An Explicit Solution for a Portfolio Selection Problem with Stochastic Volatility

In this paper, we revisit the optimal consumption and portfolio selection problem for an investor who has access to a risk-free asset (e.g. bank account) with constant return and a risky asset (e.g. stocks) with constant expected return and stochastic volatility. The main contribution of this study is twofold. Our first objective is to provide an explicit solution for dynamic portfolio choice problems, when the volatility of the risky asset returns is driven by the Ornstein-Uhlenbeck process, for an investor with a constant relative risk aversion (CRRA). The second objective is to carry out some numerical experiments using the derived solution in order to analyze the sensitivity of the optimal weight and consumption with respect to some parameters of the model, including the expected return on risky asset, the aversion risk of the investor, the mean-reverting speed, the long-term mean of the process and the diffusion coefficient of the stochastic factor of the standard Brownian motion.


Introduction
Portfolio selection is a classical problem in mathematical finance, where the main objective is to seek the best proportion of wealth to invest in risky assets in order to benefit from market opportunities. The derivation of an optimal portfolio poses a considerable challenge for market participants because they operate in an uncertain environment. The pioneer work of Markowitz [1] [2], who first introduced the so-called mean-variance (MV) model, formulated the portfolio selection problem as an optimization problem, which consists of minimizing the variance (measure of investor's risk) of the terminal wealth for a desired level of expected return. Michaud [3] found that the MV-optimized portfolios are quite unintuitive and difficult for practitioners to implement. Moreover, in a realistic setting, the Markowitz model doesn't take in account the consumption of the investor.
Another line of research in portfolio optimization is based on the Utility Theory and Expected Utility Maximization, where the preferences of an investor are described by a utility function. In this setting, the objective of the investor is to maximize the expected value of a utility function of the terminal wealth. The combined continuous-time problem of optimal portfolio selection and consumption rules was first studied by Merton [4] [5], who established the framework for dynamic portfolio choice under environmental uncertainty. Using Dynamic Programming, Merton's framework leads to a nonlinear partial differential equation (PDE), which is a general and rather complex problem, depending on the process governing the volatility; this aspect makes it difficult to find the optimal weights for the portfolio as well as the corresponding optimal consumption. Merton [4] explicitly solved the PDE under a constant volatility of the risk asset.
Recently, there has been a growth of interest in portfolio optimization problems under stochastic volatility. In [6], the authors analyzed the optimal consumption and portfolio selection problem, respectively, where the stochastic volatility is correlated with the diffusion process of the risky asset, whereas Goll and Kallsen [7] derived explicit solutions for log-optimal portfolios in complete markets in using semi-martingale characteristics of the price process. In [8], the authors established some existence and uniqueness results for the optimal investment problem, for an arbitrage-free model. Chacko and Viceira [9] obtained an exact solution in incomplete markets, when the volatility process is driven by CIR (Cox, Ingersoll and Ross) model [10]. Bae et al [11] introduced a stock market model with time-varying volatilities coupled with each other via a regime switching mechanism and a constant interaction weighting. A problem similar to the one posed by Merton was analyzed by Brennan and Xia [12] [13] [14] and Wachter [15] a well as some of the references therein. In [16], Liu established an explicit solution to a dynamic portfolio selection problem, when the returns of the risky asset are driven by a "quadratic process", which is a Markovian diffusion process and where the investor has a constant relative risk aversion (CRRA) utility function. In [17], Coulon attempted to numerically solve the Merton model using an iterative process based a finite difference scheme. However, the author acknowledged that the convergence conditions have been investigated, and the proposed algorithm required some fine-tuning of the time discretization to converge.
In this study, we reexamine the optimal consumption and portfolio selection problem for an investor who has access to a risk-free asset, e.g. bank account, with constant return and a risky asset, e.g. stocks, with constant expected return and stochastic volatility. The main contribution of the research work in this paper is twofold. First we establish an explicit solution for dynamic portfolio A. N. Sandjo et al. choice problems, when the volatility of asset returns is driven by the Ornstein-Uhlenbeck process, for an investor with CRRA. Afterwards, we carry out some sensitivity analysis of the optimal weight and consumption with respect to various parameters of the model, including the expected return on risky asset, the aversion risk of the investors, the mean-reverting speed, the long-term mean of the process and the diffusion coefficient of the stochastic factor of Brownian motion. For the derivation of the explicit solution, unlike in [16], here we provide an alternative to the tensor theory approach, thus making results more accessible to practitioners. The approach proposed in this study, provides a rigorous, relatively complete and self-contained treatment of the nonlinear PDE, as well as numerical simulations.
The outline of this paper is as follows. Section 2 presents the derivation of an explicit solution for stock portfolio problem when the stock return volatility is described by the Ornstein-Uhlenbeck model. We derive a closed-form solution for optimal portfolio selection and consumption problems for the investor with CRRA utility. Section 3 is dedicated to some sensitivity analysis of the optimal weight and consumption with respect to various parameters of the model. The effects of the financial parameters have been analyzed and economic interpretations of the optimal portfolio selection and consumption are given. The last section summarizes our findings and hints on possible improvements and future directions.

