In this paper, we consider a portfolio optimization problem with delay, in which the Cox-Ingersoll-Ross (CIR) stochastic volatility model is adopted to describe a non-constant volatility of the risky asset. The phenomenon of frowns and smiles for the volatility of stock price cannot be explained within constant volatility models, stochastic volatility (SV) is recognized recently as an important feature for asset price models. There is much literature embedding SV in assets’ returns. For example, Hull and White  assume that the volatility follows log-normal process; Scott  and Stein & Stein  assume that the volatility follows Omstein-Uhlenbeck (OU) process; Heston  and Ball & Roma  introduce CIR process to describe stochastic volatility. