Dynamic Mean-Variance Model with Borrowing Constraint under the Constant Elasticity of Variance Process

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], Teplá [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 [0,𝑇]. One asset is a bond with price 𝑃_𝑡 at time 𝑡, whose price process 𝑃_𝑡 with borrowing constraint can be expressed in the following ordinary differential equation (see Fu et al. [18]): 𝑑𝑃_𝑡=𝑟𝑃_𝑡𝑑𝑡,if𝑃_𝑡≥0,𝑡∈[0,𝑇]𝑅𝑃_𝑡𝑑𝑡,if𝑃_𝑡<0,𝑡∈[0,𝑇],𝑃₀=𝑝₀>0, (1) where the constant 𝑟>0 is the interest rate of the bond and 𝑅 is the borrowing rate being larger than 𝑟.

Letting 𝑥⁻=−min(𝑥,0), then (1) can be rewritten as 𝑑𝑃_𝑡=𝑟𝑃_𝑡−(𝑅−𝑟)𝑃⁻_𝑡𝑑𝑡,𝑃₀=𝑝₀>0. (2) The another asset is a stock with prices 𝑆_𝑡 at time 𝑡, whose price process 𝑆_𝑡 is supposed to follow the constant elasticity of variance (CEV) model: 𝑑𝑆_𝑡=𝑆_𝑡𝜇𝑑𝑡+𝑘𝑆^𝛽_𝑡𝑑𝑊_𝑡,𝑆₀=𝑠₀>0, (3) where 𝜇(𝜇>𝑅>𝑟) is the instantaneous return rate of the stock. 𝑘 and 𝛽 are constant parameters, the elasticity parameter 𝛽 satisfies the general condition: 𝛽≤0. 𝑘𝑆^𝛽_𝑡 is defined as the instantaneous volatility of the stock, and 𝑊_𝑡 is a one-dimensional standard and adapted Brownian motion defined on the filtered complete probability space (Ω,ℱ,𝑃,{ℱ_𝑡}_𝑡≥0).

Remark 1. Note that there are four special interpretations for the elasticity parameter 𝛽:(i)if 𝛽=0, the CEV model is reduced to a geometric Brownian motion (GBM);(ii)if 𝛽=−1, it is the Ornstein-Uhlenbeck process;(iii)if 𝛽=−1/2, it is the model first presented by Cox and Ross [1] as an alternative diffusion process for valuation of options;(iv)if 𝛽<0, this means that the instantaneous volatility 𝑘𝑆^𝛽_𝑡 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 𝑋_𝑡 the wealth of the investor at time 𝑡∈[0,𝑇] and by 𝑁_𝑖(𝑡) the share of asset 𝑖th held by the investor at time 𝑡, 𝑖=1,2. Let 𝜋_𝑡=𝑁₂(𝑡)𝑆_𝑡 be the amount invested in the stock at time 𝑡, 𝑡∈[0,𝑇]. Clearly, the amount invested in the bond is 𝜋⁰_𝑡=𝑁₁(𝑡)𝑃_𝑡=𝑋_𝑡−𝜋_𝑡. The wealth process 𝑋_𝑡=𝑁₁(𝑡)𝑃_𝑡+𝑁₂(𝑡)𝑆_𝑡 corresponding to trading strategy 𝜋_𝑡 is subject to the following stochastic differential equation: 𝑑𝑋_𝑡=𝑁₁(𝑡)𝑑𝑃_𝑡+𝑁₂(𝑡)𝑑𝑆_𝑡=𝑁₁(𝑡)𝑟𝑃_𝑡−(𝑅−𝑟)𝑃⁻_𝑡𝑑𝑡+𝑁₂(𝑡)𝑆_𝑡𝜇𝑑𝑡+𝑘𝑆^𝛽_𝑡𝑑𝑊_𝑡=𝑟𝑁₁(𝑡)𝑃_𝑡−(𝑅−𝑟)𝑁₁(𝑡)𝑃⁻_𝑡𝑑𝑡+𝜋_𝑡𝜇𝑑𝑡+𝑘𝑆^𝛽_𝑡𝑑𝑊_𝑡. (4) That is, we have 𝑑𝑋_𝑡=𝑟𝑋_𝑡+𝜇−𝑟𝜋_𝑡−(𝑅−𝑟)𝑋_𝑡−𝜋_𝑡⁻𝑑𝑡+𝜋_𝑡𝑘𝑆^𝛽_𝑡𝑑𝑊_𝑡,𝑋₀=𝑥₀>0. (5)
The investor’s objective is to find an optimal portfolio 𝜋_𝑡 such that the expected terminal wealth satisfies 𝔼𝑋_𝑇=𝐶, for some constant 𝐶∈ℝ, while the risk measured by the variance of the terminal wealth Var𝑋_𝑇=𝔼𝑋_𝑇−𝔼𝑋_𝑇²=𝔼𝑋_𝑇−𝐶² (6) is minimized. The problem of finding out such a portfolio 𝜋_𝑡 is referred to as the mean-variance portfolio choice problem.
In the modern portfolio selection theory, a portfolio 𝜋_𝑡 is said to be admissible if it is integrable and {ℱ_𝑡}_𝑡>0-adapted, and (5) has a unique solution corresponding to 𝜋_𝑡. In this case, we refer to (𝑋_𝑡,𝜋_𝑡) as an admissible pair. Therefore, the mean-variance problem can be formulated as a linearly constrained stochastic optimization problem: MinimizeVar𝑋_𝑇=𝔼𝑋_𝑇−𝐶²subjectto𝔼𝑋_𝑇𝑋=𝐶_𝑡,𝜋_𝑡satisfies(5). (7) Finally, an optimal investment strategy of the above problem is called an efficient portfolio corresponding to 𝐶, the corresponding (𝐶,Var𝑋_𝑇) is called an efficient point, whereas the set of all the efficient points, when the parameter 𝐶 runs over [𝑥₀𝑒^𝑟𝑇,+∞), is called an efficient frontier.