Model Formulation
In our formulation, a portfolio consists of a risk-free asset (e.g. bank account) and a risky asset (e.g. stock) whose price are driven by geometric Brownian motion.
The risk-free asset is described by where ( ) 0 S t denotes the price of one unit of the risk-free asset at time t and r is the instantaneous rate of return from the risk-free asset, and it is assumed to be constant.
The dynamics of the risky asset is driven by the following equation: where , > 0, > 0, nian motion. The parameter δ represents a long-term mean of the process, whereas κ is a value of mean-reverting speed, and σ corresponds to the diffusion coefficient of the stochastic factor ( ) dB t . The logarithms of ( ) V t were taken in order to avoid negative values of the instantaneous variance of the risky asset.
By applying Itô's Lemma to Equation (3), while setting , the proportional changes in the volatility of the Ornstein-Uhlenbeck process writes as follows: Following Chacko and Viceira [9], it is assumed that the shocks to precision, dB ν , are negatively correlated with the shocks, dB , to the return on risky asset, i.e. 0 ρ < . From Equation (4), we can deduce that the instantaneous correlation between proportional changes in variance and the return of the risky asset return is given by Applying again Itô's Lemma to Equation (3), by taking Var 1 e 1 e exp 1 exp 2 e ln 0 .
The investor starts off with an initial endowment Let us now assume that there are no transaction costs and no constraints on the structure of the portfolio. In particular, any investor in this market may instantaneously transfer funds from one account to the other and at no costs. Moreover, the investor may hold short positions of any size in both accounts. In this case, we can reparametrize the problem by introducing new variables, namely the total wealth, , and the fraction of total wealth held in The investor consumes the amount in the bank account at rate ( ) c t . Furthermore, all incomes are derived from capital gains, and the consumption is subject to the constraint that the investor must be solvent i.e. must have nonnegative net worth at all time. Under these assumptions, the market is complete, as defined in the economics literature, e.g. [18], and the investor's wealth,

( )
W t , at time t changes according to the following stochastic differential equation: Then, with initial wealth, The investor's objective is to maximize the net expected utility of consumption plus the expected utility of terminal wealth. In this study, we use a power-law utility function, which belongs to the CRRA class. The problem of optimal portfolio selection and consumption rules is then formulated as follows: subject to the budget constraint (8) and

W T
Note that the term  in (9), short for 0  , is the conditional expectation operator, given that The parameter γ , in Equation (9) is the risk aversion coefficient, which is equivalent to the inverse of the elasticity of intertemporal substitution, whereas α and λ in Equation (9) respectively denote the relative importance of the intermediate consumption and the subjective discount factor. When 0 α = , the expected utility depends solely on the terminal wealth, and the problem is then called an asset allocation problem.

