We are concerned with an optimal investment-consumption problem with stochastic affine interest rate and stochastic volatility, in which interest rate dynamics are described by the affine interest rate model including the Cox-Ingersoll-Ross model and the Vasicek model as special cases, while stock price is driven by Heston’s stochastic volatility (SV) model. Assume that the financial market consists of a risk-free asset, a zero-coupon bond (or a convertible bond), and a risky asset. By using stochastic dynamic programming principle and the technique of separation of variables, we get the HJB equation of the corresponding value function and the explicit expressions of the optimal investment-consumption strategies under power utility and logarithmic utility. Finally, we analyze the impact of market parameters on the optimal investment-consumption strategies by giving a numerical example.
As a hot topic, investment-consumption problem has abstracted increasing attention of many investment institutions which include insurance companies, pension management institutions, and commercial banks. As a milestone of investment-consumption field, Merton [
It is clear to show that the above-mentioned literatures have been achieved on the preconditions of constant interest rate and constant volatility. However, a fact has been established that some typical market parameters (such as interest rate, volatility, and inflation rate) are not invariable for a long time horizon and can be influenced by a variety of uncertain factors (e.g., disaster, war, exchange rate, and monetary policy). Hence, the introducing of stochastic interest rate or stochastic volatility makes the optimal investment strategy greatly instructive. Korn and Kraft [
Obviously, most literatures of the previous paragraph have been finished under the assumption of single uncertain factor, while real investment environment is very complicated and its interest rate and volatility should be stochastic. It is very clear that the portfolio decisions with stochastic interest rate and stochastic volatility are more practical. In recent years, some results have been obtained under the different market assumptions. For example, Liu [
This paper is organized as follows. In Section
Let
The financial market consists of a cash, a bond, and a stock. The interest rate
The risk-free asset (i.e., cash) satisfies the following equation:
The second asset is one zero-coupon bond with maturity
The maturity of the bond
Assume that price process of the convertible bond is denoted by
In fact, since the convertible bond is only correlated with interest rate, it can be reproduced by the zero-coupon bond and cash:
During the time horizon
The investment-consumption strategy
Equation (
Assume that the set of all admissible strategies could be denoted by
In this section, we obtain the HJB equation of the value function by using dynamic programming principle. We define the value function as
Using dynamic programming principle, we can get the corresponding HJB equation:
Using first-order maximizing conditions for the optimal investment and consumption strategies, we get
Putting (
In this paper, we assume that the risk aversion degree of investors can be described by power utility and logarithmic utility, respectively. We use variable change technique to investigate the optimal investment-consumption strategies under power utility and logarithmic utility.
Power utility is given by
Substituting (
Putting (
Due to
For (
Suppose that
For any function
In addition, we get
Namely, (
Assume that the solution to (
The partial derivatives of (
Plugging (
Equation (
Suppose that
Using the same method as (
Combining (
So Lemma
To sum up, we have the following conclusion.
If utility function is given by
Logarithmic utility is given by
Substituting (
Substituting (
Applying the boundary condition
Assume that the solution to (
The partial derivatives are as follows:
Putting (
Doing a further reduction for (
Under the boundary condition
Therefore, Lemma
Putting (
Under logarithmic utility
In this section, we give a numerical example to illustrate the optimal investment-consumption strategy under power utility. In order to analyze the impact of the parameters on the optimal strategies, we assume that the main parameters are given by
From the strategy expressions above, we can see that the trend of strategy can be influenced by a lot of market parameters, such as interest rate, volatility, discount rate, and risk aversion factor. In this section, we will analyze the sensitivity of the optimal investment-consumption strategies to the parameters
As Figure
The effect of
From Figure
The effect of
Figure
The effect of
Figure
The effect of
In Figures
The effect of
The effect of
The effect of
The effect of
As seen in Figure
Figure
In contrast to Figure
From Figure
In this paper, we studied optimal consumption and portfolio decision with affine interest rate and stochastic volatility. To hedge interest rate risk, we introduced a zero-coupon bond into financial market. In addition, the financial market is composed of a risk-free asset, a risky asset, and a zero-coupon bond, which can be reproduced by convertible bond. We assume that interest rate follows an affine interest rate model, while stock price is influenced by interest rate dynamics and volatility dynamics. The objective of this paper is to maximize the expected discount utility of intermediate consumption and terminal wealth. By using dynamic programming principle and method of separation of variable, we obtain the explicit expressions of the optimal investment-consumption strategy under power utility and logarithmic utility. Finally, we give a numerical example to illustrate the impact of market parameters on the optimal investment-consumption strategy and analyze economic implications of market parameters.
The authors declare that there are no competing interests regarding the publication of this paper.
This research is supported by China Postdoctoral Science Foundation Funded Project (no. 2014M560185 and no. 2016T90203) and Tianjin Natural Science Foundation of China (no. 15JCQNJC04000).