Optimal Investment Problem with Multiple Risky Assets under the Constant Elasticity of Variance (cev) Model

This paper studies the optimal investment problem for utility maximization with multiple risky assets under the constant elasticity of variance (CEV) model. By applying stochastic optimal control approach and variable change technique, we derive explicit optimal strategy for an investor with logarithmic utility function. Finally, we analyze the properties of the optimal strategy and present a numerical example.


Introduction
Optimal investment problem of utility maximization is a fundamental problem in mathematical finance and has been studied in many articles.This problem is usually studied via two approaches in literatures.One is stochastic control approach used by Merton [1,2] for the first time.By this approach, Browne [3] found the optimal investment strategy to maximize the expected exponential utility of terminal wealth for an insurance company.Yang and Zhang [4] studied a similar problem for an insurer with exponential utility via stochastic control approach.Another method is the martingale approach which was adapted to the problem of utility maximization by Pliska [5], Karatzas, Lehoczky and Shreve [6] and Cox and Huang [7].Much of this development appeared in [8,9].Applying the martingale approach, Karatzas et al. [10] investigated the utility maximization problem in an incomplete market and Zhang [11] considered a similar problem.In [12], closed-form strategies were obtained for different utilities maximization of an insurer through martingale approach.Zhou [13] applied the martingale approach to study the exponential utility maximization for an insurer in the Lévy market.
The above mentioned researches using the martingale method have provided results for general risky assets' prices, but most found specific solutions for geometric Brownian motion (GBM) model or a similar one merely.Meanwhile the works applying stochastic control theory generally supposed the risky assets' prices satisfy geometric Brownian motions.However, numerous studies (see e.g., [14] and the references therein) have shown that empirical evidence does not support the assumptions of GBM model and a model with stochastic volatility will be more practical.
The constant elasticity of variance (CEV) model with stochastic volatility is a natural extension of geometric Brownian motion and can explain the empirical bias exhibited by the GBM model, such as volatility smile.The CEV model allows the volatility to change with the underlying price and was first proposed by Cox and Ross [15].In comparison with other stochastic volatility models, the CEV model is easier to deal with analytically and the GBM model can be seen as its special case.The CEV model was usually applied for option pricing and sensitivity analysis of options in most literatures, see [16][17][18][19] for example.Recently, the CEV model has been applied in the research of optimal investment, as was done by Xiao,Zhai and Qin [20].Gao [21,22] investigated the utility maximization problem for a participant in a defined-contribution pension plan under the CEV model.Gu, Yang, Li and Zhang [23] used the CEV model for studying the optimal investment and reinsurance problems.
However, the above researches of optimization problem under the CEV model concerned only one risky asset and a risk-free asset.But actually, an investor needs to invest in multiple risky assets to disperse risk and increase his/her profit.Thus, to make the optimization problem even more realistic, we deal with the investment problem with a risk-free asset and multiple risky assets under the CEV model.Although Zhao and Rong [24] have studied portfolio selection problem with multiple of the stocks are described by the CEV model risky assets under the CEV model, they obtained closedform solutions only for special model parameters.Whereas in this paper, considering to maximize the expected logarithmic utility of an investor's terminal wealth, we derive optimal strategy explicitly for all values of the elasticity coefficient.By applying the methods of stochastic optimal control, we derive a complicated nonlinear partial differential equation (PDE).However, there are terms that contain variables concerning different assets' prices, which makes it difficult to characterize the solution structure.Therefore, we conjecture a corresponding solution to this PDE via separating variables partially and simplify it into several PDEs.The coefficient variables of these simplified PDEs are closely correlated and therefore we use a power transformation and a variable change technique to solve them.
It is noteworthy that the introduction of multiple risky assets does give rise to difficulties and this research is not a routine extension of the case of one risky asset.For portfolio selection problems concerning risky assets with the CEV price processes, the characterization of solution under dimensional case is quite different from one dimensional case.Owing to the consideration of multiple risky assets, we conjecture the solution through separating variables represented different assets' prices and combining each price variable with time variable respectively.
r Furthermore, we compare our result with that under the GBM model and that of one dimensional case.Firstly, the optimal policy for an investor with logarithmic utility under the CEV model is similar to that under the GBM model in form except for a stochastic volatility.Secondly, our solution is just the result of [20] when there is only one risky asset.Moreover, we present a numerical simulation to analyze the properties of the optimal strategy under the CEV model.This paper proceeds as follows.Section 2 proposes the utility maximization problem with multiple risky assets whose prices are driven by the CEV models and provides the general framework to solve the optimization problem.In Section 3, we derive the optimal investment strategy for logarithmic utility function and compare our result with the previous works.Section 4 provides a numerical analysis to illustrate our results.Section 5 concludes the paper.

Problem Formulation
We consider a financial market consisting of a risk-free asset (hereinafter called "bond") and risky assets (hereinafter called "stocks").The price process of the bond follows where is the interest rate.The price processes where .
t is an augmented filtration generated by the Brownian motion with T , where T is a fixed and finite time horizon. is the appreciation rate of the i th stock and .Define and , the volatility matrix is constant with respect to the stock prices and Equation ( 2) reduces to the standard Black-Scholes model.In addition, we assume that The investor is allowed to invest in those stocks as well as in the bond.Let i be the money amount invested in the i th stock at time for i n.

