Abstract
We consider a continuous-time mean-variance portfolio selection model when stock price follows the constant elasticity of variance (CEV) process. The aim of this paper is to derive an optimal portfolio strategy and the efficient frontier. The mean-variance portfolio selection problem is formulated as a linearly constrained convex program problem. By employing the Lagrange multiplier method and stochastic optimal control theory, we obtain the optimal portfolio strategy and mean-variance efficient frontier analytically. The results show that the mean-variance efficient frontier is still a parabola in the mean-variance plane, and the optimal strategies depend not only on the total wealth but also on the stock price. Moreover, some numerical examples are given to analyze the sensitivity of the efficient frontier with respect to the elasticity parameter and to illustrate the results presented in this paper. The numerical results show that the price of risk decreases as the elasticity coefficient increases.
1. Introduction
The classical mean-variance portfolio selection model, which was first proposed by Markowitz [1], is to minimize the variance of the terminal wealth subject to archiving a given mean return level in a single-period investment. This model has been widely used in financial industry and become the foundation of modern finance theory. Since the pioneer work of Markowitz, the mean-variance portfolio selection model has inspired literally hundreds of extensions and applications. One of the mainstreaming researches is extending the standard portfolio selection model to the dynamic case (see, e.g., [2–5] and the references therein). In 2000, by employing the framework of multiobjective optimization and an embedding technique, Li and Ng [6] firstly derive the exact mean-variance efficient frontier in multiperiod investment; Zhou and Li [7] used the embedding technique and linear-quadratic (LQ) optimal control theory to solve the continuous-time mean-variance problem with stocks price described by geometric Brownian motion (GBM). Recently, by exploiting the stochastic control theory, there has been a series of papers discussing the continuous-time mean-variance portfolio selection problem in different markets (see [8–14] and the references therein).
In most of the continuous-time mean-variance models, the volatility of stock price is always assumed to be independent of stock price. However, many empirical evidences (see, e.g., [15–17]) have revealed that the volatility depends on the stock price. The CEV process, which is a stochastic volatility model and a natural extension of GBM, is capable of describing this dependence. This process was originally proposed by Cox [18] to describe the stock price for European option pricing. Afterwards, owning to its ability to capture the implied volatility smile which was observed across a wide range of markets and risky assets, the CEV process was widely used to analyze the options and asset pricing formula, as was done by Beckers [15] and Davydov and Linetsky [19]. Recently, the CEV process has drawn an increasing attention in optimal investment selection. Xiao et al. [20] and Gao [21] studied the DC pension investment with the CEV risk price process and derived the optimal policy under different utility functions. Gu et al. [22] derived the optimal reinsurance-investment decision under the CRRA or CARRA utility function when stock price follows the CEV process.
Due to the support of many empirical evidences, it is clear that the introduction of CEV process will make the mean-variance model more practical. However, to the best of our knowledge, there is no paper discussing the applications of CEV processes in mean-variance investment. In this paper, we assume that the stock price follows the CEV process and try to find a mean-variance optimal portfolio strategy and efficient frontier. At first, we introduce an unconstrained stochastic control problem and derive the optimal control strategies in closed form by solving Hamilton-Jacobi-Bellman (HJB) equation. In the second, following the ideas of Lim and Zhou [10], we use the Lagrange multiplier technique to derive both the mean-variance optimal policy and the efficient frontier. Finally, we give some numerical examples to analyze the sensitivity of efficient frontier with respect to the elasticity coefficient and to illustrate the results obtained in this paper. The results in this paper show that the efficient frontier is still of a perfect square and the optimal portfolio strategy depends not only on the total wealth but also on the stock price. Moreover, the price of risk decreases as the elasticity coefficient increases.
This paper proceeds as follows. In Section 2, we give some preliminaries, formulate a continuous-time mean-variance portfolio selection model with the CEV process, and introduce an unconstrained problem. In Section 3, we derive the optimal strategies for the unconstrained problem. Section 4 presents the optimal investment strategies and efficient frontier. Finally, Section 5 gives a numerical example to analyze the sensitivity of efficient frontier with respect to the elasticity coefficient and to illustrate the results presented in this paper and Section 6 concludes this paper.
2. Problem Formulation
Throughout this paper, letbe a fixed terminal time and a fixed filtered complete probability space on which a standard-adapted 1-dimensional Brownian motionis defined. It is assumed that. We denote bythe class of all-adapted,-valued stochastic processes such thatand bythe set of all functions with continuous partial derivatives,,,,, and.
