#### Abstract

The constant elasticity of variance (*CEV*) model is used to describe the price of the risky asset. Maximizing the expected utility relating to the Hamilton-Jacobi-Bellman (*HJB*) equation which describes the optimal investment strategies, we obtain a partial differential equation. Applying the Legendre transform, we transform the equation into a dual problem and obtain an approximation solution and an optimal investment strategies for the exponential utility function.

#### 1. Introduction

The constant elasticity of variance () model is a natural extension of the geometric Brownian motion (). The Merton’s model and its extensions [1–3] were generally studied under the assumption that risky asset price was described by the . The model is originally proposed by Cox and Ross [4] as an alternative diffusion process for European option pricing [5–8]. Comparing with the , we know that the advantages of the model are that the volatility rate has correlation with the risky asset price and can explain the empirical bias such as volatility smile [9]. The model was usually applied to calculate the theoretical price, sensitivities, and implied volatility of options [10–12]. In recent years, the optimal investment problem for a pension fund has become an important subject [13–17], in which the model was applied to study the optimal investment strategy for a defined contribution () pension plan. However, the application of the model to the investment and consumption problem has not been widely reported in the existing academic articles.

In this paper, we introduce the model into an investment and consumption problem and optimally allocate the wealth between one risk-free asset and one risky asset. We use model to describe the risk asset’s price and discuss personal optimal portfolio with consumption under the framework of the classical Merton’s portfolio optimization problems. Our goal is to choose an optimal investment strategy to maximize total expected utility of wealth. By using the method of stochastic optimal control, the corresponding Hamilton-Jacobi-Bellman () equation for the value function of the optimization problem is obtained. Applying the Legendre transform and dual theory [14, 15], we obtain an approximation solution and the optimal investment strategies for the exponential utility function. The novelty of this paper is different from those of [13–17], in which the model is used to study the optimal investment strategy for pension fund, we use model to study personal optimal portfolio with consumption factors.

This paper is organized as follows. We introduce the model about the personal portfolio in Section 2. In Section 3, we turn our stochastic optimal problem into a corresponding Hamilton-Jacobi-Bellman () equation. We transform our problem into the dual problem by applying the Legendre transform and the dual theory in Section 4. In Section 5, we derive an appriximation solution by choosing the utility. Conclusions are given in Section 6.

#### 2. The *CEV* Model

In this section, we consider that the market structure consists of a risk-free asset and a single risky asset which is described by the model. With personal consumption, the dynamical equation of personal portfolio is established.

We denote the price of the risk-free asset (i.e., the bank account) at time by , which satisfies the following formula: where is a constant rate of interest.

We denote the price of the risky asset (hereinafter called “stock”) at time by , which satisfies the model. Consider where is an expected instantaneous return of the stock and satisfies the general condition . is the instantaneous volatility and is the elasticity parameter and satisfies the general condition . is a standard Brownian motion defined on a complete probability space , where is the risk-neutral probability. The filtration is a right continuous filtration of sigma-algebras on this space.

For personal portfolio, we denote personal consumption at time by which is described by an arithmetic Brownian motion as follows: where is a standard Brownian motion, and are constants. Generally, is written as a function of and , that is, , and is written as a function of . Here, for simplicity, and are regarded as constants. Assume that the Brownian motion is correlated with and the correlation coefficient is , that is, .

Let denote one’s disposable wealth at time , and let denote the proportion of one’s disposable wealth invested in the risky asset and the risk-free asset, respectively. The dynamical equation of personal portfolio is given by To seek optimal investment strategy , we maximize the expected utility of the terminal wealth where is increasing and concave.

#### 3. The Hamilton-Jacobi-Bellman Equation

In this section, we obtain the form of optimal investment strategy and the Hamilton-Jacobi-Bellman equation of the optimal problem (5) by using the dynamic programming approach.

Define the value function for the optimal problem (5) as where is the set of admissible trading strategies. Then, the wealth process becomes Markov process. Consider We define the value function Using the Markov property and the law of iterated expectation, we know that is a martingale. The value function is also a martingale.

