Optimal Investment and Consumption Decisions under the Constant Elasticity of Variance Model

Abstract

We consider an investment and consumption problem under the constant elasticity of variance (CEV) model, which is an extension of the original Merton’s problem. In the proposed model, stock price dynamics is assumed to follow a CEV model and our goal is to maximize the expected discounted utility of consumption and terminal wealth. Firstly, we apply dynamic programming principle to obtain the Hamilton-Jacobi-Bellman (HJB) equation for the value function. Secondly, we choose power utility and logarithm utility for our analysis and apply variable change technique to obtain the closed-form solutions to the optimal investment and consumption strategies. Finally, we provide a numerical example to illustrate the effect of market parameters on the optimal investment and consumption strategies.

1. Introduction

In the classical Merton’s portfolio optimization problems [1, 2], an investor dynamically allocates his wealth between one risk asset and one risk-free asset and chooses an optimal consumption rate to maximize total expected discounted utility of consumption. But in the Merton’s model, there are no transaction costs and borrowing constraints and no-shorting constraints. Since the pioneer work of Merton, the investment and consumption problems have inspired literally hundreds of extensions and applications. For example, introducing transaction costs into the investment and consumption problems, one can refer to Shreve and Soner [3], Akian et al. [4], and Janeček and Shreve [5]. Some authors as well investigated the optimal consumption problem with borrowing constraints; see Fleming and Zariphopoulou [6], Vila and Zariphopoulou [7], and Yao and Zhang [8]. However, the above mentioned models generally were studied under the assumption that risky asset price dynamics was driven by a geometric Brownian motion (GBM).

The constant elasticity of variance (CEV) model is a natural extension of the GBM. Compared with the GBM, the advantages of the CEV model are that the volatility rate has correlation with the risky asset price and can explain volatility smile. The CEV model was originally proposed by Cox and Ross [9] as an alternative diffusion process for European option pricing. The CEV model was usually applied to analyze the option pricing formula, see; for example, Schroder [10], Lo et al. [11], Phelim and Yisong [12], and Davydov and Linetsky [13]. In the recent years, the CEV model has been introduced into annuity contracts and the optimal investment strategies in the utility framework are investigated by applying dynamic programming principle. For more detailed discussion, one can refer to Xiao et al. [14], Gao [15, 16], Gu et al. [17], Lin and Li [18], Gu et al. [19], Jung and Kim [20] and Zhao and Rong [21]. However, the application of the CEV model to an investment and consumption problem has not been reported in the existing academic articles.

In this paper, we introduce a CEV model into an investment and consumption problem and optimally allocate the wealth between one risk-free asset and one risky asset, whose price process is supposed to follow a CEV model. Our goal is to maximize the expected discounted utility of consumption and terminal wealth. Dynamic programming principle is applied to obtain the HJB equation for the value function. Owing to the introducing of consumption factor and the CEV model, the HJB equation derived is much more difficult to deal with than the one obtained by Gao [15]. Inspired by the techniques of Gao [15] and Liu [22], we transform the nonlinear second-order partial differential equation into a linear one, which is easy to tackle. In the techniques of tackling the CEV model, one of the most important innovations in this paper is to suppose that the structure of the solution to (22) is of the expression form (23) and prove that (22) and (25) are equivalent. Secondly, we derive the closed-form solutions to the optimal investment and consumption strategies for power utility and logarithm utility by applying variable change technique. Finally, we propose a numerical example to illustrate the properties and sensitivities of the optimal investment and consumption strategy on market parameters. There are three main contributions in this paper: (i) we first consider an investment and consumption problem under a CEV process and obtain the closed-form solutions of the optimal investment and consumption strategies in the power and logarithm utility cases; (ii) we use the same approach as Liu [22] to solve (22), which is very difficult to solve directly; (iii) we provide a numerical example to illustrate our results.

The rest of this paper is organized as follows. In Section 2, we formulate the financial market and propose the optimization problem. In Section 3, we apply dynamic programming principle to obtain the HJB equation and investigate the optimal investment and consumption strategies in the power and logarithm utility cases. Section 4 provides a numerical example and Section 5 concludes the paper.

2. Problem Formulation

In this section, we propose the problem formulation of optimal investments and consumption decisions with a CEV process.

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 satisfies 𝑑𝑃_𝑡=𝑟𝑃_𝑡𝑑𝑡,𝑃⁰₀=𝑝₀>0, (1) where the constant 𝑟>0 is the interest rate of the bond.

The other asset is a stock with prices 𝑆_𝑡 at time 𝑡, whose price process 𝑆_𝑡 is given by the following constant elasticity of variance model (CEV):𝑑𝑆_𝑡=𝑆_𝑡𝜇𝑑𝑡+𝑘𝑆^𝛾_𝑡𝑑𝑊_𝑡,𝑆₀=𝑠₀>0, (2) where 𝜇(𝜇>𝑟) is an expected instantaneous return rate of the stock. 𝑘 and 𝛾 are constant parameters and the elasticity parameter 𝛾 satisfies the general condition: 𝛾⩽0. 𝑘𝑆^𝛾_𝑡 is defined as the instantaneous volatility of the stock. 𝑊_𝑡 is a one-dimensional standard adapted Brownian motion defined on a filtered complete probability space (Ω,ℱ,𝑃,{ℱ_𝑡}_{0⩽𝑡⩽𝑇}), where {ℱ_𝑡}_{0⩽𝑡⩽𝑇} is a 𝜎-algebra generated by Brownian motion 𝑊_𝑡.

Remark 1. Noting that there are four special interpretations for the elasticity parameter 𝛾: (i) if 𝛾=0, the CEV model is reduced to a geometric Brownian motion (GBM); (ii) if 𝛾=−1, it is the Ornstein-Uhlenbeck process; (iii) if 𝛾=−1/2, it is the model first presented by Cox and Ross [9] as an alternative diffusion process for valuation of options; (iv) if 𝛾<0, this means that the instantaneous volatility 𝑘𝑆^𝛾_𝑡 increases as the stock price decreases and can generate a distribution with a fatter left tail.

Assume that the investor invests the market value of his wealth 𝜋_𝑡 into the stock at time 𝑡, 𝑡∈[0,𝑇]. Clearly, the amount invested in the bond is 𝑋_𝑡−𝜋_𝑡, in which 𝑋_𝑡 represents the wealth of the investor at time 𝑡. Suppose that short-selling of stocks and borrowing at the interest rate of the bond are allowed and transaction cost is not taken into consideration. We also introduce the consumption rate denoted by 𝐶_𝑡. The wealth process 𝑋_𝑡 corresponding to trading strategy (𝜋_𝑡,𝐶_𝑡) is subject to the following equation: 𝑑𝑋_𝑡=𝑋_𝑡−𝜋_𝑡𝑑𝑃_𝑡𝑃_𝑡+𝜋_𝑡𝑑𝑆_𝑡𝑆_𝑡−𝐶_𝑡𝑑𝑡; (3) namely, we have 𝑑𝑋_𝑡=𝑟𝑋_𝑡+𝜇−𝑟𝜋_𝑡−𝐶_𝑡𝑑𝑡+𝜋_𝑡𝑘𝑆^𝛾_𝑡𝑑𝑊_𝑡,𝑋₀=𝑥₀>0. (4)

Definition 2 (admissible strategy). An investment and consumption (𝜋_𝑡,𝐶_𝑡) is admissible if the following conditions are satisfied:(i)(𝜋_𝑡,𝐶_𝑡) is ℱ_𝑡-progressively measurable and ∫^𝑇₀𝜋²_𝑡𝑑𝑡<∞, ∫^𝑇₀𝐶_𝑡𝑑𝑡<∞, a.s. ∀𝑇>0;(ii)𝐸[∫^𝑇₀(𝜋_𝑡𝑘𝑆^𝛾_𝑡)²𝑑𝑡]<∞;(iii)For ∀(𝜋_𝑡,𝐶_𝑡), stochastic differential equation (4) has a unique solution.Assume that the set of all admissible investment and consumption strategies (𝜋_𝑡,𝐶_𝑡) is denoted by Γ={(𝜋_𝑡,𝐶_𝑡)∶0⩽𝑡⩽𝑇}. Mathematically, an investor wishes to maximize the following objective function: Maximize_{𝜋𝑡,𝐶𝑡∈Γ}𝔼𝛼^𝑇₀𝑒^−𝛽𝑡𝑈₁𝐶_𝑡𝑑𝑡+(1−𝛼)𝑒^−𝛽𝑇𝑈₂𝑋_𝑇, (5) where utility function 𝑈(⋅) is assumed to be strictly concave and continuously differentiable on (−∞,+∞) and 𝛽 is the subjective discount. The parameter 𝛼 determines the relative importance of the intermediate consumption and the terminal wealth. Since 𝑈(⋅) is strictly concave, there exists a unique optimal trading strategy (𝜋_𝑡,𝐶_𝑡) which maximizes (5).
In this paper, we consider two-type utility functions 𝑈(⋅). One is power utility function defined by 𝑈(𝑥)=𝑥^𝜂/𝜂, 𝜂<1 and 𝜂≠0. The other is logarithm utility function given by 𝑈(𝑥)=ln𝑥.

3. The Closed-Form Solution

In this section, we apply dynamic programming principle to derive the HJB equation for the value function and investigate the optimal investment and consumption policies for problem (5) in the power and logarithm utility cases.

The value function is defined as 𝐻(𝑡,𝑠,𝑥)=sup_{𝜋𝑡,𝐶𝑡∈Γ}𝔼𝛼^𝑇₀𝑒^−𝛽𝑡𝑈₁𝐶_𝑡+()𝑒𝑑𝑡1−𝛼^−𝛽𝑇𝑈₂𝑋_𝑇∣𝑆_𝑡=𝑠,𝑋_𝑡=𝑥, (6) with boundary condition given by 𝐻(𝑇,𝑠,𝑥)=(1−𝛼)𝑒^−𝛽𝑇𝑈₂(𝑥).

Using dynamic programming principle, one can get the HJB equation as follows: sup_{𝜋𝑡,𝐶𝑡∈Γ}𝜕𝐻+𝜋𝜕𝑡𝜇−𝑟_𝑡+𝑟𝑥−𝐶_𝑡𝜕𝐻+1𝜕𝑥2𝜋_𝑡𝑘𝑠^𝛾2𝜕²𝐻𝜕𝑥²+𝜇𝑠𝜕𝐻+1𝜕𝑠2𝑘𝑠^𝛾+12𝜕²𝐻𝜕𝑠²+𝜋_𝑡𝑘²𝑠^2𝛾+1𝜕²𝐻𝜕𝑥𝜕𝑠+𝛼𝑒^−𝛽𝑡𝑈₁𝐶_𝑡=0, (7) where 𝜕𝐻/𝜕𝑡, 𝜕𝐻/𝜕𝑥, 𝜕²𝐻/𝜕𝑥², 𝜕𝐻/𝜕𝑠, 𝜕²𝐻/𝜕𝑠², and 𝜕²𝐻/𝜕𝑥𝜕𝑠 denote first-order and second-order partial derivatives with respect to the variables 𝑡, 𝑠, and 𝑥, respectively.

The first-order maximizing conditions for the optimal value is given by 𝜋^∗_𝑡=−𝜇−𝑟(𝑘𝑠^𝛾)²⋅𝜕𝐻/𝜕𝑥𝜕²𝐻/𝜕𝑥²𝜕−𝑠⋅²𝐻/𝜕𝑥𝜕𝑠𝜕²𝐻/𝜕𝑥²,𝑈^′𝐶^∗_𝑡=𝜕𝐻/𝜕𝑥𝛼𝑒^−𝛽𝑡. (8) Introducing (8) into (7), we obtain 𝜕𝐻𝜕𝑡+𝑟𝑥𝜕𝐻𝜕𝑥+𝜇𝑠𝜕𝐻+1𝜕𝑠2𝑘𝑠^𝛾+12𝜕²𝐻𝜕𝑠²−12⋅𝜇−𝑟𝜕𝐻/𝜕𝑥+𝑘²𝑠^2𝛾+1𝜕²𝐻/𝜕𝑥𝜕𝑠²𝑘²𝑠^2𝛾𝜕²𝐻/𝜕𝑥²−𝐶^∗_𝑡𝜕𝐻𝜕𝑥+𝛼𝑒^−𝛽𝑡𝑈₁𝐶^∗_𝑡=0. (9)

Here, we notice that the stochastic control problem has been transformed into a nonlinear second-order partial differential equation; yet it is difficult to solve it. In the following subsection, we choose power utility and logarithm utility for our analysis, respectively, and try to obtain the closed-form solutions to (9).

3.1. Power Utility

Power utility function is defined as𝑈₁(𝑥)=𝑈₂(𝑥)=𝑥^𝜂𝜂,𝜂<1,𝜂≠0. (10) For (9), we can conjecture a solution with the following structure: 𝐻(𝑡,𝑠,𝑥)=𝑒^−𝛽𝑡𝑥^𝜂𝜂𝑓(𝑡,𝑠),𝑓(𝑇,𝑠)=1−𝛼. (11) Then 𝜕𝐻𝜕𝑡=−𝛽𝑒^−𝛽𝑡𝑥^𝜂𝜂𝑓(𝑡,𝑠)+𝑒^−𝛽𝑡𝑥^𝜂𝜂⋅𝜕𝑓,𝜕𝑡𝜕𝐻𝜕𝑥=𝑒^−𝛽𝑡𝑥^𝜂−1𝜕𝑓,²𝐻𝜕𝑥²=𝜂−1𝑒^−𝛽𝑡𝑥^𝜂−2𝑓,𝜕𝐻𝜕𝑠=𝑒^−𝛽𝑡𝑥^𝜂𝜂⋅𝜕𝑓,𝜕𝜕𝑠²𝐻𝜕𝑠²=𝑒^−𝛽𝑡𝑥^𝜂𝜂⋅𝜕²𝑓𝜕𝑠²,𝜕²𝐻𝜕𝑠𝜕𝑥=𝑒^−𝛽𝑡𝑥^𝜂−1𝜕𝑓.𝜕𝑠 (12) Therefore, (8) is rewritten as 𝜋^∗_𝑡=1⋅1−𝜂𝜇−𝑟(𝑘𝑠^𝛾)²𝑠𝑥+⋅1−𝜂𝜕𝑓/𝜕𝑠𝑓𝐶𝑥,_∗𝑡=𝛼^1/(1−𝜂)𝑓^{−1/(1−𝜂)}𝑥. (13)

Putting these partial derivatives and the optimal policy (13) into (9), we get 𝑒^−𝛽𝑡𝑥^𝜂𝜂𝜕𝑓+𝜕𝑡𝜂𝑟−𝛽𝑓+𝜇𝑠𝜕𝑓+1𝜕𝑠2𝑘²𝑠^2𝛾+2𝜕²𝑓𝜕𝑠²−𝜂2𝜂−1𝜇−𝑟𝑘𝑠^𝛾2𝜂𝑓−𝑠𝜂−1𝜇−𝑟𝜕𝑓−𝜂𝜕𝑠2𝑘𝜂−1²𝑠^2𝛾+2𝜕𝑓/𝜕𝑠𝑓+1−𝜂𝛼^1/(1−𝜂)𝑓^{𝜂/(𝜂−1)}=0. (14) Eliminating the dependence on 𝑥, we obtain 𝑓_𝑡+𝜂𝑟−𝛽𝑓+𝜇𝑠𝑓_𝑠+12𝑘²𝑠^2𝛾+2𝑓_𝑠𝑠−𝜂2𝜂−1𝜇−𝑟𝑘𝑠^𝛾2𝜂𝑓−𝜂−1𝜇−𝑟𝑠𝑓_𝑠−𝜂2𝑘𝜂−1²𝑠^2𝛾+2𝜕𝑓/𝜕𝑠𝑓+1−𝜂𝛼^1/(1−𝜂)𝑓^{𝜂/(𝜂−1)}=0. (15)

Inspired by the approach of Gao [15], we can use the following power transform and variable change technique. So letting 𝑓(𝑡,𝑠)=𝑔𝑡,𝑦,𝑦=𝑠^−2𝛾, (16) we get 𝜕𝑓=𝜕𝑡𝜕𝑔,𝜕𝑡𝜕𝑓=𝜕𝑠𝜕𝑔𝑠𝜕𝑦−2𝛾^−2𝛾−1,𝜕²𝑓𝜕𝑠²=𝜕²𝑔𝜕𝑦²4𝛾²𝑠^−4𝛾−2+𝜕𝑔𝑠𝜕𝑦−2𝛾−2𝛾−1^−2𝛾−2. (17) Introducing these derivatives into (15), we obtain 𝜕𝑔+𝜂𝜕𝑡𝜂𝑟−𝛽𝑔−2𝜂−1𝜇−𝑟𝑘²+𝜂𝑦𝑔𝑦𝜂−12𝛾𝜇−𝑟−2𝛾𝜇𝜕𝑔𝑘𝜕𝑦+𝛾2𝛾+1²𝜕𝑔𝜕𝑦+2𝛾²𝑘²𝑦𝜕²𝑔𝜕𝑦²−2𝜂𝛾𝜂−1²𝑘²𝑦𝜕𝑔/𝜕𝑦²𝑔+1−𝜂𝛼^{−1/(𝜂−1)}𝑔^{𝜂/(𝜂−1)}=0,𝑔𝑇,𝑦=1−𝛼. (18)