Remark 2. If 𝜋_𝑡<0, this means that the investor is short-selling the stock. If 𝑋_𝑡−𝜋_𝑡<0, then the investor needs borrowing the money from the bank at interest rate 𝑅 and the amount to borrow is |𝑋_𝑡−𝜋_𝑡|. 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 𝔼𝑋_𝑇=𝐶, we introduce a Lagrange multiplier 2𝜆∈𝑅 and arrive at a new objective function: 𝐽̂𝜋_𝑡,𝜆=𝔼𝑋_𝑇−𝐶²+2𝜆𝑋_𝑇𝑋−𝐶=𝔼_𝑇−(𝐶−𝜆)²−𝜆². (8) Letting 𝛾=𝐶−𝜆, we obtain the following stochastic control problem: Minimize𝐽𝜋_𝑡,𝛾=𝔼𝑋_𝑇−𝛾²−𝐶−𝛾²subjectto𝑋_𝑡,𝜋_𝑡satisfies(5). (9) The link between problem (7) and (9) is provided by Lagrange duality theorem (see Fu et al. [18] and Luenberger [19]): MinimizeVar𝑋_𝑇=Max_𝜆∈ℝMin_𝜋𝑡𝐽̂𝜋_𝑡,𝜆=Max_𝛾∈ℝMin_𝜋𝑡𝐽𝜋_𝑡,𝛾. (10)

For a fixed constant 𝛾, the problem (9) is clearly equivalent to Minimize𝔼𝑋_𝑇−𝛾²subjectto𝑋_𝑡,𝜋_𝑡satisfies(5). (11) We define the value function 𝐻(𝑡,𝑠,𝑥) as 𝐻(𝑡,𝑠,𝑥)=Min_𝜋𝑡𝔼𝑋_𝑇−𝛾²∣𝑆_𝑡=𝑠,𝑋_𝑡=𝑥 (12) with boundary condition 𝐻(𝑇,𝑠,𝑥)=(𝑥−𝛾)².

According to dynamic programming principle, 𝐻(𝑡,𝑠,𝑥) can be taken as the continuous solution to the following HJB equation: 𝐻_𝑡+𝜇𝑠𝐻_𝑠+12𝑘²𝑠^2𝛽+2𝐻_𝑠𝑠+Min_𝜋𝑡𝑟𝑥+𝜇−𝑟𝜋_𝑡−(𝑅−𝑟)𝑥−𝜋_𝑡⁻𝐻_𝑥+12𝜋_𝑡𝑘𝑠^𝛽2𝐻_𝑥𝑥+𝑘²𝑠^2𝛽+1𝜋_𝑡𝐻_𝑥𝑠=0, (13) where 𝐻_𝑡, 𝐻_𝑠, 𝐻_𝑠𝑠, 𝐻_𝑥, 𝐻_𝑥𝑥, and 𝐻_𝑥𝑠 denote first-order and second-order partial derivatives with respect to time 𝑡, stock price 𝑆_𝑡, and wealth process 𝑋_𝑡.

