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

Authors: Hao Chang; Xi-min Rong
Journal: Journal of Applied Mathematics
 (2013)

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## 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.


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## Body

## 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 Section2, 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 Pt at time t, whose price process Pt with borrowing constraint can be expressed in the following ordinary differential equation (see Fu et al. [18]):
(1)dPt={rPtdt,ifPt≥0,t∈[0,T]RPtdt,ifPt<0,t∈[0,T],P0=p0>0,
where the constant r>0 is the interest rate of the bond and R is the borrowing rate being larger than r.Lettingx-=-min(x,0), then (1) can be rewritten as
(2)dPt=(rPt-(R-r)Pt-)dt,P0=p0>0.
The another asset is a stock with prices St at time t, whose price process St is supposed to follow the constant elasticity of variance (CEV) model:
(3)dSt=St[μdt+kStβdWt],S0=s0>0,
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. kStβ is defined as the instantaneous volatility of the stock, and Wt is a one-dimensional standard and adapted Brownian motion defined on the filtered complete probability space (Ω,ℱ,P,{ℱt}t≥0).Remark 1.
Note that there are four special interpretations for the elasticity parameterβ:(i)
ifβ=0, the CEV model is reduced to a geometric Brownian motion (GBM);(ii)
ifβ=-1, it is the Ornstein-Uhlenbeck process;(iii)
ifβ=-1/2, it is the model first presented by Cox and Ross [1] as an alternative diffusion process for valuation of options;(iv)
ifβ<0, this means that the instantaneous volatility kStβ 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 byXt the wealth of the investor at time t∈[0,T] and by Ni(t) the share of asset ith held by the investor at time t, i=1,2. Let πt=N2(t)St be the amount invested in the stock at time t, t∈[0,T]. Clearly, the amount invested in the bond is πt0=N1(t)Pt=Xt-πt. The wealth process Xt=N1(t)Pt+N2(t)St corresponding to trading strategy πt is subject to the following stochastic differential equation:
(4)dXt=N1(t)dPt+N2(t)dSt=N1(t)(rPt-(R-r)Pt-)dt+N2(t)St(μdt+kStβdWt)=(rN1(t)Pt-(R-r)N1(t)Pt-)dt+πt(μdt+kStβdWt).
That is, we have
(5)dXt=(rXt+(μ-r)πt-(R-r)(Xt-πt)-)dt+πtkStβdWt,X0=x0>0.
The investor’s objective is to find an optimal portfolioπt such that the expected terminal wealth satisfies 𝔼XT=C, for some constant C∈ℝ, while the risk measured by the variance of the terminal wealth
(6)VarXT=𝔼[XT-𝔼XT]2=𝔼(XT-C)2
is minimized. The problem of finding out such a portfolio πt is referred to as the mean-variance portfolio choice problem.
In the modern portfolio selection theory, a portfolioπt is said to be admissible if it is integrable and {ℱt}t>0-adapted, and (5) has a unique solution corresponding to πt. In this case, we refer to (Xt,πt) as an admissible pair. Therefore, the mean-variance problem can be formulated as a linearly constrained stochastic optimization problem:
(7)MinimizeVarXT=𝔼(XT-C)2subjectto𝔼XT=C(Xt,πt)satisfies(5).
Finally, an optimal investment strategy of the above problem is called an efficient portfolio corresponding to C, the corresponding (C,VarXT) is called an efficient point, whereas the set of all the efficient points, when the parameter C runs over [x0erT,+∞), is called an efficient frontier.Remark 2.
Ifπt<0, this means that the investor is short-selling the stock. If Xt-πt<0, then the investor needs borrowing the money from the bank at interest rate R and the amount to borrow is |Xt-πt|. 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 𝔼XT=C, we introduce a Lagrange multiplier 2λ∈R and arrive at a new objective function:
(8)J^(πt,λ)=𝔼[(XT-C)2+2λ(XT-C)]=𝔼(XT-(C-λ))2-λ2.
Letting γ=C-λ, we obtain the following stochastic control problem:
(9)MinimizeJ¯(πt,γ)=𝔼(XT-γ)2-(C-γ)2subjectto(Xt,πt)satisfies(5).