In addition, we use the following variable change technique. Assume that 𝑔𝑡,𝑦=ℎ𝑡,𝑦^1−𝜂,ℎ𝑇,𝑦=(1−𝛼)^1/(1−𝜂). (19) Then 𝜕𝑔=ℎ𝜕𝑡1−𝜂^−𝜂𝜕ℎ,𝜕𝑡𝜕𝑔=ℎ𝜕𝑦1−𝜂^−𝜂𝜕ℎ,𝜕𝜕𝑦₂𝑔𝜕𝑦²=1−𝜂−𝜂ℎ^−𝜂−1𝜕ℎ𝜕𝑦²+1−𝜂ℎ^−𝜂𝜕²ℎ𝜕𝑦². (20)

Substituting these derivatives back into (18), we have 1−𝜂ℎ^−𝜂𝜕ℎ+𝜕𝑡𝜂𝑟−𝛽+𝜂1−𝜂2𝜂−1²𝜇−𝑟𝑘²𝑦ℎ+𝜂𝑦𝜂−12𝛾𝜇−𝑟−2𝛾𝜇𝜕ℎ𝑘𝜕𝑦+𝛾2𝛾+1²𝜕ℎ𝜕𝑦+2𝛾²𝑘²𝑦𝜕²ℎ𝜕𝑦²+𝛼^1/(1−𝜂)=0. (21) So we obtain the following partial differential equation: 𝜕ℎ+𝜕𝑡𝜂𝑟−𝛽+𝜂1−𝜂2𝜂−1²𝜇−𝑟𝑘²𝑦ℎ+𝜂𝑦𝜂−12𝛾𝜇−𝑟−2𝛾𝜇𝜕ℎ𝑘𝜕𝑦+𝛾2𝛾+1²𝜕ℎ𝜕𝑦+2𝛾²𝑘²𝑦𝜕²ℎ𝜕𝑦²+𝛼^1/(1−𝜂)ℎ=()=0,𝑇,𝑦1−𝛼^1/(1−𝜂). (22)

Noting that (22) has been a linear second-order partial differential equation, it is still very difficult to solve it directly. Inspired by the approach proposed by Liu [22], we try to fit a solution to (22) and we have the following Lemma.

Lemma 3. Assume that ℎ(𝑡,𝑦)=𝛼^1/(1−𝜂)∫^𝑇_𝑡ℎ(𝑢,𝑦)𝑑𝑢+(1−𝛼)^1/(1−𝜂)ℎ(𝑡,𝑦) is a solution of (22); then one can prove that ℎ(𝑡,𝑦) satisfies the equation: 𝜕ℎ+𝜕𝑡𝜂𝑟−𝛽+𝜂1−𝜂2𝜂−1²𝜇−𝑟𝑘²𝑦ℎ+𝜂𝑦𝜕ℎ𝜂−12𝛾𝜇−𝑟−2𝛾𝜇𝑘𝜕𝑦+𝛾2𝛾+1²𝜕ℎ𝜕𝑦+2𝛾²𝑘²𝑦𝜕²ℎ𝜕𝑦²=0,ℎ𝑇,𝑦=1. (23)

Proof. Define differential operator ∇ on any function ℎ(𝑡,𝑦) by ∇ℎ𝑡,𝑦=2𝛾²𝑘²𝑦𝜕²ℎ𝜕𝑦²+𝜂𝑦𝜂−12𝛾𝜇−𝑟−2𝛾𝜇𝜕ℎ𝑘𝜕𝑦+𝛾2𝛾+1²𝜕ℎ+𝜕𝑦𝜂𝑟−𝛽+𝜂1−𝜂2𝜂−1²𝜇−𝑟𝑘²𝑦ℎ. (24) Then (22) can be rewritten as 𝜕ℎ𝑡,𝑦𝜕𝑡+∇ℎ𝑡,𝑦+𝛼^1/(1−𝜂)ℎ=()=0,𝑇,𝑦1−𝛼^1/(1−𝜂). (25)
On the other hand, we find that 𝜕ℎ𝑡,𝑦𝜕𝑡=−𝛼^1/(1−𝜂)ℎ𝑡,𝑦+(1−𝛼)^1/(1−𝜂)𝜕ℎ𝑡,𝑦𝜕𝑡=𝛼^{1/(1−𝜂)𝑇}_𝑡𝜕ℎ𝑢,𝑦ℎ+()𝜕𝑢𝑑𝑢−𝑇,𝑦1−𝛼^1/(1−𝜂)𝜕ℎ𝑡,𝑦,𝜕𝑡∇ℎ𝑡,𝑦=𝛼^{1/(1−𝜂)𝑇}_𝑡∇ℎ+()𝑢,𝑦𝑑𝑢1−𝛼^1/(1−𝜂)⋅∇ℎ𝑡,𝑦. (26) Further, (25) is reduced to 𝛼^{1/(1−𝜂)𝑇}_𝑡𝜕ℎ𝑢,𝑦ℎℎ+()𝜕𝑢+∇𝑢,𝑦𝑑𝑢−𝑇,𝑦+11−𝛼^1/(1−𝜂)𝜕ℎ𝑡,𝑦ℎ𝜕𝑡+∇𝑡,𝑦=0. (27)
Then we obtain 𝜕ℎ𝑡,𝑦ℎℎ𝜕𝑡+∇𝑡,𝑦=0,𝑇,𝑦=1. (28)
Therefore, (23) holds.