Now we consider a financial market consisting of two assets, where one is bond and the other is stock (we can also treat it as equity index). The bond pricesatisfies the following ordinary differential equation: where the interest rateis a constant. The stock priceis described by the CEV model (see Yuen et al. [17]) whereis an expected instantaneous rate of return on the stock,is the instantaneous volatility, andis the elasticity parameter. We assume throughout thatandsatisfy the general conditions; that is,and.
Remark 1. Note that the CEV process covers a number of well-known processes. When, it reduces to the GBM; when, it becomes the absolute diffusion process; when, it becomes the square root process proposed by Cox and Ross [23].
Suppose that the trading of shares takes place continuously in a self-financing fashion (i.e., there is no consumption or income) and that there are no transaction costs. We denote bythe total wealth of an investor at time and bythe total stock value of the investor. Then We calla portfolio of the investor. If, that means the investor is short selling in stock; otherwise, that means the investor is borrowing the amountat rate. For a prescribed mean terminal wealth, mean-variance portfolio problem is to determine a portfoliominimizing the variance of the terminal wealth, that is, minimizing For the risk-free portfolio, the wealth processsatisfies and and has, for solution,. This implies that the investor should expect a mean return above. Thus, we give the following natural assumption:
Definition 2. A portfoliois said to be admissible if. In this case, one callsan admissible pair, whereis the solution to (3).
By Definition 2, the mean-variance problem can be formulated as follows: where.
An admissible portfoliois said to be a feasible portfolio for (6) if it satisfies the constraints in (6) and then (6) is said to be feasible.
Definition 3. An admissible portfoliois called an efficient portfolio if it is the optimal strategy for (6). The pairis called an efficient point, whereis the wealth process corresponding to the efficient portfolio, whereas the set of all efficient points is called an efficient frontier.
It is clear that (6) is a linearly constrained convex program problem. Thus, it can be reduced to an unconstrained problem by introducing a Lagrange multiplier. So, in Section 3, we will consider the following unconstrained problem parameterized by, and solve it by using HJB approach. Furthermore, in Section 4, we employ the results in Section 3 and the Lagrange multiplier method to obtain the mean-variance efficient portfolio and efficient frontier.
3. Solutions to the Unconstrained Problem
In this section, we consider deriving the optimal solution for the unconstrained problem (7). In order to solve it, at first, we establish the HJB equation for (7). In the next, we try to solve the HJB equation via power transformation and variable change method. Finally, we derive the optimal strategy and optimal value function in closed form for (7).
Now we define the value function and the variational operator
By using stochastic optimal control theory, we obtain the following result.
Theorem 4 (verification theorem). Letsatisfy the following HJB equation: Then. Moreover, if there existssuch that for almost, whereis the solution of (3) when, then andis the optimal control of (7).
Proof. By applying Itô’s formula, we have
Then, the solution of (13) is given by
Since the last term of (14) is a martingale with zero expectation, taking expectation on both sides of (14) yields
For any, (10) implies
for all.
From (10), (15), and (16), we get
and so
When, (16), (17), and (18) become equalities; that is, , which meansis the optimal control of problem (7). This completes the proof.
By Theorem 4, the HJB equation for the optimal value functionis as follows: for all, with the boundary condition
Thus, we are now in the position to solve the above HJB equation.
Suppose that (19) and (20) have a solutionwith. Then (19) attains its minimum at Putting (21) in (19), we obtain a partial differential equation (PDE) for the value functionas follows: with Inspired by Gao [21], we conjecture a solution to (22) with the following form: where the boundary conditions are given byand. Thus, wheredenotes the derivative of. Submitting these derivatives into (22), we have Therefore,andshould satisfy the following differential equations: Taking into account the boundary condition, the solution of (27) is as follows:
In order to solve the nonlinear second-order partial differential equation (28), we employ the power transformation and variable change technique which were proposed by Cox [18]. Let with the boundary condition. Then Putting those partial derivatives in (28), we know thatsatisfies the following equation: Suppose that the solution of (32) is in the following form: with the boundary conditionsand. It follows from (32) and (33) that By matching coefficients, we derive Taking into account the boundary conditions, we get the solutions to (35) and (36) as follows (see the Appendix for proof details): where
From the above discussions, we get the solution to the HJB equations (19) and (20).