Denote by 1 n
and each    -predictable process for .Corresponding to a trading strategy and an initial capital V , the investor's wealth process where is an vector.
n Suppose that the investor has a utility function U which is strictly concave and continuously differentiable on   By applying the classical tools of stochastic optimal control, we define the value function as , , , , , The Hamilton-Jacobi-Bellman (HJB) equation associated with the portfolio selection problem under the CEV model is where , whose th component is 1.Differentiating with respect to in Equation ( 6) gives the optimal policy Putting Equation ( 7) into HJB Equation ( 6), after simplification, we have The problem now is to solve the nonlinear partial dif-ferential equation (PDE) (8) for H and recover from derivatives of * π H .

Optimal Strategy for the Logarithmic Utility
In this paper, we consider the investment problem for logarithmic utility function A solution to Equation ( 8) is conjectured in the following form: , , , , = ln , , with the boundary conditions given by . Then = ln , = . Plugging these derivatives into Equation (8) gives In order to eliminate the dependence on .
x , we can split Equation ( 11) into two equations: For Equation ( 12), we use a power transform and a va 2 and = riable change technique proposed by Cox [16] to solve it.Let with the bounda Bringing these derivatives into Equation ( 12), we ob We conjecture a solution to Equation ( 15) in the follo with the boundary conditions given by utting Equation E get: Again to eliminate the , =1, , Taking into account the boundary conditi tio ons, the soluns to Equations ( 18) and ( 19) are Subsequently, we have the following the rithmic ut orem.Theorem 1.The optimal strategy for the loga ility maximization with multiple stocks under the CEV model is given by and the value function is given by , ,


Proof.Equations ( 7) and (10) leads to . Due to Equations ( 14), ( 16) ), we have for each According to Equations ( 14), ( 16) and (20), we ob .This together with Equations ( 10) and ( 13) imcompletes the proof.Remark 2. From Equation ( optimal investment proportion     * π t X t is independent of the wealth.This can ed by the relative risk tolerance , which is a constant for logarithmic wealth has no influence on the optimal proportion invested in stocks. Remark 3.For a logarithmic utility function, the op utility.Thus, the under a geom timal policy under the CEV model is similar to that etric Brownian motion (GBM) model.How-

 
In this section, we provide so analyze the properties of the trate the dynamic behavior of the optimal strategy.We assume that there are two stocks and one bond in the market during the time horizon = 10 T (years = 2 .Throughout the numerical analysis, we use the optimal proportion invested in stoc time t , i.e.,   * X t to denote the optimal strategy.Let on the optimal strategies.As expected, the optimal proportion invested in stock 1 increases with respect to its appreciation rate 1  .Since multiple stocks are considered, we can analyze the impact of one stock on the other stock.From Figure 1, we find that there is an inverse relationship between 1  and the optimal strategy of stock 2.This is consistent with intuition.When the appreciation rate of one stock increases constantly, it is optimal to increase the proportion of wealth in this stock and reduce investment in the other stock.Furthermore, Figure 1 also shows that the total proportion invested in two stocks changes moderately with 1  .This implies that the influence of one stock's appreciation rate on the total investment is not obvious. In Figure 2, the parameters are given by:   fluctuates with the overall tendency of optimal proportion invested in ure 2(b) also indicates that sometimes it is optimal to sell short stock 1.On the contrary, the optimal strategy of stock 2 increases in general due to the rising tendency of its price.Conse-quently, the total proportion invested in stocks is relatively steady over time.

Conclusion
By considering multiple risky assets and a risk-free asset in a financial market, this paper extends the port der the constant elasticity of varil.We propose the framework of port- folio selection problem un ance (CEV) mode folio selection problem with multiple risky assets under the CEV model.Explicit solution for the logarithmic utility maximization has been derived via stochastic control approach.It is shown that for portfolio selection problems concerning risky assets with the CEV price processes, there are differences in solution characterization and calculations between one dimensional case and n dimensional case.The optimal policy under the CEV model is the same as that under the GBM model in form except for a stochastic volatility matrix.Finally, numerical results demonstrate the properties of the multidimenonal optimal strategy under the CEV model.


denotes the adjoint matrix of here  , ij  is the element of  in the i th row and th column and j T  T represents the determinant of the matrix   

Figure 1
Figure1shows the effects of the appreciation rate 1 

1 Figure 1 .
Figures 2(a) and (b) plot the evolution of the stocks' prices and the optimal strategy over time under the CEV model, respectively.Unlike the GBM model, the optimal proportion invested in each stock under the CEV model

Figure 2 .
Figure 2. (a) Evolution of stocks' prices over time; (b) Evolution of optimal strategy over tim .

e
stocks' prices.As shown in Figure 2(b), over time (see Figure 2(a)).Moreover, Fig stock 1 decreases with respect to time.This is because that the actual price of stock 1 has a decreasing trend under grant ts' innovation training fountainty: The Continuous-Time Case," The Review of Economics and Sta , pp. 247-257.doi:10.2307/19