The link between problem (7) and (9) is provided by Lagrange duality theorem (see Fu et al. [18] and Luenberger [19]):
(10)MinimizeVarXT=Maxλ∈ℝMinπtJ^(πt,λ)=Maxγ∈ℝMinπtJ¯(πt,γ).For a fixed constantγ, the problem (9) is clearly equivalent to
(11)Minimize𝔼(XT-γ)2subjectto(Xt,πt)satisfies(5).
We define the value function H(t,s,x) as
(12)H(t,s,x)=Minπt𝔼((XT-γ)2∣St=s,Xt=x)
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:
(13)Ht+μsHs+12k2s2β+2Hss+Minπt{12[rx+(μ-r)πt-(R-r)(x-πt)-]Hx+12(πtksβ)2Hxx+k2s2β+1πtHxs}=0,
where Ht, Hs, Hss, Hx, Hxx, and Hxs denote first-order and second-order partial derivatives with respect to time t, stock price St, and wealth process Xt.Let us firstly describe borrowing situation. Not borrowing and investing in the bond means thatx-πt≥0 and borrowing to invest in the stock means that x-πt<0. We define the nonborrowing region Θ in the (t,x)-plane to be
(14)Θ={(t,x)∈[0,T]×ℝ∣x-πt≥0}.
Taking borrowing situation into consideration, we rewrite the HJB equation (13) as
(15)Ht+μsHs+12k2s2β+2Hss+Minπt{[rx+(μ-r)πt]Hx+12(πtksβ)2Hxx12+k2s2β+1πtHxs}=0,if(t,x)∈Θ,Ht+μsHs+12k2s2β+2Hss+Minπt{[Rx+(μ-R)πt]Hx+12(πtksβ)2Hxx12+k2s2β+1πtHxs}=0,if(t,x)∉Θ.
The optimal value πt* of (15) is given by
(16)πt*={-(μ-r)Hx-k2s2β+1Hxsk2s2βHxx,if(t,x)∈Θ,-(μ-R)Hx-k2s2β+1Hxsk2s2βHxx,if(t,x)∉Θ.Putting (16) into (15), we have
(17)Ht+μsHs+12k2s2β+2Hss+rxHx-12k2s2βHxx×[(μ-r)Hx+k2s2β+1Hxs]2=0,if(t,x)∈Θ,Ht+μsHs+12k2s2β+2Hss+RxHx-12k2s2βHxx×[(μ-R)Hx+k2s2β+1Hxs]2=0,if(t,x)∉Θ.
Letting τ=r, if (t,x)∈Θ; τ=R, if (t,x)∉Θ, we get
(18)Ht+μsHs+12k2s2β+2Hss+τxHx-12k2s2βHxx[(μ-τ)Hx+k2s2β+1Hxs]2=0.According to the convexity of the value function, we can define a Legendre transform:(19)H^(t,s,z)=supx>0{H(t,s,x)-zx},
where z>0 denotes the dual variable to x.The value ofx which the maximum in the above equation will be attained at is denoted by g(t,s,z), so we have
(20)g(t,s,z)=infx>0{x∣H(t,s,x)≥zx+H^(t,s,z)}.
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 functionH^(t,s,z) is related to g(t,s,z) by g(t,s,z)=-H^z(t,s,z).Noting thatH(T,s,x)=(x-γ)2, we can define the following Legendre transform at terminal time:
(21)H^(T,s,z)=supx>0{(x-γ)2-zx},g(T,s,z)=infx>0{(x∣x-γ)2≥zx+H^(T,s,z)}.
So we have
(22)g(T,s,z)=12z+γ.
According to (19), we have Hx(t,s,x)=z, and this leads to
(23)g(t,s,z)=x,H^(t,s,z)=H(t,s,g)-zg.Referring to Jonsson and Sircar [20], Xiao et al. [6], and Gao [7], we get the following transformation rules:
(24)Ht=H^t,Hx=z,Hxx=-1H^zz,Hs=H^s,Hss=H^ss-H^sz2H^zz,Hxs=-H^szH^zz.