Using It formula, we have

Considering the equation , we have the following Hamilton-Jacobi-Bellman () equation associated with the optimization problem where , , , , , , , , , and denote partial derivatives of first and second orders with respect to time, stock price, wealth, and consumption.

Differentiating (10) with respect to , we have

The optimal strategy is Putting (12) into (10), we obtain a partial differential equation for the value function

#### 4. The Legendre Transform and Dual Theory

*Definition 1. *Let be a convex function. For , we write the Legendre transform
The function is called the Legendre dual of the function (see [16]).

If is strictly convex, the maximum in the above equation will be attained at just one point, which is denoted by . It arrives at the unique solution to the first-order condition, namely,
Therefore, we write

According to Definition 1, we take advantage of the convexity of value function to define the Legendre transform
where denotes the dual variable to . The value of where this optimum is attained is denoted by . Namely,

Using (16) and (17), we have
From (18) and (4), the function related to is given by . Therefore, we take one of the two functions and as the dual of . Here, we work mainly with the function , as it is easy to be computed numerically for the purpose of computing optimal strategies.

Differentiating (18) and (4) with respect to , , and , we know that the transformation rules for the derivatives of the value function and the dual function are given by (see [13, 16])

At the terminal time , we define
Kramkov and Schachermayer [18] and Cox and Huang [19] have shown that the functions and can be obtained from each other by using the Legendre transform as follows:
Thus, the primary problem is turned into a dual problem.

Putting (21) into (13) yields

Differentiating on both sides with respect to , , , and , we have the following first order and second order partial derivatives:
Therefore, differentiating (24) for with respect to and using the above derivatives, we derive

Thus, the optimal strategy (12) denoted by is rewritten as

We solve (26) to obtain the dual variable . Then, replacing into (27) we obtain the optimal strategy.

#### 5. An Approximation Solution for * CARA* Utility

Consider the utility where is the coefficient of absolute risk aversions. We have We try to find a solution for (26) in the following form: Then Substituting the above derivatives into (26), we obtain where We decompose (32) into the three equations We state that and are found from (34) and (35). We know that (36) is a nonlinear partial differential equation, which has not a general solution.

In order to find solutions of (36), we let vary slowly and a small parameter satisfy Substituting (37) into (36), namely, replacing and with and , respectively in (36), we know that the corresponding in (36) is rewritten in the form

Theorem 2. *If a solution to (38) takes in the form
**
then the approximation solutions to (36) in the slow-fluctuating regime are , , and . Moreover,
**
where , ;
**
where , ;
**
where , .*

*Proof. *Substituting (39) into (38), we derive
Collecting the same order of the terms, we obtain the following three equations.

Zero-order term:
term of order :
term of order Considering the boundary condition, we will obtain solutions to (44)–(46).

(i) From (44), we have
We decompose (47) into two equations
From (48), we obtain . It is

Putting (50) into (49), we obtain
where .

(ii) From (45), we have

We decompose (52) into two equations

From (53), we obtain . It is

Putting into (54), we obtain the solution
where , .

(iii) From (46), we have
We decompose (57) into the following two equations
From (58), we obtain . It is
Putting into (59), we get the solution
where . Therefore, (51), (56), and (61) are the solutions to (44)–(46) which are the solutions to (36).

Considering from (27), (51), (56), and (61), we have

When and according to (50) and (51), we have

#### 6. Conclusion

In the optimal investment strategies, we consider personal wealth with consumption invested in risk-free asset and risky asset. We apply the model to the optimal personal portfolio and obtain the dual solution by using the Legendre transform and dual theory. Maximizing the expected utility and using the corresponding Hamilton-Jacobi-Bellman equation, we obtain an approximation solution and the optimal investment strategies for the exponential utility function. The results obtained in this paper are only applicable for the exponential utility function. For other cases, it would extend this result for further research in our forthcoming paper.

#### Conflict of Interests

The authors declare that they have no conflict of interests.