Lemma 4. Suppose that a solution to (23) is of the form ℎ(𝑡,𝑦)=𝑒^{𝐴(𝑡)+𝐵(𝑡)𝑦}, with terminal conditions 𝐴(𝑇)=0 and 𝐵(𝑇)=0; then 𝐴(𝑡) and 𝐵(𝑡) are given by (31) and (32), respectively.

Proof. Putting ℎ(𝑡,𝑦)=𝑒^{𝐴(𝑡)+𝐵(𝑡)𝑦} into (23) yields ℎ𝑦(𝑡)𝑡,𝑦𝑑𝐵𝑑𝑡+2𝛾²𝑘²𝐵²(𝑡)+𝜂𝐵(𝑡)+1𝜂−12𝛾𝜇−𝑟−2𝛾𝜇2𝜂𝜂−1²𝜇−𝑟𝑘²+𝑑𝐴(𝑡)+𝑑𝑡𝜂𝑟−𝛽𝑘1−𝜂+𝛾2𝛾+1²𝐵(𝑡)=0. (29) We can decompose (29) into two equations in order to eliminate the dependence on 𝑦:𝑑𝐵(𝑡)𝑑𝑡+2𝛾²𝑘²𝐵²(𝑡)+𝜂𝐵(𝑡)+1𝜂−12𝛾𝜇−𝑟−2𝛾𝜇2⋅𝜂𝜂−1²𝜇−𝑟𝑘²(𝑇)(𝑡)=0,𝐵=0;𝑑𝐴+𝑑𝑡𝜂𝑟−𝛽𝑘1−𝜂+𝛾2𝛾+1²𝐵(𝑡)=0,𝐴(𝑇)=0. (30)
Solving the above two equations by using the same approach as Gao [15], we obtain𝐴(𝑡)=𝜆₁𝛾2𝛾+1𝑘²+𝜂𝑟−𝛽)𝜆1−𝜂(𝑇−𝑡+ln²−𝜆₁𝜆₂−𝜆₁𝑒^{−2𝛾2𝑘2(𝜆1−𝜆2)(𝑇−𝑡)(2𝛾+1)/2𝛾},𝐵(𝑡)=𝑘⁻²𝐼(𝑡), (31)(32) where 𝐼(𝑡)=𝜆₁𝜆₂1−𝑒^{−2𝛾2𝑘2(𝜆1−𝜆2)(𝑇−𝑡)}𝜆₂−𝜆₁𝑒^{−2𝛾2𝑘2(𝜆1−𝜆2)(𝑇−𝑡)},𝜆_1,2=𝜇−𝑟𝜂±1−𝜂𝜇²−𝑟²𝜂2𝛾𝑘².1−𝜂 (33)
As a result, Lemma 4 is completed.

Taking 𝑓(𝑡,𝑠)=𝑔(𝑡,𝑦)=ℎ(𝑡,𝑦)^1−𝜂 and their relationships into consideration, we derive 𝜕𝐻/𝜕𝑥𝜕²𝐻/𝜕𝑥²1=−𝜕1−𝜂𝑥,₂𝐻/𝜕𝑥𝜕𝑠𝜕²𝐻/𝜕𝑥²=𝑥⋅𝜂−1𝜕𝑓/𝜕𝑠𝑓=𝑥⋅𝑠𝜂−1−2𝛾^−2𝛾−1𝜕𝑔/𝜕𝑦𝑔=2𝛾𝑠^−2𝛾−1𝜕ℎ/𝜕𝑦ℎ𝑥. (34)

Therefore, the optimal investment strategy is 𝜋^∗_𝑡=1⋅1−𝜂𝜇−𝑟𝑘𝑆^𝛾_𝑡²𝑋_𝑡−2𝛾𝑆_𝑡^−2𝛾𝜕ℎ/𝜕𝑦ℎ𝑋_𝑡. (35)

So, we can summarize the above results in the following theorem.

Theorem 5. If utility function is given by 𝑈₁(𝑥)=𝑈₂(𝑥)=𝑥^𝜂/𝜂,  𝜂<1 and 𝜂≠0, the optimal investment and consumption strategy of the problem (5) is 𝜋^∗_𝑡=1⋅1−𝜂𝜇−𝑟𝑘𝑆^𝛾_𝑡²𝑋_𝑡𝑘1−2𝛾1−𝜂²⋅𝜇−𝑟𝜕ℎ/𝜕𝑦ℎ,𝐶^∗_𝑡=𝛼^1/(1−𝜂)ℎ⁻¹𝑋_𝑡, (36) where ℎ=ℎ(𝑡,𝑦)=𝛼^1/(1−𝜂)∫^𝑇_𝑡𝑒^{𝐴(𝑡)+𝐵(𝑡)𝑦}𝑑𝑡+(1−𝛼)^1/(1−𝜂)𝑒^{𝐴(𝑡)+𝐵(𝑡)𝑦}, 𝑦=𝑆_𝑡^−2𝛾, and 𝐴(𝑡) and 𝐵(𝑡) are determined by (31) and (32), respectively.

Remark 6. Noting that a CEV model includes several existing stochastic processes as special cases, for example, geometric Brownian motion (GBM), Ornstein-Uhlenbeck process, and an alternative diffusion process investigated by Cox and Ross [9], the closed-form expression (36) obtained is the general framework of the optimal investment and consumption strategies when stock price dynamics is given by the above stochastic processes.

Remark 7. It can be seen from (36) that the optimal investment strategy 𝜋^∗_𝑡 can be decomposed into two terms. One term is (1/(1−𝜂))⋅((𝜇−𝑟)/(𝑘𝑆^𝛾_𝑡)²)𝑋_𝑡, which has an analogical form of the optimal policy under a GBM model. The other term is 1−((2𝛾(1−𝜂)𝑘²)/(𝜇−𝑟))⋅((𝜕ℎ/𝜕𝑦)/ℎ), which can be called as modification factor denoted by 𝑀(𝑡). In addition, we notice that two terms are influenced by stock price 𝑆_𝑡.
In order to compare our results with those in the existing literature, we discuss several special cases for Theorem 5.