Putting transformation rules (24) into (18), we get
(25)H^t+μsH^s+12k2s2β+2H^ss+(τg)z+(μ-τ)2z22k2s2βH^zz-(μ-τ)szH^sz=0.DifferentiatingH^ with respect to z, we derive the following dual equation:
(26)gt+τsgs+12k2s2β+2gss+((μ-τ)2k2s2β-τ)zgz+(μ-τ)22k2s2βz2gzz-(μ-τ)szgsz-τg=0,
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:
(27)g(t,s,z)={fr(t,y)z+hr(t),y=s-2β,if(t,x)∈Θ;fR(t,y)z+hR(t),y=s-2β,if(t,x)∉Θ.
Considering τ=r, if (t,x)∈Θ; τ=R, if (t,x)∉Θ, we can rewrite g(t,s,z) as
(28)g(t,s,z)=fτ(t,y)z+hτ(t),y=s-2β,
with boundary conditions given by fτ(T,y)=1/2 and hτ(T)=γ.Further, we have(29)gt=ftτz+htτ,gs=fyτ(-2β)s-2β-1z,gsz=fyτ(-2β)s-2β-1,gss=fyyτ((-2β)s-2β-1)2·z+fyτ(-2β)(-2β-1)s-2β-2z,gz=fτ,gzz=0.
Putting the above partial derivatives into (26), we get
(30)[(μ-τ)2k2ftτ+(2β(μ-2τ)y+β(2β+1)k2)fyτ+2β2k2yfyyτ+((μ-τ)2k2y-2τ)fτ]z+htτ-τhτ=0.
Eliminating the dependence on z, we obtain
(31)htτ-τhτ=0,h(T)=γ;(32)ftτ+(2β(μ-2τ)y+β(2β+1)k2)fyτ+2β2k2yfyyτ+((μ-τ)2k2y-2τ)fτ=0,fτ(T,y)=12.
The solution to (31) is
(33)h(t)=γe-τ(T-t).Lemma 3.
Assume that the structure of the solution to (32) is fτ(t,y)=Aτ(t)eBτ(t)y, with the boundary conditions given by Aτ(T)=1/2 and Bτ(T)=0; then Aτ(t) and Bτ(t) are given by (43) and (42), respectively.Proof .
Puttingfτ(t,y)=Aτ(t)eBτ(t)y into (32), we have
(34)[Aτ(t)dBτ(t)dt+2β(μ-2τ)Aτ(t)Bτ(t)+2β2k2Aτ(t)Bτ2(t)+(μ-τ)2k2Aτ(t)]y+dAτ(t)dt+β(2β+1)k2Aτ(t)Bτ(t)-2τAτ(t)=0.
By matching the coefficients, we get
(35)dAτ(t)dt+β(2β+1)k2Aτ(t)Bτ(t)-2τAτ(t)=0,Aτ(T)=12,(36)dBτ(t)dt+2β(μ-2τ)Bτ(t)+2β2k2Bτ2(t)+(μ-τ)2k2=0,Bτ(T)=0.
Equation (36) can be reduced to
(37)dBτ(t)dt=-2β2k2Bτ2(t)-2β(μ-2τ)Bτ(t)-(μ-τ)2k2,Bτ(T)=0.
Let Δ denote the discriminant of the quadratic equation -2β2k2Bτ2(t)-2β(μ-2τ)Bτ(t)-((μ-τ)2/k2)=0.
Easy calculation leads toΔ=4β2(2τ2-μ2). Assume that Δ>0, that is, -2τ<μ<2τ, then the quadratic equation has two real roots:
(38)m1τ=-(μ-2τ)+2τ2-μ22βk2,m2τ=-(μ-2τ)-2τ2-μ22βk2.
So (37) can be rewritten as
(39)1m1τ-m2τ∫tT‍(1Bτ(t)-m1τ-1Bτ(t)-m2τ)dBτ(t)=-2β2k2(T-t).
Further, we obtain(40)Bτ(t)=m1τm2τ(1-e-2β2k2(m1τ-m2τ)(T-t))m1τ-m2τe-2β2k2(m1τ-m2τ)(T-t).