Let us firstly describe borrowing situation. Not borrowing and investing in the bond means that 𝑥−𝜋_𝑡≥0 and borrowing to invest in the stock means that 𝑥−𝜋_𝑡<0. We define the nonborrowing region Θ in the (𝑡,𝑥)-plane to be Θ=(𝑡,𝑥)∈[0,𝑇]×ℝ∣𝑥−𝜋_𝑡≥0. (14) Taking borrowing situation into consideration, we rewrite the HJB equation (13) as 𝐻_𝑡+𝜇𝑠𝐻_𝑠+12𝑘²𝑠^2𝛽+2𝐻_𝑠𝑠+Min_𝜋𝑡𝑟𝑥+𝜇−𝑟𝜋_𝑡𝐻_𝑥+12𝜋_𝑡𝑘𝑠^𝛽2𝐻_𝑥𝑥+𝑘²𝑠^2𝛽+1𝜋_𝑡𝐻_𝑥𝑠()𝐻=0,if𝑡,𝑥∈Θ,_𝑡+𝜇𝑠𝐻_𝑠+12𝑘²𝑠^2𝛽+2𝐻_𝑠𝑠+Min_𝜋𝑡𝑅𝑥+𝜇−𝑅𝜋_𝑡𝐻_𝑥+12𝜋_𝑡𝑘𝑠^𝛽2𝐻_𝑥𝑥+𝑘²𝑠^2𝛽+1𝜋_𝑡𝐻_𝑥𝑠=0,if(𝑡,𝑥)∉Θ. (15) The optimal value 𝜋^∗_𝑡 of (15) is given by 𝜋^∗_𝑡=−𝜇−𝑟𝐻_𝑥−𝑘²𝑠^2𝛽+1𝐻_𝑥𝑠𝑘²𝑠^2𝛽𝐻_𝑥𝑥()−𝐻,if𝑡,𝑥∈Θ,𝜇−𝑅_𝑥−𝑘²𝑠^2𝛽+1𝐻_𝑥𝑠𝑘²𝑠^2𝛽𝐻_𝑥𝑥,if(𝑡,𝑥)∉Θ. (16)

Putting (16) into (15), we have 𝐻_𝑡+𝜇𝑠𝐻_𝑠+12𝑘²𝑠^2𝛽+2𝐻_𝑠𝑠+𝑟𝑥𝐻_𝑥−12𝑘²𝑠^2𝛽𝐻_𝑥𝑥×𝜇−𝑟𝐻_𝑥+𝑘²𝑠^2𝛽+1𝐻_𝑥𝑠²()𝐻=0,if𝑡,𝑥∈Θ,_𝑡+𝜇𝑠𝐻_𝑠+12𝑘²𝑠^2𝛽+2𝐻_𝑠𝑠+𝑅𝑥𝐻_𝑥−12𝑘²𝑠^2𝛽𝐻_𝑥𝑥×𝜇−𝑅𝐻_𝑥+𝑘²𝑠^2𝛽+1𝐻_𝑥𝑠²=0,if(𝑡,𝑥)∉Θ. (17) Letting 𝜏=𝑟, if (𝑡,𝑥)∈Θ; 𝜏=𝑅, if (𝑡,𝑥)∉Θ, we get 𝐻_𝑡+𝜇𝑠𝐻_𝑠+12𝑘²𝑠^2𝛽+2𝐻_𝑠𝑠+𝜏𝑥𝐻_𝑥−12𝑘²𝑠^2𝛽𝐻_𝑥𝑥𝜇−𝜏𝐻_𝑥+𝑘²𝑠^2𝛽+1𝐻_𝑥𝑠²=0. (18)

According to the convexity of the value function, we can define a Legendre transform: 𝐻(𝑡,𝑠,𝑧)=sup_𝑥>0{𝐻(𝑡,𝑠,𝑥)−𝑧𝑥}, (19) where 𝑧>0 denotes the dual variable to 𝑥.

The value of 𝑥 which the maximum in the above equation will be attained at is denoted by 𝑔(𝑡,𝑠,𝑧), so we have 𝑔(𝑡,𝑠,𝑧)=inf_𝑥>0𝑥∣𝐻(𝑡,𝑠,𝑥)≥𝑧𝑥+𝐻(𝑡,𝑠,𝑧). (20) The functions 𝑔(𝑡,𝑠,𝑧) and 𝐻(𝑡,𝑠,𝑧) are closely related and we will refer to either one of them as the dual function of 𝐻(𝑡,𝑠,𝑥). In this paper, we will work mainly with the function 𝑔(𝑡,𝑠,𝑧).

The function 𝐻(𝑡,𝑠,𝑧) is related to 𝑔(𝑡,𝑠,𝑧) by 𝑔(𝑡,𝑠,𝑧)=−𝐻_𝑧(𝑡,𝑠,𝑧).

Noting that 𝐻(𝑇,𝑠,𝑥)=(𝑥−𝛾)², we can define the following Legendre transform at terminal time: 𝐻(𝑇,𝑠,𝑧)=sup_𝑥>0𝑥−𝛾²,𝑔()−𝑧𝑥𝑇,𝑠,𝑧=inf_𝑥>0𝑥∣𝑥−𝛾²≥𝑧𝑥+𝐻(𝑇,𝑠,𝑧). (21) So we have 𝑔()=1𝑇,𝑠,𝑧2𝑧+𝛾. (22) According to (19), we have 𝐻_𝑥(𝑡,𝑠,𝑥)=𝑧, and this leads to 𝑔(𝑡,𝑠,𝑧)=𝑥,𝐻(𝑡,𝑠,𝑧)=𝐻𝑡,𝑠,𝑔−𝑧𝑔. (23)

