratic and is hence no longer a LQ control problem (see [18]). In addition, the introduction of the CEV model gives rise to some new difficulties, which are not easily dealt with in solving the associated HJB equation. In this paper, we firstly apply Lagrange duality theorem to change an original mean-variance problem into an equivalent optimization one. Secondly, we use dynamic programming principle to get the Hamilton-Jacobi-Bellman (HJB) equation for the value function, which is a more sophisticated nonlinear second-order partial differential equation. Further, we use Legendre transform and dual theory to transform the HJB equation into its dual one. Finally, the closed-form solutions to the optimal investment strategy and efficient frontier are derived by applying variable change technique. There are several innovations in this paper: (i) stock price is supposed to follow the CEV model, which is a natural extension of geometric Brownian motion; (ii) we consider a dynamic mean-variance portfolio selection problem with borrowing constraints under a CEV model and assume that the borrowing rate is larger than the risk-free interest rate; (iii) the closed-form solutions to the optimal investment strategy and the efficient frontier are obtained.

The paper is organized as follows. In Section 2, we introduce a CEV model and describe portfolio selection problem with borrowing constraint in a mean-variance framework. In Section 3, the closed-form solution to optimal investment strategy is derived by applying Legendre transform and dual theory. Section 4 gives the main results on the optimal strategy and the efficient frontier. Section 5 concludes this paper.

#### 2. Mean-Variance Model

In this paper, we consider a financial market where two assets are traded continuously over . One asset is a bond with price at time , whose price process with borrowing constraint can be expressed in the following ordinary differential equation (see Fu et al. [18]): where the constant is the interest rate of the bond and is the borrowing rate being larger than .

Letting , then (1) can be rewritten as The another asset is a stock with prices at time , whose price process is supposed to follow the constant elasticity of variance (CEV) model: where is the instantaneous return rate of the stock. and are constant parameters, the elasticity parameter satisfies the general condition: . is defined as the instantaneous volatility of the stock, and is a one-dimensional standard and adapted Brownian motion defined on the filtered complete probability space .

*Remark 1. *Note that there are four special interpretations for the elasticity parameter :(i)if , the CEV model is reduced to a geometric Brownian motion (GBM);(ii)if , it is the Ornstein-Uhlenbeck process;(iii)if , it is the model first presented by Cox and Ross [1] as an alternative diffusion process for valuation of options;(iv)if , this means that the instantaneous volatility increases as the stock price decreases and can generate a distribution with a fatter left tail (referring to Gao [7]).

Suppose that short-selling of the stock is allowed and transaction cost and consumption are not considered. We denote by the wealth of the investor at time and by the share of asset th held by the investor at time , . Let be the amount invested in the stock at time , . Clearly, the amount invested in the bond is . The wealth process corresponding to trading strategy is subject to the following stochastic differential equation:
That is, we have

The investorâ€™s objective is to find an optimal portfolio such that the expected terminal wealth satisfies , for some constant , while the risk measured by the variance of the terminal wealth
is minimized. The problem of finding out such a portfolio is referred to as the mean-variance portfolio choice problem.

In the modern portfolio selection theory, a portfolio is said to be admissible if it is integrable and -adapted, and (5) has a unique solution corresponding to . In this case, we refer to as an admissible pair. Therefore, the mean-variance problem can be formulated as a linearly constrained stochastic optimization problem:
Finally, an optimal investment strategy of the above problem is called an efficient portfolio corresponding to , the corresponding is called an efficient point, whereas the set of all the efficient points, when the parameter runs over , is called an efficient frontier.

*Remark 2. *If , this means that the investor is short-selling the stock. If , then the investor needs borrowing the money from the bank at interest rate and the amount to borrow is . Otherwise, we do not borrow the money to run the portfolio.

#### 3. The Optimal Portfolio

To find out an optimal investment strategy for the problem (7) corresponding to the constraint , we introduce a Lagrange multiplier and arrive at a new objective function: Letting , we obtain the following stochastic control problem: The link between problem (7) and (9) is provided by Lagrange duality theorem (see Fu et al. [18] and Luenberger [19]):

For a fixed constant , the problem (9) is clearly equivalent to We define the value function as with boundary condition .

According to dynamic programming principle, can be taken as the continuous solution to the following HJB equation: where , , , , , and denote first-order and second-order partial derivatives with respect to time , stock price , and wealth process .

Let us firstly describe borrowing situation. Not borrowing and investing in the bond means that and borrowing to invest in the stock means that . We define the nonborrowing region in the -plane to be Taking borrowing situation into consideration, we rewrite the HJB equation (13) as The optimal value of (15) is given by

Putting (16) into (15), we have Letting , if ; , if , we get

According to the convexity of the value function, we can define a Legendre transform: where denotes the dual variable to .