Letting
(41)λ1τ=-(μ-2τ)+2τ2-μ22β,λ2τ=-(μ-2τ)-2τ2-μ22β.
We get
(42)Bτ(t)=k-2Iτ(t),Iτ(t)=λ1τλ2τ(1-e-2β2(λ1τ-λ2τ)(T-t))λ1τ-λ2τe-2β2(λ1τ-λ2τ)(T-t).
Plugging (42) into (35) yields
(43)Aτ(t)=12e-∫tT‍(2τ-β(2β+1)Iτ(t))dt.
Therefore, Lemma3 is completed.Finally, summarizing the above results, we obtain the optimal trading strategy for the problem (11).Theorem 4.
For a givenλ, T and C⩾x0erT, the optimal investment strategy with borrowing constraint under a mean-variance criterion corresponding to the problem (11) is
(44)πt*={-(μ-r)k2St2β(Xt-γe-r(T-t))K~r(t),ifXt⩾γρ(t),r<μ<2r,-(μ-R)k2St2β(Xt-γe-R(T-t))K~R(t),ifXt<γρ(t),R<μ<2R,
where Kτ(t) and ρ(t) are given by (48) and (50), respectively.Proof .
Under the transformation rules, the optimal strategy (16) is derived as follows:
(45)πt*=-(μ-τ)Hx-k2s2β+1Hxsk2s2βHxx=-(μ-τ)Hx/Hxx-k2s2β+1Hxs/Hxxk2s2β=(μ-τ)zH^zz-k2s2β+1H^szk2s2β=-(μ-τ)zgz+k2s2β+1gsk2s2β.
Taking (33),(35), and Lemma 3 into consideration, we have
(46)πt*=-(μ-τ)zgz+k2s2β+1gsk2s2β=-(μ-τ)zfτ+k2s2β+1zfyτ(-2β)s-2β-1k2s2β=-(μ-τ)(g-hτ(t))+k2zAτ(t)Bτ(t)eBτ(t)y(-2β)k2s2β=-(μ-τ)(x-hτ(t))+k2Bτ(t)(-2β)(x-hτ(t))k2s2β=-(μ-τ)(x-hτ(t))+Iτ(t)(-2β)(x-hτ(t))k2s2β=-(μ-τ)+2βIτ(t)k2s2β(x-hτ(t)).
Therefore, the optimal strategy is reduced to
(47)πt*={-(μ-r)k2St2β(Xt-γe-r(T-t))K~r(t),if(t,x)∈Θ,r<μ<2r,-(μ-R)k2St2β(Xt-γe-R(T-t))K~R(t),if(t,x)∉Θ,R<μ<2R,
where
(48)K~τ(t)=[1+2βIτ(t)μ-τ],τ=r,if(t,x)∈Θ,τ=R,if(t,x)∉Θ.
As the boundary condition of borrowing the money from the bank isXt-πt*=0; that is,
(49)Xt+(μ-R)k2St2β(Xt-γe-R(T-t))K~R(t)=0.
Denoting by γρ(t) the borrowing curve, we yield
(50)ρ(t)=((μ-R)/k2St2β)e-R(T-t)K~R(t)1+((μ-R)/k2St2β)K~R(t).
Therefore, nonborrowing region Θ in the (t,x)-plane can be rewritten as
(51)Θ={(t,x)∈[0,T]×ℝ∣Xt⩾γρ(t)}.
Hence, the proof of the Theorem4 is completed.Remark 5.
We can draw some conclusions from (44).(i)
IfXt⩾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 Xt-πt*.(ii)
IfXt<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 |Xt-πt*|, 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
(52)θτ=μ-τkStβ,
where τ=r, if Xt⩾γρ(t) and τ=R, if Xt<γρ(t).In both cases above, the wealth equation (5) is reduced to
(53)dXt=(τXt+(μ-τ)πt)dt+πtkStβdWt,X0=x0>0.