Referring to Jonsson and Sircar [20], Xiao et al. [6], and Gao [7], we get the following transformation rules: 𝐻_𝑡=𝐻_𝑡,𝐻_𝑥=𝑧,𝐻_𝑥𝑥1=−𝐻_𝑧𝑧,𝐻_𝑠=𝐻_𝑠,𝐻_𝑠𝑠=𝐻_𝑠𝑠−𝐻²_𝑠𝑧𝐻_𝑧𝑧,𝐻_𝑥𝑠𝐻=−^𝑠𝑧𝐻_𝑧𝑧. (24) Putting transformation rules (24) into (18), we get 𝐻_𝑡+𝜇𝑠𝐻_𝑠+12𝑘²𝑠^2𝛽+2𝐻_𝑠𝑠+𝑧+𝜏𝑔𝜇−𝜏²𝑧²2𝑘²𝑠^2𝛽𝐻_𝑧𝑧−𝜇−𝜏𝑠𝑧𝐻_𝑠𝑧=0. (25)

Differentiating 𝐻 with respect to 𝑧, we derive the following dual equation: 𝑔_𝑡+𝜏𝑠𝑔_𝑠+12𝑘²𝑠^2𝛽+2𝑔_𝑠𝑠+𝜇−𝜏²𝑘²𝑠^2𝛽−𝜏𝑧𝑔_𝑧+𝜇−𝜏²2𝑘²𝑠^2𝛽𝑧²𝑔_𝑧𝑧−𝜇−𝜏𝑠𝑧𝑔_𝑠𝑧−𝜏𝑔=0, (26) where 𝜏=𝑟, if (𝑡,𝑥)∈Θ; 𝜏=𝑅, if (𝑡,𝑥)∉Θ.

Taking (22) into consideration, we can fit a solution to (26) with the following structure: 𝑔()=𝑓𝑡,𝑠,𝑧^𝑟𝑡,𝑦𝑧+ℎ^𝑟(𝑡),𝑦=𝑠^−2𝛽()𝑓,if𝑡,𝑥∈Θ;_𝑅𝑡,𝑦𝑧+ℎ^𝑅(𝑡),𝑦=𝑠^−2𝛽,if(𝑡,𝑥)∉Θ. (27) Considering 𝜏=𝑟, if (𝑡,𝑥)∈Θ; 𝜏=𝑅, if (𝑡,𝑥)∉Θ, we can rewrite 𝑔(𝑡,𝑠,𝑧) as 𝑔(𝑡,𝑠,𝑧)=𝑓^𝜏𝑡,𝑦𝑧+ℎ^𝜏(𝑡),𝑦=𝑠^−2𝛽, (28) with boundary conditions given by 𝑓^𝜏(𝑇,𝑦)=1/2 and ℎ^𝜏(𝑇)=𝛾.

Further, we have 𝑔_𝑡=𝑓^𝜏_𝑡𝑧+ℎ^𝜏_𝑡,𝑔_𝑠=𝑓^𝜏_𝑦−2𝛽𝑠^−2𝛽−1𝑔𝑧,_𝑠𝑧=𝑓^𝜏_𝑦−2𝛽𝑠^−2𝛽−1,𝑔_𝑠𝑠=𝑓^𝜏_𝑦𝑦−2𝛽𝑠^{−2𝛽−12}⋅𝑧+𝑓^𝜏_𝑦−2𝛽−2𝛽−1𝑠^−2𝛽−2𝑔𝑧,_𝑧=𝑓^𝜏,𝑔_𝑧𝑧=0. (29) Putting the above partial derivatives into (26), we get 𝑓^𝜏_𝑡+2𝛽𝜇−2𝜏𝑦+𝛽2𝛽+1𝑘²𝑓^𝜏_𝑦+2𝛽²𝑘²𝑦𝑓^𝜏_𝑦𝑦+𝜇−𝜏²𝑘²𝑦−2𝜏𝑓^𝜏𝑧+ℎ^𝜏_𝑡−𝜏ℎ^𝜏=0. (30) Eliminating the dependence on 𝑧, we obtain ℎ^𝜏_𝑡−𝜏ℎ^𝜏(𝑇)𝑓=0,ℎ=𝛾;_𝜏𝑡+2𝛽𝜇−2𝜏𝑦+𝛽2𝛽+1𝑘²𝑓^𝜏_𝑦+2𝛽²𝑘²𝑦𝑓^𝜏_𝑦𝑦+𝜇−𝜏²𝑘²𝑦−2𝜏𝑓^𝜏=0,𝑓^𝜏=1𝑇,𝑦2. (31)(32) The solution to (31) is ℎ(𝑡)=𝛾𝑒^{−𝜏(𝑇−𝑡)}. (33)

Lemma 3. Assume that the structure of the solution to (32) is 𝑓^𝜏(𝑡,𝑦)=𝐴^𝜏(𝑡)𝑒^{𝐵𝜏(𝑡)𝑦}, with the boundary conditions given by 𝐴^𝜏(𝑇)=1/2 and 𝐵^𝜏(𝑇)=0; then 𝐴^𝜏(𝑡) and 𝐵^𝜏(𝑡) are given by (43) and (42), respectively.