Proposition 5. The solution to the HJB equations (19) and (20) is given as follows: whereandare given by (38) and (37), respectively.
Proof. We need to verify that the assumptionis true. Noting that
we only need to verify thatholds for all.
It is obvious thatwhen. When, let
Then it is easy to know that
and sois decreasing inwith
Thus,when. Similarly,holds when.
Now putting (29), (30), and (33) in (24), it follows that (41) is true. This completes the proof.
Finally, from Theorem 4, we obtain the optimal strategy and optimal value for (7) in the following result.
Theorem 6. The optimal value for unconstrained problem (7) is given by (41) and the optimal policy is whereis given by (37) andis the solution to (3) when.
Proof. It follows from Theorem 4 that (41) is the optimal value for (7). From (21), (24), (29), (30), (33), (37), and (38), we know that (46) is the optimal policy of unconstrained problem (7). This completes the proof.
Remark 7. If we takeas a certain predefined benchmark and measure deviations fromby the squared distance, then (7) becomes a benchmark portfolio selection problem (see [24, 25] and the references therein).
Remark 8. Taking, it follows from (41) that which is the efficient frontier for the benchmark portfolio selection problem. By Theorem 6, we know thatdefined by (46) is the optimal portfolio strategy for the benchmark portfolio selection problem.
4. Efficient Portfolios and Efficient Frontier
In this section, we try to derive the efficient frontier and the optimal portfolio strategy for (6) by using the results in the above section and the Lagrange multiplier method. On the feasibility of (6), we have the following result.
Theorem 9. For any prescribed mean target, problem (6) is feasible.
Proof. For the mean target, we consider the following admissible portfolio: Putting (48) in (3) and taking expectation on both sides of (3), we conclude thatsatisfies the following ordinary differential equation: Solving (49), we can get the explicit expression ofas follows: This completes the proof.
Theorem 9 implies that the investor can archive any prescribed mean target. The following result will show the risk that the investor has to take.
Theorem 10. If,, andare given by (33), (38), and (37), respectively, then where. The efficient frontier of problem (6) exists and is given by where. The optimal portfolio associated with the mean returnis as follows: where
Proof. It is easy to verify that the mean-variance problem (6) has a convex cost which is bounded below and a convex constrained set, which means that (6) is a linearly constrained convex problem. Since the problem (6) is feasible, it follows from Luenberger [26] that
For any fixed, the unconstrained problem
is equivalent to
in the sense thatsolves (56) if and only ifsolves (57) and. Therefore, it follows from Theorem 6 that
and the optimal policy of (57) is
Becauseis quadratic inandis finite, we can get that. In fact, if, thencan only be finite whenfor all, which is a contradiction. So it must be the case that.
Thus, we can get the optimalfor (55) as follows:
Puttingin (58) and (59), we obtain that
This completes the proof.
Remark 11. Theorem 10 shows that the efficient frontier (52) is a parabola and tells the risk that the investor has to bear to archive a prescribed mean target. In particular, if the investor does not want to take any risk, namely,, then the expectation of his/her terminal wealth must be, which implies that the optimal strategy of the investor is investing his/her total money in the bond.
Letbe the standard deviation of the terminal wealth. It follows from (52) that
Hence, the efficient frontier in the mean-standard deviation plane is a straight line, which is also called the capital market line. The slope of this line,, is termed the price of risk.
Before obtaining the efficient frontier with, we give the following proposition.
Proposition 12. Consider whereis given by (37) and functionis given by (33).
Proof. In the case of,
whereandare given by (39) and (40), respectively. Putting (40) in (39), we get
From l’Hôspital’s rule, we get that
Thus,
which means that (63) is true when.
Let
Then it follows from (70) that
By (71), we have
Finally, it follows from (68) and (72) that
that is, (64) holds when.
We can prove other two cases in a similar way. So, we omit it here.
This completes the proof.
From Theorem 10 and Proposition 12, it is very easy to get the following result.
Corollary 13. When, the mean-variance efficient frontier and efficient portfolio are given by respectively.
Remark 14. In a financial market consisting of a risk-free bond and a stock, the results of Corollary 13 are the same as that shown in Zhou and Li [7].
