Optimal Consumption and Portfolio Decision with Convertible Bond in Affine Interest Rate and Heston ’ s SV Framework

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.


Introduction
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 [1,2] studied a continuous-time consumption and portfolio selection problem and obtained optimal investment strategies under power utility and logarithmic utility by using dynamic programming principle.Subsequently, more and more scholars paid their attentions to the investment-consumption problems.Vila and Zariphopoulou [3] researched an investmentconsumption problem with borrowing constraints.On this basis, Yao and Zhang [4] studied the investmentconsumption problem with housing risky.Duffie et al. [5] investigated the investment-consumption problem with HARA utility in incomplete markets.Dai et al. [6] investigated the investment-consumption problem with transaction costs in finite time horizon.Peng et al. [7] studied the optimal investment-consumption-proportional reinsurance problem with option type payoff.By complicated deliberation and calculation, Zhao et al. [8] got the optimal investment-consumption policies with nonexponential discounting and logarithmic utility.In order to further investigate investment-consumption problems, Palacios-Huertay and Prez-Kakabadsez [9] and de-Paz et al. [10] introduced discounting function into stochastic hyperbolic discounting function and heterogeneous discounting function, respectively.Kronborg and Steffensen [11] devoted themselves to an inconsistent investment-consumption problem and received some instructive results.
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 [12] studied the portfolio selection problem with stochastic interest rate by applying stochastic control approach.Deelstra et al. [13] researched the optimal investment problem with minimum guarantee.Fleming and Pang [14] introduced stochastic control theory to investigate the consumption-investment problem with stochastic interest rate.H. Chang and K.

Problem Formulation
Let (Ω, F, {F  } ≥0 , ) be a filtered complete probability space; F  represents information available at time  in the market.
We assume that all processes introduced below are well defined and adapted to {F  } ≥0 .

Financial Market.
The financial market consists of a cash, a bond, and a stock.The interest rate () is supposed to be a stochastic process and to be governed by where , ,  1 , and  2 are positive real constants and   () is a standard Brownian motion on (Ω, F, {F  } ≥0 , ).Notice that (1) consists of the Vasicek model ( 1 = 0) and CIR model ( 2 = 0) as special cases.In the case of ( 2 = 0), the condition 2 >  is required to ensure that () > 0.
The risk-free asset (i.e., cash) satisfies the following equation: The second asset is one zero-coupon bond with maturity , whose price process (, ) is given by the following stochastic differential equation (SDE): where the boundary condition (, ) = 1 and   √ 1 () +  2 is the market price of risk resulting from   ().In addition, The maturity of the bond (, ) is  − , which varies continuously over time.Since there does not exist zerocoupon bond with any maturity  > 0 in the market, it is unrealistic to invest in (, ).So we introduced a convertible bond with a constant maturity .We can invest in the convertible bond to hedge the risk of interest rate.
Assume that price process of the convertible bond is denoted by   () and follows the following stochastic differential equations (SDE): In fact, since the convertible bond is only correlated with interest rate, it can be reproduced by the zero-coupon bond and cash: The price process of risky asset (i.e., stock) is denoted by (); then () satisfies the following SDE: where where  1 (⋅) and  2 (⋅) are utility functions which are strictly concave. is the subject discount rate and  determines the relative importance of the intermediate consumption and the bequest.When  = 0, the expected utility only depends on the terminal wealth and problem ( 11) is reduced to an asset allocation problem.