Proof . Putting 𝑓^𝜏(𝑡,𝑦)=𝐴^𝜏(𝑡)𝑒^{𝐵𝜏(𝑡)𝑦} into (32), we have 𝐴^𝜏(𝑡)𝑑𝐵^𝜏(𝑡)𝐴𝑑𝑡+2𝛽𝜇−2𝜏^𝜏(𝑡)𝐵^𝜏(𝑡)+2𝛽²𝑘²𝐴^𝜏(𝑡)𝐵^𝜏2(𝑡)+𝜇−𝜏²𝑘²𝐴^𝜏(𝑡𝑦+)𝑑𝐴^𝜏(𝑡)𝑘𝑑𝑡+𝛽2𝛽+1²𝐴^𝜏(𝑡)𝐵^𝜏(𝑡)−2𝜏𝐴^𝜏(𝑡)=0. (34) By matching the coefficients, we get 𝑑𝐴^𝜏(𝑡)𝑘𝑑𝑡+𝛽2𝛽+1²𝐴^𝜏(𝑡)𝐵^𝜏(𝑡)−2𝜏𝐴^𝜏(𝑡)=0,𝐴^𝜏(𝑇)=12,𝑑𝐵^𝜏(𝑡)𝐵𝑑𝑡+2𝛽𝜇−2𝜏^𝜏(𝑡)+2𝛽²𝑘²𝐵^𝜏2(𝑡)+𝜇−𝜏²𝑘²=0,𝐵^𝜏(𝑇)=0. (35)(36) Equation (36) can be reduced to 𝑑𝐵^𝜏(𝑡)𝑑𝑡=−2𝛽²𝑘²𝐵^𝜏2(𝑡)−2𝛽𝜇−2𝜏𝐵^𝜏(𝑡)−𝜇−𝜏²𝑘²,𝐵^𝜏(𝑇)=0. (37) Let Δ denote the discriminant of the quadratic equation −2𝛽²𝑘²𝐵^𝜏2(𝑡)−2𝛽(𝜇−2𝜏)𝐵^𝜏(𝑡)−((𝜇−𝜏)²/𝑘²)=0.
Easy calculation leads to Δ=4𝛽²(2𝜏²−𝜇²). Assume that Δ>0, that is, −√2𝜏<𝜇<√2𝜏, then the quadratic equation has two real roots: 𝑚^𝜏₁=−𝜇−2𝜏+2𝜏²−𝜇²2𝛽𝑘²,𝑚^𝜏₂=−𝜇−2𝜏−2𝜏²−𝜇²2𝛽𝑘². (38) So (37) can be rewritten as 1𝑚^𝜏₁−𝑚^𝜏₂^𝑇_𝑡1𝐵^𝜏(𝑡)−𝑚^𝜏₁−1𝐵^𝜏(𝑡)−𝑚^𝜏₂𝑑𝐵^𝜏(𝑡)=−2𝛽²𝑘²(𝑇−𝑡). (39)
Further, we obtain 𝐵^𝜏(𝑡)=𝑚^𝜏₁𝑚^𝜏₂1−𝑒^{−2𝛽2𝑘2(𝑚𝜏1−𝑚𝜏2)(𝑇−𝑡)}𝑚^𝜏₁−𝑚^𝜏₂𝑒^{−2𝛽2𝑘2(𝑚𝜏1−𝑚𝜏2)(𝑇−𝑡)}. (40) Letting 𝜆^𝜏₁=−𝜇−2𝜏+2𝜏²−𝜇²,𝜆2𝛽_𝜏2=−𝜇−2𝜏−2𝜏²−𝜇².2𝛽 (41) We get 𝐵^𝜏(𝑡)=𝑘⁻²𝐼^𝜏(𝑡),𝐼^𝜏(𝑡)=𝜆^𝜏₁𝜆^𝜏₂1−𝑒^{−2𝛽2(𝜆𝜏1−𝜆𝜏2)(𝑇−𝑡)}𝜆^𝜏₁−𝜆^𝜏₂𝑒^{−2𝛽2(𝜆𝜏1−𝜆𝜏2)(𝑇−𝑡)}. (42) Plugging (42) into (35) yields 𝐴^𝜏(𝑡)=12𝑒^{−∫𝑇𝑡(2𝜏−𝛽(2𝛽+1)𝐼𝜏(𝑡))𝑑𝑡}. (43)
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 𝜆, 𝑇 and 𝐶⩾𝑥₀𝑒^𝑟𝑇, the optimal investment strategy with borrowing constraint under a mean-variance criterion corresponding to the problem (11) is 𝜋^∗_𝑡=−𝜇−𝑟𝑘²𝑆_𝑡^2𝛽𝑋_𝑡−𝛾𝑒^{−𝑟(𝑇−𝑡)}𝐾^𝑟(𝑡),𝑖𝑓𝑋_𝑡⩾𝛾𝜌(𝑡),𝑟<𝜇<√−2𝑟,𝜇−𝑅𝑘²𝑆_𝑡^2𝛽𝑋_𝑡−𝛾𝑒^{−𝑅(𝑇−𝑡)}𝐾^𝑅(𝑡),𝑖𝑓𝑋_𝑡<𝛾𝜌(𝑡),𝑅<𝜇<√2𝑅, (44) where 𝐾^𝜏(𝑡) and 𝜌(𝑡) are given by (48) and (50), respectively.