Dynamic Optimization Problem
In this section, we derive expressions for optimal policies. We apply the dynamic programming principle of optimality by rewriting (9) into a dynamic programming form. For this aim, we define the indirect utility function, ( ) , , J t W V , as follows: Following Merton [4] [5], we derive the Hamilton-Jacobi-Bellman (HJB) equation for J : where the differential operator [ ] 2  2  2 2  2   2  2  2  2  2  2   , ,  , ,  , ,  :   , ,  , ,  1  ln  2  2 , , , , , 2 with boundary condition: Taking as trial solution where the function ( ) , K t V needs to be found, we can see that ( ) , , J W V t must satisfy the following partial differential equation: with boundary condition ( ) with boundary condition ( )

Exact Solution of Portfolio and Consumption Rules
In the sequel, we present our main results on an explicit solution for the optimal portfolio. The details on the derivation and the proof of the results can be found in the appendix.
It is worth mentioning that Liu [16] established a framework for a general solution of the optimal portfolio. However, in order to achieve this, some restrictions have been imposed on the dynamics of the state variables, including the assumption that these variables must follow a "quadratic process" as well as the application of the solution to the Heston's model [19]. Furthermore, although Liu's solution is in an explicit form, the abstraction of the results obtained using the tensor calculus does not seem easily accessible for the non-mathematician.
As mentioned earlier, Coulon [17] attempted an iterative process, using a finite propositions may look similar to those in [16]. This doesn't come as surprise since the Ornstein-Uhlenbeck model is a quadratic process. However, the details of the problem and the method of analysis are substantially different.
The results within the following propositions characterize the optimal consumption policy and the optimal portfolio choice. γ ≠ At time t with 0 t T ≤ ≤ , the optimal consumption policy c * is given by (19) and the optimal portfolio choice w * is given by where the function ( ) , K t V satisfies Equation (16) with the boundary condition ( ) , and , , Ξ Θ Γ and Σ are given in (17) if 0 α ≠ and where the function ( ) , K t V satisfies Equation (18) with the boundary condition ( )

α =
Note that Equation (19) and Equation (20) In [17], the author found the following result in the case of a stochastic volatility, using a similar utility function.

Proposition 3 ([17]
). The optimal management is given by F V t is a solution for the following nonlinear partial differential equation with the boundary condition ( ) , 1 F V T = , for all . V As we have mentioned earlier, without analytical solutions available, Coulon [17] had to rely on numerical approximations that necessitated a fine tuning of the discretization time. By providing explicit analytical solutions to the problem, we fill this gap. Corollary 1. Reusing notations from Propositions 2 and 3, the optimal portfolio choice * w is given by Equation (22) while the optimal consumption policy * c is given by Equation (21).

Numerical Experiments and Economic Interpretations
In this section we carry out some numerical experiments on the model and analyze the qualitative changes in the solution with respect to shifts in the financial parameters. From Equation (22) and Equation (21), the optimal weight and the optimal consumption, respectively, are not always bounded. However for practical purposes, the quantities need to be bounded. Therefore, in order to highlight the practical features of the model, we consider the following set-up for the numerical experiments, which corresponds to the case without short-selling i.e.
where ( ) d t and ( ) g t can be found in Proposition 2.

Sensitivity of the Optimal Weight and Consumption with Respect to the Expected Return on the Risky Asset
It can be observed that the higher the return of the risky asset, the greater the proportion invested in the risky asset depending on investor risk tolerance. But when the return on the risky asset is smaller than the risk free rate, it is wise to borrow and invest at the risk-free rate, see Figure 1(a) and Figure 1(b). The yields of the risky asset exert a significant impact on investors. The so-called risk-return trade-off is validated, that is the principle that potential return increases as the risk increases. In other words, low levels of uncertainty (low-risk) are associated with low potential returns, whereas high levels of uncertainty (high-risk) are associated with high potential returns. We also observe that higher expected return in the risky asset is likely to lead the consumer to increase current consumption and reduce current savings, see Figure 1(c) and

Sensitivity of the Optimal Weight and Consumption with Respect to the Return of Risk-Free Asset
The ups and downs in the return of risk-free asset are an important source of A. N. Sandjo et al.