For any fixed γ, under the efficient strategy in the Theorem 4, the dynamics of the wealth equation (5) are
(54)dXt=((τ-(θτ)2K~τ(t))Xt+(θτ)2γe-τ(T-t)K~τ(t))dt-θτ(Xt-γe-τ(T-t))K~τ(t)dWt,X0=x0>0.ApplyingItô's lemma to the wealth process (54), we yield
(55)dXt2=((γe-τ(T-t)K~τ(t))2[2τ+(θτ)2(K~2τ(t)-2K~τ(t))]Xt2-(θτ)22γe-τ(T-t)(K~2τ(t)-K~τ(t))Xt+(θτ)2(γe-τ(T-t)K~τ(t))2)dt-θτ(Xt-γe-τ(T-t))K~τ(t)2XtdWt,X02=x02>0.Taking expectations on both sides of (54) and (55), respectively, one has
(56)d𝔼Xt=((τ-(θτ)2K~τ(t))𝔼Xt+(θτ)2γe-τ(T-t)K~τ(t))dt,𝔼X0=x0>0,(57)d𝔼Xt2=([2τ+(θτ)2(K~2τ(t)-2K~τ(t))]𝔼Xt2-(θτ)22γe-τ(T-t)(K~2τ(t)-K~τ(t))𝔼Xt+(θτ)2(γe-τ(T-t)K~τ(t))2)dt,𝔼X02=x02>0.The solution of the linear ordinary differential equation (56) is
(58)𝔼Xt=x0e∫0t‍(τ-(θτ)2K~τ(t))dt+γe-τ(T-t)[1-e-∫0t‍(θτ)2K~τ(t)dt],
and it results in
(59)𝔼XT=x0e∫0T‍(τ-(θτ)2K~τ(t))dt+γ[1-e-∫0T‍(θτ)2K~τ(t)dt].Similarly, by solving (57), one has
(60)𝔼XT2=e∫0T‍(θτ)2(K~2τ(t)-2K~τ(t))dt(x0eτT-γ)2+2γe-∫0T‍(θτ)2K~τ(t)dt(x0eτT-γ)+γ2.Therefore, the objective function of the problem (9), as a explicit function of parameter γ, is given by
(61)J¯min(πt*,γ)=𝔼(XT-γ)2-(C-γ)2=𝔼XT2-2γ𝔼XT+2γC-C2=γ2(e∫0T‍(θτ)2(K~2τ(t)-2K~τ(t))dt-1)+2γ(C-x0eτTe∫0T‍(θτ)2(K~2τ(t)-2K~τ(t))dt)+x02e2τTe∫0T‍(θτ)2(K~2τ(t)-2K~τ(t))dt-C2.
Using (10) obtained by Lagrange duality theorem, the minimum variance VarXT is achieved for
(62)γτ*=C-x0eτTe∫0T‍(θτ)2(K~2τ(t)-2K~τ(t))dt1-e∫0T‍(θτ)2(K~2τ(t)-2K~τ(t))dt.
In addition, we obtain
(63)J¯maxmin(πt*,γτ*)=(C-x0eτT)2e∫0T‍(θτ)2(K~2τ(t)-2K~τ(t))dt-1.LettingVarτXT=J¯maxmin(πt*,γτ*). The optimal value of γ and the minimum variance VarXT are
(64)VarXT=Max{VarrXT,VarRXT}=Max{(C-x0erT)2e∫0T‍(θr)2(K~2r(t)-2K~r(t))dt-1,(C-x0eRT)2e∫0T‍(θR)2(K~2R(t)-2K~R(t))dt-1},(65)γ*={γr*,ifVarrXT=VarXT,γR*,ifVarRXT=VarXT.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
(66)πt*={-(μ-r)k2St2β(Xt-γ*e-r(T-t))K~r(t),ifXt⩾γ*ρ(t),r<μ<2r,-(μ-R)k2St2β(Xt-γ*e-R(T-t))K~R(t),ifXt<γ*ρ(t),R<μ<2R.
Moreover, the efficient frontier is given by
(67)VarXT=Max{(C-x0erT)2e∫0T‍(θr)2(K~2r(t)-2K~r(t))dt-1,(C-x0eRT)2e∫0T‍(θR)2(K~2R(t)-2K~R(t))dt-1},
where Kτ(t) and ρ(t) are given by (48) and (50), respectively.Remark 7.
Whenβ=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.



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