Proof . Under the transformation rules, the optimal strategy (16) is derived as follows: 𝜋^∗_𝑡=−𝜇−𝜏𝐻_𝑥−𝑘²𝑠^2𝛽+1𝐻_𝑥𝑠𝑘²𝑠^2𝛽𝐻_𝑥𝑥=−𝜇−𝜏𝐻_𝑥/𝐻_𝑥𝑥−𝑘²𝑠^2𝛽+1𝐻_𝑥𝑠/𝐻_𝑥𝑥𝑘²𝑠^2𝛽=𝜇−𝜏𝑧𝐻_𝑧𝑧−𝑘²𝑠^2𝛽+1𝐻_𝑠𝑧𝑘²𝑠^2𝛽=−𝜇−𝜏𝑧𝑔_𝑧+𝑘²𝑠^2𝛽+1𝑔_𝑠𝑘²𝑠^2𝛽. (45) Taking (33),(35), and Lemma 3 into consideration, we have 𝜋^∗_𝑡=−𝜇−𝜏𝑧𝑔_𝑧+𝑘²𝑠^2𝛽+1𝑔_𝑠𝑘²𝑠^2𝛽=−𝜇−𝜏𝑧𝑓^𝜏+𝑘²𝑠^2𝛽+1𝑧𝑓^𝜏_𝑦−2𝛽𝑠^−2𝛽−1𝑘²𝑠^2𝛽=−𝜇−𝜏𝑔−ℎ^𝜏(𝑡)+𝑘²𝑧𝐴^𝜏(𝑡)𝐵^𝜏(𝑡)𝑒^{𝐵𝜏(𝑡)𝑦}−2𝛽𝑘²𝑠^2𝛽=−𝜇−𝜏𝑥−ℎ^𝜏(𝑡)+𝑘²𝐵^𝜏(𝑡)−2𝛽𝑥−ℎ^𝜏(𝑡)𝑘²𝑠^2𝛽=−𝜇−𝜏𝑥−ℎ^𝜏(𝑡)+𝐼^𝜏(𝑡)−2𝛽𝑥−ℎ^𝜏(𝑡)𝑘²𝑠^2𝛽=−𝜇−𝜏+2𝛽𝐼^𝜏(𝑡)𝑘²𝑠^2𝛽𝑥−ℎ^𝜏(𝑡). (46) Therefore, the optimal strategy is reduced to 𝜋^∗_𝑡=−𝜇−𝑟𝑘²𝑆_𝑡^2𝛽𝑋_𝑡−𝛾𝑒^{−𝑟(𝑇−𝑡)}𝐾^𝑟(𝑡),if(𝑡,𝑥)∈Θ,𝑟<𝜇<√−2𝑟,𝜇−𝑅𝑘²𝑆_𝑡^2𝛽𝑋_𝑡−𝛾𝑒^{−𝑅(𝑇−𝑡)}𝐾^𝑅(𝑡),if(𝑡,𝑥)∉Θ,𝑅<𝜇<√2𝑅, (47) where 𝐾^𝜏(𝑡)=1+2𝛽𝐼^𝜏(𝑡)()()𝜇−𝜏,𝜏=𝑟,if𝑡,𝑥∈Θ,𝜏=𝑅,if𝑡,𝑥∉Θ. (48)
As the boundary condition of borrowing the money from the bank is 𝑋_𝑡−𝜋^∗_𝑡=0; that is, 𝑋_𝑡+𝜇−𝑅𝑘²𝑆_𝑡^2𝛽𝑋_𝑡−𝛾𝑒^{−𝑅(𝑇−𝑡)}𝐾^𝑅(𝑡)=0. (49) Denoting by 𝛾𝜌(𝑡) the borrowing curve, we yield 𝜌(𝑡)=𝜇−𝑅/𝑘²𝑆_𝑡^2𝛽𝑒^{−𝑅(𝑇−𝑡)}𝐾^𝑅(𝑡)1+𝜇−𝑅/𝑘²𝑆_𝑡^2𝛽𝐾^𝑅(𝑡). (50) Therefore, nonborrowing region Θ in the (𝑡,𝑥)-plane can be rewritten as Θ=(𝑡,𝑥)∈[0,𝑇]×ℝ∣𝑋_𝑡⩾𝛾𝜌(𝑡). (51)
Hence, the proof of the Theorem 4 is completed.

