The CEV Model and Its Application in a Study of Optimal Investment Strategy

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.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Aiyin Wang 1, 2 and Ls Yong 1 and Yang Wang 1 and Xuanjun Luo 1

Academic Editor:Shouming Zhong

1, The School of Finance, Southwestern University of Finance and Economics, Chengdu 610074, China
2, Department of Mathematics, Xinjiang University of Finance and Economics, Urumqi 830012, China

Received 2 June 2014; Revised 7 July 2014; Accepted 22 July 2014; 13 August 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The constant elasticity of variance ( C E V ) model is a natural extension of the geometric Brownian motion ( G B M ). The Merton's model and its extensions [1-3] were generally studied under the assumption that risky asset price was described by the G B M . The C E V model is originally proposed by Cox and Ross [4] as an alternative diffusion process for European option pricing [5-8]. Comparing with the G B M , we know that the advantages of the C E V 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 C E V 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 C E V model was applied to study the optimal investment strategy for a defined contribution ( D C ) pension plan. However, the application of the C E V model to the investment and consumption problem has not been widely reported in the existing academic articles.

In this paper, we introduce the C E V model into an investment and consumption problem and optimally allocate the wealth between one risk-free asset and one risky asset. We use C E V 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 ( H J B ) 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 C E V model is used to study the optimal investment strategy for pension fund, we use C E V model to study personal optimal portfolio with consumption factors.

This paper is organized as follows. We introduce the C E V model about the personal portfolio in Section 2. In Section 3, we turn our stochastic optimal problem into a corresponding Hamilton-Jacobi-Bellman ( H J B ) 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 C A R A 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 C E V 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 t by _{B t} , which satisfies the following formula: [figure omitted; refer to PDF] where r is a constant rate of interest.

We denote the price of the risky asset (hereinafter called "stock") at time t by _{S t} , which satisfies the C E V model. Consider [figure omitted; refer to PDF] where μ is an expected instantaneous return of the stock and satisfies the general condition μ > β . k ^{S t β} is the instantaneous volatility and β is the elasticity parameter and satisfies the general condition β < 0 . _{W t} : t ...5; 0 is a standard Brownian motion defined on a complete probability space ( Ω , F , P ) , where P is the risk-neutral probability. The filtration F = { _{F t} } is a right continuous filtration of sigma-algebras on this space.

For personal portfolio, we denote personal consumption at time t by _{λ t} which is described by an arithmetic Brownian motion as follows: [figure omitted; refer to PDF] where _{M t} is a standard Brownian motion, a and b are constants. Generally, b is written as a function of t and _{λ t} , that is, b = b ( _{λ t} , t ) , and a is written as a function of t . Here, for simplicity, a and b are regarded as constants. Assume that the Brownian motion _{W t} is correlated with _{M t} and the correlation coefficient is ρ , that is, d _{W t} d _{M t} = ρ .

Let _{V t} denote one's disposable wealth at time t ∈ [ 0 , T ] , _{π t} and let 1 - _{π t} denote the proportion of one's disposable wealth _{V t} invested in the risky asset and the risk-free asset, respectively. The dynamical equation of personal portfolio is given by [figure omitted; refer to PDF] To seek optimal investment strategy _{π t} , we maximize the expected utility of the terminal wealth [figure omitted; refer to PDF] where U ( · ) is increasing and concave.

3. The Hamilton-Jacobi-Bellman Equation

In this section, we obtain the form of optimal investment strategy _{π t} 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 [figure omitted; refer to PDF] where A is the set of admissible trading strategies. Then, the wealth process becomes Markov process. Consider [figure omitted; refer to PDF] We define the value function [figure omitted; refer to PDF] Using the Markov property and the law of iterated expectation, we know that ^{H π} ( t , s , v , λ ) is a martingale. The value function H ( t , s , v , λ ) = _{sup ... π t ∈ A} ^{H π} ( t , s , v , λ ) is also a martingale.

Using It o formula, we have [figure omitted; refer to PDF]

Considering the equation H ( T , s , v , λ ) = U ( _{V T} ) , we have the following Hamilton-Jacobi-Bellman ( H J B ) equation associated with the optimization problem [figure omitted; refer to PDF] where _{H t} , _{H s} , _{H v} , _{H λ} , _{H s s} , _{H λ λ} , _{H v v} , _{H s v} , _{H v λ} , and _{H s λ} denote partial derivatives of first and second orders with respect to time, stock price, wealth, and consumption.

Differentiating (10) with respect to π , we have [figure omitted; refer to PDF]

The optimal strategy ^{π t *} is [figure omitted; refer to PDF] Putting (12) into (10), we obtain a partial differential equation for the value function H [figure omitted; refer to PDF]

4. The Legendre Transform and Dual Theory

Definition 1.

Let f : ^{R n} [arrow right] R be a convex function. For z > 0 , we write the Legendre transform [figure omitted; refer to PDF] The function L ( z ) is called the Legendre dual of the function f ( x ) (see [16]).

If f ( x ) is strictly convex, the maximum in the above equation will be attained at just one point, which is denoted by _{x 0} . It arrives at the unique solution to the first-order condition, namely, [figure omitted; refer to PDF] Therefore, we write [figure omitted; refer to PDF]

According to Definition 1, we take advantage of the convexity of value function H ( t , s , λ , v ) to define the Legendre transform [figure omitted; refer to PDF] where z > 0 denotes the dual variable to v . The value of v where this optimum is attained is denoted by g ( t , s , λ , z ) . Namely, [figure omitted; refer to PDF]

Using (16) and (17), we have [figure omitted; refer to PDF] From (18) and (4), the function H ^ related to g is given by g = - _{H ^ z} . Therefore, we take one of the two functions g and H ^ as the dual of H . Here, we work mainly with the function g , as it is easy to be computed numerically for the purpose of computing optimal strategies.