The value of which the maximum in the above equation will be attained at is denoted by , so we have The functions and are closely related and we will refer to either one of them as the dual function of . In this paper, we will work mainly with the function .

The function is related to by .

Noting that , we can define the following Legendre transform at terminal time: So we have According to (19), we have , and this leads to

Referring to Jonsson and Sircar [20], Xiao et al. [6], and Gao [7], we get the following transformation rules: Putting transformation rules (24) into (18), we get

Differentiating with respect to , we derive the following dual equation: where , if ; , if .

Taking (22) into consideration, we can fit a solution to (26) with the following structure: Considering , if ; , if , we can rewrite as with boundary conditions given by and .

Further, we have Putting the above partial derivatives into (26), we get Eliminating the dependence on , we obtain The solution to (31) is

Lemma 3. *Assume that the structure of the solution to (32) is , with the boundary conditions given by and ; then and are given by (43) and (42), respectively.*

* Proof . *Putting into (32), we have
By matching the coefficients, we get
Equation (36) can be reduced to
Let denote the discriminant of the quadratic equation .

Easy calculation leads to . Assume that , that is, , then the quadratic equation has two real roots:
So (37) can be rewritten as

Further, we obtain
Letting
We get
Plugging (42) into (35) yields

Therefore, Lemma 3 is completed.

Finally, summarizing the above results, we obtain the optimal trading strategy for the problem (11).

Theorem 4. *For a given , and , the optimal investment strategy with borrowing constraint under a mean-variance criterion corresponding to the problem (11) is
**
where and are given by (48) and (50), respectively.*

*Proof . *Under the transformation rules, the optimal strategy (16) is derived as follows:
Taking (33),(35), and Lemma 3 into consideration, we have
Therefore, the optimal strategy is reduced to
where

As the boundary condition of borrowing the money from the bank is ; that is,
Denoting by the borrowing curve, we yield
Therefore, nonborrowing region in the -plane can be rewritten as

Hence, the proof of the Theorem 4 is completed.

*Remark 5. *We can draw some conclusions from (44).(i)If and , the investor need not borrowing the money from the bank and the optimal amount invested in the stock can be calculated by the first equation of (44), while the amount invested in the bond is .(ii)If and , the optimal amount invested in the stock is given by the second equation of (44). In addition, investors need to borrow the money to invest the stock and the amount to borrow is , while the bond need not be invested.

#### 4. The Efficient Frontier

In this section, we apply Lagrange duality theorem to derive the efficient frontier for the mean-variance portfolio selection problem (7). To simplify the presentation, we denote by either the interest rate or the borrowing rate , and letting where , if and , if .

In both cases above, the wealth equation (5) is reduced to For any fixed , under the efficient strategy in the Theorem 4, the dynamics of the wealth equation (5) are

Applying *ItĂ´'s* lemma to the wealth process (54), we yield

Taking expectations on both sides of (54) and (55), respectively, one has

The solution of the linear ordinary differential equation (56) is and it results in

Similarly, by solving (57), one has

Therefore, the objective function of the problem (9), as a explicit function of parameter , is given by Using (10) obtained by Lagrange duality theorem, the minimum variance is achieved for In addition, we obtain

Letting . The optimal value of and the minimum variance are

Putting (65) into the optimal strategy (44) in Theorem 4, we can summarize our main results in the following theorem.

Theorem 6. *The optimal investment strategy for the mean-variance portfolio selection problem (7) with borrowing constraint under a CEV process is
**
Moreover, the efficient frontier is given by
**
where and are given by (48) and (50), respectively.*

*Remark 7. *When , the results in the Theorem 6 are reduced to the ones under a geometric Brownian motion model, which is obtained by [18]. When and , the corresponding results are all given by (66) and (67). Therefore, extending a geometric Brownian motion to a CEV model is the most important innovation in our paper.

#### 5. Conclusions

This paper is concerned with a continuous-time dynamic portfolio selection problem in a mean-variance framework, in which the constraint of the borrowing rate higher than the lending rate is allowed and stock price process is supposed to follow the constant elasticity of variance (CEV) model. The closed-form solution to the optimal investment strategy is derived by applying Legendre transform and dual theory. In addition, the efficient strategy and efficient frontier are derived by using Lagrange duality theorem.

In future research, we will continue to concentrate on continuous-time portfolio selection problems under a CEV model. It would be interesting to extend our model to those with more sophisticated cases, such as introducing consumption and transaction cost, short-selling constraint, and liability process. We leave these problems and corresponding verification theorem for future research.

#### Acknowledgments

This research is supported by the Humanities and Social Science Research Youth Foundation of Ministry of Education (no. 11YJC790006) and the Higher School Science and Technology Development Foundation of Tianjin (no. 20100821).