Remark 5. We can draw some conclusions from (44).(i)If 𝑋_𝑡⩾𝛾𝜌(𝑡) and 𝑟<𝜇<√2𝑟, 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 𝑋_𝑡−𝜋^∗_𝑡.(ii)If 𝑋_𝑡<𝛾𝜌(𝑡) and 𝑅<𝜇<√2𝑅, 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 |𝑋_𝑡−𝜋^∗_𝑡|, 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 𝜏 either the interest rate 𝑟 or the borrowing rate 𝑅, and letting 𝜃^𝜏=𝜇−𝜏𝑘𝑆^𝛽_𝑡, (52) where 𝜏=𝑟, if 𝑋_𝑡⩾𝛾𝜌(𝑡) and 𝜏=𝑅, if 𝑋_𝑡<𝛾𝜌(𝑡).

In both cases above, the wealth equation (5) is reduced to 𝑑𝑋_𝑡=𝜏𝑋_𝑡+𝜇−𝜏𝜋_𝑡𝑑𝑡+𝜋_𝑡𝑘𝑆^𝛽_𝑡𝑑𝑊_𝑡,𝑋₀=𝑥₀>0. (53) For any fixed 𝛾, under the efficient strategy in the Theorem 4, the dynamics of the wealth equation (5) are 𝑑𝑋_𝑡=𝜏−𝜃^𝜏2𝐾^𝜏(𝑡)𝑋_𝑡+𝜃^𝜏2𝛾𝑒^{−𝜏(𝑇−𝑡)}𝐾^𝜏(𝑡)𝑑𝑡−𝜃^𝜏𝑋_𝑡−𝛾𝑒^{−𝜏(𝑇−𝑡)}𝐾^𝜏(𝑡)𝑑𝑊_𝑡,𝑋₀=𝑥₀>0. (54)

Applying Itô's lemma to the wealth process (54), we yield 𝑑𝑋²_𝑡=2𝜏+𝜃^𝜏2𝐾^2𝜏(𝑡)−2𝐾^𝜏(𝑡)𝑋²_𝑡−𝜃^𝜏22𝛾𝑒^{−𝜏(𝑇−𝑡)}𝐾^2𝜏(𝑡)−𝐾^𝜏(𝑡)𝑋_𝑡+𝜃^𝜏2𝛾𝑒^{−𝜏(𝑇−𝑡)}𝐾^𝜏(𝑡)²𝑑𝑡−𝜃^𝜏𝑋_𝑡−𝛾𝑒^{−𝜏(𝑇−𝑡)}𝐾^𝜏(𝑡)2𝑋_𝑡𝑑𝑊_𝑡,𝑋²₀=𝑥²₀>0. (55)

Taking expectations on both sides of (54) and (55), respectively, one has 𝑑𝔼𝑋_𝑡=𝜏−𝜃^𝜏2𝐾^𝜏(𝑡)𝔼𝑋_𝑡+𝜃^𝜏2𝛾𝑒^{−𝜏(𝑇−𝑡)}𝐾^𝜏(𝑡)𝑑𝑡,𝔼𝑋₀=𝑥₀>0,𝑑𝔼𝑋²_𝑡=2𝜏+𝜃^𝜏2𝐾^2𝜏(𝑡)−2𝐾^𝜏(𝑡)𝔼𝑋²_𝑡−𝜃^𝜏22𝛾𝑒^{−𝜏(𝑇−𝑡)}𝐾^2𝜏(𝑡)−𝐾^𝜏(𝑡)𝔼𝑋_𝑡+𝜃^𝜏2𝛾𝑒^{−𝜏(𝑇−𝑡)}𝐾^𝜏(𝑡)²𝑑𝑡,𝔼𝑋²₀=𝑥²₀>0. (56)(57)

The solution of the linear ordinary differential equation (56) is 𝔼𝑋_𝑡=𝑥₀𝑒^{∫𝑡0(𝜏−(𝜃𝜏)2}𝐾^{𝜏(𝑡))𝑑𝑡}+𝛾𝑒^{−𝜏(𝑇−𝑡)}1−𝑒^{−∫𝑡0(𝜃𝜏)2}𝐾^{𝜏(𝑡)𝑑𝑡}, (58) and it results in 𝔼𝑋_𝑇=𝑥₀𝑒^{∫𝑇0(𝜏−(𝜃𝜏)2}𝐾^{𝜏(𝑡))𝑑𝑡}+𝛾1−𝑒^{−∫𝑇0(𝜃𝜏)2}𝐾^{𝜏(𝑡)𝑑𝑡}. (59)

Similarly, by solving (57), one has 𝔼𝑋²_𝑇=𝑒^{∫𝑇0(𝜃𝜏)2(}𝐾^2𝜏_(𝑡)−2𝐾^𝜏_{(𝑡))𝑑𝑡}𝑥₀𝑒^𝜏𝑇−𝛾²+2𝛾𝑒^{−∫𝑇0(𝜃𝜏)2}𝐾^𝜏_{(𝑡)𝑑𝑡}𝑥₀𝑒^𝜏𝑇−𝛾+𝛾². (60)