Special Case  1. If 𝛾→0, the CEV model is reduced to a GBM. In addition, we obtain 𝑦=1 and 𝐵(𝑡)=𝜂2𝜂−1²𝜇−𝑟𝑘²(),𝐴(𝑡)=𝑇−𝑡𝜂𝑟−𝛽().1−𝜂𝑇−𝑡 (37) Letting 𝜂𝜃=2𝜂−1²𝜇−𝑟𝑘²+𝜂𝑟−𝛽,1−𝜂 (38) then we have ℎ𝑡,𝑦=ℎ(𝑡)=𝛼^1/(1−𝜂)⋅1𝜃𝑒^{𝜃(𝑇−𝑡)}+()−11−𝛼^1/(1−𝜂)𝑒^{𝜃(𝑇−𝑡)}, (39) and (𝜕ℎ/𝜕𝑦)/ℎ=0. Therefore, (36) is reduced to 𝜋^∗_𝑡=1⋅1−𝜂𝜇−𝑟𝑘²𝑋_𝑡,𝐶^∗_𝑡=𝛼^1/(1−𝜂)ℎ⁻¹(𝑡)𝑋_𝑡, (40) where ℎ(𝑡) is given by (39).

Special Case  2. If 𝛼→0, then ℎ(𝑡,𝑦)=𝑒^{𝐴(𝑡)+𝐵(𝑡)𝑦} and (𝜕ℎ/𝜕𝑦)/ℎ=𝐵(t). As a result, the optimal consumption policy is 𝐶^∗_𝑡=0 and the optimal investment strategy is given by 𝜋^∗_𝑡=1⋅1−𝜂𝜇−𝑟𝑘𝑆^𝛾_𝑡²𝑋_𝑡𝐼(𝑡)1−2𝛾1−𝜂,𝜇−𝑟 (41) where 𝐼(𝑡) is given by (33). This is just the optimal policy for power utility maximizing when consumption is not considered and stock price follows a CEV process.

Special Case  3. If 𝜂→0, then 𝐵(𝑡)=0 and 𝐴(𝑡)=𝛽(𝑡−𝑇). In addition, we have ℎ=𝛼𝑡,𝑦𝛽1−𝑒^{−𝛽(𝑇−𝑡)}+(1−𝛼)𝑒^{−𝛽(𝑇−𝑡)} (42) and (𝜕ℎ/𝜕𝑦)/ℎ=0. Therefore, (36) is reduced to 𝜋^∗_𝑡=𝜇−𝑟𝑘𝑆^𝛾_𝑡²𝑋_𝑡,𝐶^∗_𝑡=𝛼𝛼/𝛽1−𝑒^{−𝛽(𝑇−𝑡)}+(1−𝛼)𝑒^{−𝛽(𝑇−𝑡)}𝑋_𝑡. (43) It is well known that power utility is degenerated to logarithm utility if 𝜂→0. Furthermore, we find that the optimal investment and consumption strategy obtained by (43) is just that under logarithm utility function.

3.2. Logarithm Utility

Logarithm utility function is defined as 𝑈₁(𝑥)=𝑈₂(𝑥)=ln𝑥. (44) We can assume that the solution to the HJB equation (9) is of the following structure: 𝐻(𝑡,𝑠,𝑥)=𝑣(𝑡)𝑒^−𝛽𝑡(),𝑣(𝑇)()ln𝑥+𝑤𝑡,𝑠=1−𝛼,𝑤𝑇,𝑠=0. (45)

Then, partial derivatives with respect to 𝑡, 𝑠, and 𝑥 are given by 𝜕𝐻𝜕𝑡=𝑣^′(𝑡)𝑒^−𝛽𝑡ln𝑥−𝛽𝑣(𝑡)𝑒^−𝛽𝑡ln𝑥+𝜕𝑤,𝜕𝑡𝜕𝐻(𝑡)𝑒𝜕𝑥=𝑣^−𝛽𝑡1𝑥,𝜕²𝐻𝜕𝑥²=𝑣(𝑡)𝑒^−𝛽𝑡−1𝑥²,𝜕𝐻=𝜕𝑠𝜕𝑤,𝜕𝜕𝑠²𝐻𝜕𝑠²=𝜕²𝑤𝜕𝑠²,𝜕²𝐻𝜕𝑠𝜕𝑥=0. (46) Substituting these partial derivatives back into (9) yields 𝜕𝑤𝜕𝑡+𝜇𝑠𝜕𝑤+1𝜕𝑠2𝑘²𝑠^2𝛾+2𝜕²𝑤𝜕𝑠²+1𝑟+2𝜇−𝑟𝑘𝑠^𝛾2𝑒^−𝛽𝑡𝑣(𝑡)+𝛼𝑒^−𝛽𝑡(ln𝛼−ln𝑣(𝑡)−1)+𝑒^−𝛽𝑡(𝑡)ln𝑥𝑑𝑣(𝑡)𝑑𝑡−𝛽𝑣+𝛼=0. (47)

We can split this equation into the following two equations: 𝑑𝑣(𝑡)(𝑡)(𝑇)𝑑𝑡−𝛽𝑣+𝛼=0,𝑣=1−𝛼;𝜕𝑤𝜕𝑡+𝜇𝑠𝜕𝑤+1𝜕𝑠2𝑘²𝑠^2𝛾+2𝜕²𝑤𝜕𝑠²+1𝑟+2𝜇−𝑟𝑘𝑠^𝛾2𝑒^−𝛽𝑡𝑣(𝑡)+𝛼𝑒^−𝛽𝑡(ln𝛼−ln𝑣(𝑡)−1)=0,𝑤(𝑇,𝑠)=0. (48)(49) Solving (48), we get 𝑣(𝑡)=(1−𝛼)𝑒^{−𝛽(𝑇−𝑡)}−𝛼𝛽𝑒^{−𝛽(𝑇−𝑡)}−1. (50)

Lemma 8. For (49), one tries to fit a solution of the form 𝑤(𝑡,𝑠)=𝐷(𝑡)+𝐸(𝑡)𝑦, 𝑦=𝑠^−2𝛾, with boundary conditions 𝐷(𝑇)=0 and 𝐸(𝑇)=0; then 𝐸(𝑡) and 𝐷(𝑡) are given by (55) and (56), respectively.