Sensitivity of the Optimal Weight and Consumption with Respect to the Other Parameters of the Model
In this section, we investigate the impact of the other parameters of the model on the optimal weight and consumption.

Impact of the Mean-Reverting Speed κ
We observed that the greater the speed of mean reversion in volatility, the greater the proportion invested in the risky asset for an investor with some risk tolerance. This proportion is very high at beginning of the period, see Figure 3

Impact of the Diffusion Coefficient of the Stochastic Factor σ
The diffusion coefficient has noticeable effects on the investor. The greater the diffusion coefficient, the greater the proportion invested in the risky asset for an investor with some risk tolerance, see Figure 4(a) and Figure 4(c). By contrast, when the volatility is high, a high risk-averter investor will always choose to increase the present consumption, see Figure 4(b) and Figure 4(d).

Impact of the Risk Aversion Coefficient γ
A high risk-averter investor will reduce his/her present consumption if the expected return in the risky asset is high and will invest even more in that asset, see Figure 5(a) and Figure 5(b). Similarly, a low risk-averter investor will change his/her consumption as the return in the risk-free asset decreases and will also invest more in the risky asset. For a certain degree of relative richness, the investor will give up some present consumption to attain an expected higher return in investment, see Figure 5(c) and Figure 5(d).

Impact of the Correlation Coefficient ρ in the Portfolio
The correlation between the risky assets and the volatility of an individual asset can change and is often negative [9]. An investor may wish to periodically re-A. N. Sandjo et al.

Concluding Remarks
In this study we derive an explicit solution of a model for optimal portfolio selection under stochastic volatility. Our main result is a characterization of optimal portfolio weights and consumption. The major technical difficulties come from the nonlinearity of the model due to the market parameters and constraints. These difficulties have been overcome using a specific exponential form of the trial solution: a natural theoretical approach is to transform the resulting PDE into a more tractable one, namely a Riccarti equation. After a complete PDE characterization of the value function, we carried out some numerical experiments on the model to draw economic interpretations.
Furthermore, we also analyzed the dynamics of the desired consumption, namely its response to various factors, such as interest rates, mean reverting A. N. Sandjo et al. viduals. An important result is the confirmation of the separation theorem proved by Fisher [20] stating that, the portfolio-selection decision is independent of the consumption decision and the consumption decision is independent of the financial parameters and only depends upon the level of wealth.
Since the proposed model offers a framework, which is numerically tractable, future work will consider the incorporation of accurate estimate of the model parameters. In particular, risky assets such as stocks depend on parameters that have to be estimated from data. These will provide more specific economic interpretation, depending on the characteristics of the risky assets, enabling to investigate the response of the model not only to predictable events such as dividend policy announcements or macroeconomic data releases, but also the contagion effect in international markets.
The first-order conditions for a regular interior extremum Throughout this section, we use a power-law utility function that belongs to , , J t W V must satisfy the partial differential equation Equation (16) where the coefficients are defined in (17).
Taking into account the fact that .
Therefore, the consumption (31) and portfolio selection (32) are given by Therefore, a sufficient condition on ( ) d t and on ( )

K t V is also a solution for Equation (36) is given by
and ( ) ( ) In addition, the boundary condition ( ) We perform a variable change to transform Equation (39) into a second order ordinary equation. To do this, we consider a new function F defined by It is actually an easy task to solve Equation (40) since we are now facing a second order linear ordinary differential equation.
The characteristic equation of (40) is given by We then distinguish three cases according to the sign of the discriminant ( )