Differentiating (18) and (4) with respect to t , s , and z , we know that the transformation rules for the derivatives of the value function H and the dual function H ^ are given by (see [13, 16]) [figure omitted; refer to PDF]

At the terminal time T , we define [figure omitted; refer to PDF] Kramkov and Schachermayer [18] and Cox and Huang [19] have shown that the functions U ^ ( z ) and U ( v ) can be obtained from each other by using the Legendre transform as follows: [figure omitted; refer to PDF] Thus, the primary problem is turned into a dual problem.

Putting (21) into (13) yields [figure omitted; refer to PDF]

Differentiating g ( t , s , λ , z ) = _{H ^ z} on both sides with respect to t , s , λ , and z , we have the following first order and second order partial derivatives: [figure omitted; refer to PDF] Therefore, differentiating (24) for H ^ with respect to z and using the above derivatives, we derive [figure omitted; refer to PDF]

Thus, the optimal strategy (12) denoted by g is rewritten as [figure omitted; refer to PDF]

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

5. An Approximation Solution for CARA Utility

Consider the C A R A utility [figure omitted; refer to PDF] where q is the coefficient of absolute risk aversions. We have [figure omitted; refer to PDF] We try to find a solution for (26) in the following form: [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] Substituting the above derivatives into (26), we obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF] We decompose (32) into the three equations [figure omitted; refer to PDF] We state that m and n 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 λ vary slowly and a small parameter [varepsilon] ( 0 < [varepsilon] ...a; 1 ) satisfy [figure omitted; refer to PDF] Substituting (37) into (36), namely, replacing b and a with [varepsilon] b and [varepsilon] a , respectively in (36), we know that the corresponding ^{h [varepsilon]} in (36) is rewritten in the form [figure omitted; refer to PDF]

Theorem 2.

If a solution to (38) takes in the form [figure omitted; refer to PDF] then the approximation solutions to (36) in the slow-fluctuating regime are ^{h ( 0 )} ( t , y , λ ) , ^{h ( 1 )} ( t , y , λ ) , and ^{h ( 2 )} ( t , y , λ ) . Moreover, [figure omitted; refer to PDF] where _{p 0} = ( _{m t} - r m ) / m , _{q 0} = ^{k 2} β ( 2 β + 1 ) ^{e - ( ( m ( μ - r ) + μ ) / k m β ) y} ; [figure omitted; refer to PDF] where _{p 1} = ( _{m t} - r m ) / m , _{q 1} = ^{k 2} β ( 2 β + 1 ) ^{e - ( ( ( μ - r ) m + μ ) / k m β ) y} + ( μ - r ) a ρ q y / k m ; [figure omitted; refer to PDF] where _{p 2} = _{( m t} - r m ) / m , _{q 2} = ^{k 2} β ( 2 β + 1 ) ^{e ( ( μ - m ( μ - r ) ) / m k ) y} + ( μ - r ) a ρ q y / m k - b q / m - ( 1 / 2 ^{m 2} ) ^{a 2} ( 1 - ^{ρ 2} ) ^{q 2} .

Proof.

Substituting (39) into (38), we derive [figure omitted; refer to PDF] Collecting the same order of the terms, we obtain the following three equations.

Zero-order term: [figure omitted; refer to PDF]

: term of order [varepsilon] : [figure omitted; refer to PDF]

: term of order [varepsilon] [figure omitted; refer to PDF]

Considering the boundary condition, we will obtain solutions to (44)-(46).

(i) From (44), we have [figure omitted; refer to PDF] We decompose (47) into two equations [figure omitted; refer to PDF] From (48), we obtain ^{h y ( 0 )} . It is [figure omitted; refer to PDF]

Putting (50) into (49), we obtain [figure omitted; refer to PDF] where _{p 0} = ( _{m t} - r m ) / m , _{q 0} = ^{k 2} β ( 2 β + 1 ) ^{e - ( ( m ( μ - r ) + μ ) / k m β ) y} .

(ii) From (45), we have [figure omitted; refer to PDF]

We decompose (52) into two equations [figure omitted; refer to PDF]

From (53), we obtain ^{h y ( 1 )} . It is [figure omitted; refer to PDF]

Putting ^{h y ( 1 )} into (54), we obtain the solution [figure omitted; refer to PDF] where _{p 1} = ( _{m t} - r m ) / m , _{q 1} = ^{k 2} β ( 2 β + 1 ) ^{e - ( ( ( μ - r ) m + μ ) / k m β ) y} + ( μ - r ) a ρ q y / k m .

(iii) From (46), we have [figure omitted; refer to PDF] We decompose (57) into the following two equations [figure omitted; refer to PDF] From (58), we obtain ^{h y ( 2 )} . It is [figure omitted; refer to PDF] Putting ^{h y ( 2 )} into (59), we get the solution [figure omitted; refer to PDF] where _{p 2} = ( _{m t} - r m ) / m , _{q 2} = ^{k 2} β ( 2 β + 1 ) ^{e ( μ - m ( μ - r ) / m k ) y} + ( μ - r ) a ρ q y / m k - b q / m - ( 1 / 2 ^{m 2} ) ^{a 2} ( 1 - ^{ρ 2} ) ^{q 2} . Therefore, (51), (56), and (61) are the solutions to (44)-(46) which are the solutions to (36).

Considering [figure omitted; refer to PDF] from (27), (51), (56), and (61), we have [figure omitted; refer to PDF]

When [varepsilon] [arrow right] 0 and according to (50) and (51), we have [figure omitted; refer to PDF]

6. Conclusion

In the optimal investment strategies, we consider personal wealth with consumption invested in risk-free asset and risky asset. We apply the C E V 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 ( H J B ) 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.

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