Proof. Note that 𝜕𝑤=(𝑡)𝜕𝑡𝑑𝐷+(𝑡)𝑑𝑡𝑑𝐸𝑑𝑡𝑦,𝜕𝑤(𝑡)𝑠𝜕𝑠=𝐸−2𝛾^−2𝛾−1,𝜕²𝑤𝜕𝑠²=𝐸(𝑡)−2𝛾−2𝛾−1𝑠^−2𝛾−2. (51) Introducing these derivatives into (49), we get 𝑦𝑑𝐸(𝑡)(𝑡)+1𝑑𝑡−2𝛾𝜇𝐸2𝜇−𝑟𝑘²𝑒^−𝛽𝑡𝑣(𝑡+(𝑡))𝑑𝐷𝑑𝑡+𝑟𝑒^−𝛽𝑡𝑣(𝑡)+𝛾2𝛾+1𝑘²𝐸(𝑡)+𝛼𝑒^−𝛽𝑡(ln𝛼−ln𝑣(𝑡)−1)=0. (52)
Then we can decompose this equation into two equations: 𝑑𝐸(𝑡)(𝑡)+1𝑑𝑡−2𝛾𝜇𝐸2𝜇−𝑟𝑘²𝑒^−𝛽𝑡𝑣(𝑡)(𝑇)(𝑡)=0,𝐸=0;𝑑𝐷𝑑𝑡+𝑟𝑒^−𝛽𝑡𝑣(𝑡)+𝛾2𝛾+1𝑘²𝐸(𝑡)+𝛼𝑒^−𝛽𝑡(ln𝛼−ln𝑣(𝑡)−1)=0,𝐷(𝑇)=0. (53)(54) Solving (53) and (54) yields 𝐸(𝑡)=12𝜇−𝑟𝑘²𝛽−𝛼𝛽−𝛼𝑒2𝛽𝛾𝜇^−𝛽𝑇1−𝑒^{−2𝛾𝜇(𝑇−𝑡)}+12𝜇−𝑟𝑘²𝛼𝛽𝑒𝛽+2𝛾𝜇^−𝛽𝑇𝑒^{𝛽(𝑇−𝑡)}−𝑒^{−2𝛾𝜇(𝑇−𝑡)},𝐷(𝑡)=^𝑇_𝑡𝑟𝑒^−𝛽𝑡𝑣(𝑡)𝑑𝑡+𝛾2𝛾+1𝑘^2𝑇_𝑡𝐸(𝑡)×𝑑𝑡−𝛼_𝑇𝑡𝑒^−𝛽𝑡(𝑡)𝛼()ln𝑣𝑑𝑡−ln𝛼−1𝛽𝑒^−𝛽𝑇−𝑒^−𝛽𝑡. (55)(56)
At the end, Lemma 8 is proved.

Noting that 𝜕𝐻/𝜕𝑥𝜕²𝐻/𝜕𝑥²𝜕=−𝑥,²𝐻/𝜕𝑥𝜕𝑠𝜕²𝐻/𝜕𝑥²=0, (57) we can rewrite the optimal investment and consumption strategy as 𝜋^∗_𝑡=𝜇−𝑟(𝑘𝑠^𝛾)²𝑥,𝐶^∗_𝑡=𝛼𝑣(𝑡)𝑥. (58)

Finally, we can conclude the optimal investment and consumption strategy for logarithm utility in the following theorem.

Theorem 9. If utility function is given by 𝑈₁(𝑥)=𝑈₂(𝑥)=ln𝑥, the optimal investment and consumption policy of problem (5) is 𝜋^∗_𝑡=𝜇−𝑟𝑘𝑆^𝛾_𝑡²𝑋_𝑡,𝐶^∗_𝑡=𝛼(1−𝛼)𝑒^{−𝛽(𝑇−𝑡)}−𝛼/𝛽𝑒^{−𝛽(𝑇−𝑡)}𝑋−1^𝑡. (59)

Remark 10. It can be seen from (59) that the optimal investment strategy 𝜋^∗_𝑡 is not correlated with the parameters 𝛼 and 𝛽. And when 𝛼→0, the optimal consumption policy is 𝐶^∗_𝑡=0 and the optimal portfolios are still given by 𝜋^∗_𝑡=((𝜇−𝑟)/(𝑘𝑆^𝛾_𝑡)²)𝑋_𝑡.

4. Numerical Analysis

This section provides a numerical example to illustrate the dynamic behavior of the optimal investment and consumption strategy in the power utility case. Assume that there are two assets in the financial market; one is a bond and the other is a stock. Throughout the numerical analysis, unless otherwise stated, the basic parameters are given by 𝑟=0.03, 𝜇=0.12, 𝑘=16.16, 𝛾=−1, 𝑆₀=67, 𝑡=0, 𝑇=2, 𝑥₀=100, and 𝜂=−2. Notice that the first five parameters are taken from the paper of Gao [15]; without loss of a generality, we consider the initial time 𝑡=0 and the initial wealth 𝑥₀=100.