Optimal Consumption and Portfolio Decision
In this section, we obtain the HJB equation of the value function by using dynamic programming principle.We define the value function as with the boundary condition (, , , ) = (1 − ) −  2 ().
Using dynamic programming principle, we can get the corresponding HJB equation: sup where   ,   ,   ,   ,   ,   ,   ,   , and   are the first-order, second-order, and mixed partial derivatives of the value function with respect to the variables , , , and .We use similar symbol to represent partial derivatives of the other functions in latter part of this paper.Using first-order maximizing conditions for the optimal investment and consumption strategies, we get Putting ( 14) into (13), we obtain the HJB equation as follows: 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.
Power utility is given by  1 () =  2 () =   /,  < 1 and  ̸ = 0, where  is the risk aversion factor.We conjecture the solution to problem (15) with the following form: The partial derivatives of ( 16) are as follows: Therefore, we get the optimal consumption policy as follows: .
Substituting ( 17) and ( 18) back into (15), we derive Eliminating the dependence on , we obtain Assume that the solution to ( 20) is given by The partial derivatives of the above function are as follows: Putting ( 22) into (20), we can get Due to (1 − ) − ̸ = 0, we can get the partial differential equation as follows: For (24), we find that this equation is similar to (22) in the paper of Chang and Rong [24].Inspired by the method of Chang and Rong [24], we give Lemma 2.
Proof.The partial derivatives of (32) are as follows: Plugging ( 33) into (26), we can get Eliminating the dependence on  and , we obtain the following three equations: Equation ( 35) can be written as follows: 2 . (38) The discriminant of quadratic equation is denoted by Δ  .Then Suppose that Δ  > 0; we can get Combining with the premise condition  < 1 and  ̸ = 0, we get Doing an integral calculation for (38), we obtain where  1 and  2 are two different roots of the above quadratic equation and  1 and  2 are given by Under the boundary condition () = 0, the solution to (35) is as follows: Using the same method as (35), we rewrite (36) in the following form: (46) The discriminant of quadratic equation is denoted by Δ  .Then Suppose that Δ  > 0, and then we can get  <  2 /((V  +   ) 2 +  2 ( 2  − 1)).We can easily see that Combining with the premise conditions  < 1 and  ̸ = 0, we get Using the same calculation as (35), we obtain where Combining (42) and (49), we can get Solving (37), we get So Lemma 3 is proved.
To sum up, we have the following conclusion.
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 ,  1 , , and , respectively.
As Figure 1 shows, the wealth proportion invested in stock, bond, and cash displays different trends with the increasing of .The dotted line means that  has no effect on  *  ().In addition, the astroid line shows that  *  () has a decreasing trend and the broken line shows that  * 0 () has an increasing trend.From (1), we see that a larger  leads to a smaller expectation of interest rate.Therefore, the proportion invested in convertible bond will decrease.From the views of practical investment, this conclusion is consistent with our intuition.
From Figure 2, we see a similar trend as Figure 1.That is to say, investors will reduce the wealth proportion invested in bond and increase the wealth proportion invested in cash with the growth of  1 .As a matter of fact, an increasing  1 means a decreasing volatility of stock price, which displays that the volatility of convertible bond is increasing.So, investors can reduce the proportion in the convertible bond in order to avoid the risk of investment.However, we find that the optimal amount in the stock remains fixed, although  1  has an effect on the dynamics of stock price from (7).This conclusion surprised us.We should keep it in mind in the practical investments.
Figure 3 shows us that  *  () increases with respect to the parameter , while  *  () remains fixed value and  * 0 () decreases with respect to the parameter .From (8), we can see that the expectation value of volatility will decrease when the value of  is increasing.It means that the risks from stochastic volatility will descend.Therefore, the decreasing risks lead to the increasing wealth proportion invested in the stock.This conclusion conforms to the economic implication of .From the views of utility theory, the risk aversion coefficient for power utility is given by 1 − .This means that the degree of risk aversion of investors will descend when the value of  is ascending.Therefore, investors would invest more money in the stock and reduce the proportion of wealth in cash and convertible bond.This conclusion agrees with that obtained by Guan and Liang [25].
In Figures 5-8, -axis represents the optimal consumption rate.
As seen in Figure 5, the optimal consumption rate increases with the increasing of the parameter . Figure 6 describes the effect of  1 on the optimal consumption rate.We can see that there is a positive correlation between  1 and the optimal consumption rate when  1 ≤ 0.75, while there exists a negative correlation when  1 > 0.75.
In contrast to Figure 5, Figure 7 shows that the optimal consumption rate will decrease with the growth of .It tells us that investors would cut down consumption expenses as  gets larger.
From Figure 8, we can learn the impact of  on the optimal consumption rate.Moreover, we obtain a strictly monotonic increasing function in Figure 8.It can be seen that the wealth of investors will also increase when the value of  is increasing.It means that investors may possess more wealth to consume.Therefore, the consumption rate of investors will increase accordingly as  is rising.This conclusion is consistent with that in the paper of Chang and Rong [24].

Conclusions
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.

Figure 1 :
Figure1: The effect of  on the optimal investment strategy.

Figure 2 :
Figure 2: The effect of  1 on the optimal investment strategy.

Figure 3 :Figure 4 :
Figure3: The effect of  on the optimal investment strategy.

Figure 4
Figure 4 tells us that  *  () increases in , while  *  () and  * 0 () decrease in .From the views of utility theory, the risk aversion coefficient for power utility is given by 1 − .This means that the degree of risk aversion of investors will descend when the value of  is ascending.Therefore, investors would invest more money in the stock and reduce the proportion of wealth in cash and convertible bond.This conclusion agrees with that obtained by Guan and Liang[25].In Figures5-8, -axis represents the optimal consumption rate.As seen in Figure5, the optimal consumption rate increases with the increasing of the parameter .

Figure 5 :k 1 Figure 6 :
Figure5: The effect of  on the optimal consumption rate.

Figure 7 :
Figure7: The effect of  on the optimal consumption rate.

Figure 8 :
Figure8: The effect of  on the optimal consumption rate.