5. Numerical Analysis
In this section, we give some numerical examples to analyze the sensitivity of the mean-variance efficient frontier with respect to the elasticity coefficient and to illustrate the dynamic behaviors of the mean-variance optimal portfolio strategy and the total wealth. The basic parameters in the model are given by,,,, and, which are the same as that in Yuen et al. [17] who estimated the CEV model parameters for Hong Kong stock option market. We consider an investor with an initial endowmentmillion and wishing to achieve an expected return ofin three years; that is,million.
5.1. Sensitivity Analysis of the Efficient Frontier
From the efficient frontier in Theorem 10, it is hard to find what role the elasticity coefficientplays. Thus, we try to examine the relations between the efficient frontier and the elasticity coefficient. Now, we consider the following five cases:,,,, and. Then, the efficient frontiers under the five cases are given, respectively, by
Figures 1 and 2 present the efficient frontiers in mean-variance plane and mean-standard deviation plane, respectively.
Similarly, the efficient frontiers are plotted in Figures 3 and 4 for the case with,, andand in Figures 5 and 6 for another case with,, and.
Clearly, efficient frontiers in mean-variance plane are increasing and concave with respect to the variance of the terminal wealth. Moreover, from Figures 1, 3, and 5, we can also observe that the expected return decreases as the elasticity coefficient increases under the same risk level or the risk increases as the elasticity coefficient increases under the same mean profit level. Figures 2, 4, and 6 show that the efficient frontier in mean-standard deviation is a straight line and the price of risk (which is the slope of this line) increases as the elasticity coefficientdecreases.
5.2. The Optimal Strategy and the Total Wealth
According to Theorem 10, we carry out 600 simulations to get the evolutions of the stock price, the optimal strategy, and total wealth. Figure 7 presents the evolution of stock price.
Putting the basic parameters andin (37) and (38), respectively, and taking, we get Thus,and the price of risk is. From (52), we get that the standard deviation of the investor’s goal ismillion, which implies that the minimized standard deviation is as high as.
Then, we get the investor’s optimal portfolio. By the definition of, we obtain. Thus, the amount that investor should invest in the stock is which is a function of time, stock price, and the wealth. In particular, at the initial time, the investor should investmillion in the stock and the rest of his initial endowment in the bond. The dynamic behaviors of the optimal portfolio strategy and total wealth are plotted in Figure 8.
Similarly, the evolutions of the stock price, the optimal strategy, and total wealth are plotted in Figures 9 and 10 for the case with,, andand in Figures 11 and 12 for the case with,, and.
6. Conclusion
In this paper, we employ the CEV process to describe the dynamic evolution of the stock price in the mean-variance portfolio selection problem. The results in this paper show that the mean-variance efficient frontier is still a parabola in the mean-variance plane and the optimal strategies are not independent of the stock price. Moreover, under the same risk level, the expected return decreases as the elasticity coefficient increases.
Our results show differences after comparing with some well-known results. Firstly, in most of the papers (see, e.g., [7–9, 12]) where stock price is described by GBM, the optimal mean-variance portfolio is independent of stock price, while our optimal investment strategy depends on the stock price, which is caused by the fact that the volatility of CEV price process depends on stock price. Secondly, although there are many discussions in mean-variance model with stochastic volatility (see, e.g., [10, 11]), the optimal portfolio is given in terms of the solutions of backward stochastic differential equations, which is difficult to apply in practice. However, our results are completely determined and therefore can be easily applied in practice.
Appendix
Proof Details for Solving (35) and (36)
For the sake of simplicity, we define Then (35) can be rewritten as Integrating (A.2) on both sides with respect to the time, we have whereis a constant.
When, we have and so the quadratic equation has two different real solutions; namely, Then, we get Putting (A.7) in (A.3) and taking into account the boundary condition, we get where
When, we have and so (A.5) has a unique real solution: Then, we get Putting (A.13) in (A.3) and taking into account the boundary condition, we get
When, we have and so (A.5) has no real solution. Thus, it follows that Putting (A.16) in (A.3) and taking into account the boundary condition, we get Putting (A.8), (A.14), and (A.17) in (36), respectively, together with the boundary condition, we have whereandare given by (A.10).
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author is grateful to two anonymous referees for their comments and suggestions. This work was supported by the National Natural Science Foundation of China (11171237 and 71101099) and by the Construction Foundation of Southwest University for Nationalities for the Subject of Applied Economics (2011XWD-S0202).