Therefore, the objective function of the problem (9), as a explicit function of parameter 𝛾, is given by 𝐽_min𝜋^∗_𝑡,𝛾=𝔼𝑋_𝑇−𝛾²−𝐶−𝛾²=𝔼𝑋²_𝑇−2𝛾𝔼𝑋_𝑇+2𝛾𝐶−𝐶²=𝛾²𝑒^{∫𝑇0(𝜃𝜏)2(}𝐾^{2𝜏(𝑡)−2}𝐾^{𝜏(𝑡))𝑑𝑡}−1+2𝛾𝐶−𝑥₀𝑒^𝜏𝑇𝑒^{∫𝑇0(𝜃𝜏)2(}𝐾^2𝜏_(𝑡)−2𝐾^𝜏_{(𝑡))𝑑𝑡}+𝑥²₀𝑒^2𝜏𝑇𝑒^{∫𝑇0(𝜃𝜏)2(}𝐾^2𝜏_(𝑡)−2𝐾^𝜏_{(𝑡))𝑑𝑡}−𝐶². (61) Using (10) obtained by Lagrange duality theorem, the minimum variance Var𝑋_𝑇 is achieved for 𝛾^∗_𝜏=𝐶−𝑥₀𝑒^𝜏𝑇𝑒^{∫𝑇0(𝜃𝜏)2(}𝐾^2𝜏_(𝑡)−2𝐾^𝜏_{(𝑡))𝑑𝑡}1−𝑒^{∫𝑇0(𝜃𝜏)2(}𝐾^2𝜏_(𝑡)−2𝐾^𝜏_{(𝑡))𝑑𝑡}. (62) In addition, we obtain 𝐽_maxmin𝜋^∗_𝑡,𝛾^∗_𝜏=𝐶−𝑥₀𝑒^𝜏𝑇2𝑒^{∫𝑇0(𝜃𝜏)2(}𝐾^2𝜏_(𝑡)−2𝐾^𝜏_{(𝑡))𝑑𝑡}.−1 (63)

Letting Var^𝜏𝑋_𝑇=𝐽_maxmin(𝜋^∗_𝑡,𝛾^∗_𝜏). The optimal value of 𝛾 and the minimum variance Var𝑋_𝑇 are Var𝑋_𝑇=MaxVar^𝑟𝑋_𝑇,Var^𝑅𝑋_𝑇=Max𝐶−𝑥₀𝑒^𝑟𝑇2𝑒^{∫𝑇0(𝜃𝑟)2(}𝐾^{2𝑟(𝑡)−2}𝐾^{𝑟(𝑡))𝑑𝑡},−1𝐶−𝑥₀𝑒^𝑅𝑇2𝑒^{∫𝑇0(𝜃𝑅)2(}𝐾^2𝑅_(𝑡)−2𝐾^𝑅_{(𝑡))𝑑𝑡},𝛾−1_∗=𝛾^∗_𝑟,ifVar^𝑟𝑋_𝑇=Var𝑋_𝑇,𝛾^∗_𝑅,ifVar^𝑅𝑋_𝑇=Var𝑋_𝑇. (64)(65)

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 𝜋^∗_𝑡=−𝜇−𝑟𝑘²𝑆_𝑡^2𝛽𝑋_𝑡−𝛾^∗𝑒^{−𝑟(𝑇−𝑡)}𝐾^𝑟(𝑡),𝑖𝑓𝑋_𝑡⩾𝛾^∗𝜌(𝑡),𝑟<𝜇<√−2𝑟,𝜇−𝑅𝑘²𝑆_𝑡^2𝛽𝑋_𝑡−𝛾^∗𝑒^{−𝑅(𝑇−𝑡)}𝐾^𝑅(𝑡),𝑖𝑓𝑋_𝑡<𝛾^∗𝜌(𝑡),𝑅<𝜇<√2𝑅. (66) Moreover, the efficient frontier is given by Var𝑋_𝑇=Max𝐶−𝑥₀𝑒^𝑟𝑇2𝑒^{∫𝑇0(𝜃𝑟)2(}𝐾^2𝑟_(𝑡)−2𝐾^𝑟_{(𝑡))𝑑𝑡},−1𝐶−𝑥₀𝑒^𝑅𝑇2𝑒^{∫𝑇0(𝜃𝑅)2(}𝐾^2𝑅_(𝑡)−2𝐾^𝑅_{(𝑡))𝑑𝑡},−1 (67) where 𝐾^𝜏(𝑡) and 𝜌(𝑡) are given by (48) and (50), respectively.

Remark 7. When 𝛽=0, the results in the Theorem 6 are reduced to the ones under a geometric Brownian motion model, which is obtained by [18]. When 𝛽=−1/2 and 𝛽=−1, 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).