4.1. Sensitivity Analysis for the Optimal Investment Strategy

Figures 1, 2, and 3 provide some sensitivity analysis of market parameters on the optimal investment strategy. Some results and economic implications can be summarized as follows.(a1) The optimal investment strategy 𝜋^∗_𝑡 decreases with respect to interest rate 𝑟. When risk-free interest rate is increasing, the income from investment in the risk-free asset will be more and more bigger; accordingly, the investor will reduce the investment amount in the stock in order to avoid the risk form investments. It is consistent with the intuition.(a2)𝜋^∗_𝑡 is increasing as the parameter 𝜇 becomes bigger. In fact, as the value of 𝜇 is increasing, the appreciate rate of the stock is increasing; the investor would invest more money in the stock in order to gain more revenue.(a3)𝜋^∗_𝑡 is decreasing in 𝛾. As a matter of fact, when elasticity parameter 𝛾 (under the constraints 𝛾<0 and 𝑟<𝜇) becomes larger, the instantaneous volatility 𝑘𝑆^𝛾_𝑡 of the stock will be more and more bigger, which leads to the more risk of investment. Therefore, the investor needs to reduce the amount invested in the stock. Moreover, we can draw the conclusion that the amount invested in the stock under a CEV model is more than that under a GBM model.(a4)𝜋^∗_𝑡 is increasing with the aversion factor of risk 𝜂 increasing. In reality, the aversion coefficient of risk is given by 1−𝜂 in the power utility case. It means that the bigger the value of 𝜂 is, the less the risk aversion of the investor is. It leads to that the investor would invest more money in the stock in order to get more income.(a5)𝜋^∗_𝑡 increases with respect to stock price 𝑠₀. It is completely different from the optimal investment strategy under a geometric Brownian motion. From Special Case 1, we can get that the optimal policy under a GBM is not correlated with stock price. In fact, under the CEV model, the instantaneous volatility 𝑘𝑆^𝛾_𝑡 of the stock decreases when stock price is increasing. So the risk of investments is decreasing; accordingly, the investor has more willing to invest more money in the stock in order to get more wealth.(a6)𝜋^∗_𝑡 has opposite trend with the value of 𝑘. Actually, the larger the value of 𝑘 is, the bigger the instantaneous volatility 𝑘𝑆^𝛾_𝑡 of the stock is, which leads to the more risk of investment. Therefore, the investor should invest less money in the stock.

(a)

(b)

Figure 1 

The impact of 𝑟 and 𝜇 on 𝜋∗𝑡.

(a)

(b)

Figure 2 

The impact of 𝛾 and 𝜂 on 𝜋∗𝑡.

(a)

(b)

Figure 3 

The impact of 𝑠0 and 𝑘 on 𝜋∗𝑡.

4.2. Sensitivity Analysis for the Optimal Consumption Strategy

From Figures 4, 5, and 6, we can draw the following conclusions.(b1) The optimal consumption strategy 𝐶^∗_𝑡 is increasing in interest rate 𝑟. In fact, the bigger the value of risk-free interest rate 𝑟 is, the more the amount invested in the risk-free asset is, and meantime the less the amount in the stock is. But the total expected wealth of the investor will become more and more larger. Therefore, The amount the investor can consume will increase accordingly.(b2)𝐶^∗_𝑡 has same trend with the appreciation rate of the stock 𝜇. It implies that when the value of 𝜇 is increasing, the amount invested in the stock is rising, which leads to that the expected wealth of an investor will increase as well. Therefore, the optimal consumption amount will become more and more bigger.(b3)𝐶^∗_𝑡 is decreasing in 𝛾. This displays that the bigger the value of elasticity parameter 𝛾 is, the less the optimal consumption amount is. As a matter of fact, when the value of 𝛾 becomes larger, the amount invested in the stock is decreasing, which leads to that the expected wealth of an investor will become less. So the amount to consume will decrease.(b4)𝐶^∗_𝑡 increases with respect to the risk aversion factor 𝜂. According to the above analysis, when the value of 𝜂 is increasing, the amount invested in the stock is also increasing, which results in that the expected wealth of an investor is rising meantime. Therefore, an investor can consume more money.(b5)𝐶^∗_𝑡 is increasing as the value of 𝛼 is rising. When the parameter 𝛼 becomes larger, the relative importance of consumption will become bigger between consumption and terminal wealth. Therefore, an investor must add the amount to consume.(b6)𝐶^∗_𝑡 has the same trend with discount factor 𝛽. It implies that the larger the value of 𝛽 is, the more the amount to consume is.

(a)

(b)

Figure 4 

The impact of 𝑟 and 𝜇 on 𝐶∗𝑡.

(a)

(b)

Figure 5 

The impact of 𝛾 and 𝜂 on 𝐶∗𝑡.

(a)

(b)

Figure 6 

The impact of 𝛼 and 𝛽 on 𝐶∗𝑡.

5. Conclusions

In this paper, we have investigated an investment and consumption problem, in which stock price is assumed to follow a CEV process. The CEV process is a natural extension of a geometric Brownian motion. It is one of the most important innovations. By applying dynamic programming principle and variable change technique, we derive the closed-form solutions to the optimal investment and consumption strategies in the power and logarithm utility cases. Finally, we propose a numerical example to illustrate the impact of market parameters on the optimal investment and consumption strategies and give some economic implications. Some important conclusions are found: (i) 𝜋^∗_𝑡 and 𝐶^∗_𝑡 have opposite trend in risk-free interest rate 𝑟; (ii) 𝜋^∗_𝑡 and 𝐶^∗_𝑡 have same trend in the parameters 𝜇, 𝛾, and 𝜂.

In future research on the investment and consumption problem, we focus on the cases of more sophisticated situations such as introducing transaction costs, stochastic affine interest rate, and the other uncertain factors, which would result in more sophisticated nonlinear second-order partial differential equations and cannot deal with them. We leave these points to future research.

Acknowledgments

This research is supported by National Natural Science Foundation of China (no. 11301376), Humanities and Social Science Research Youth Foundation of Ministry of Education of China (no. 11YJC790006), and Higher School Science and Technology Development Foundation of Tianjin (no. 20100821).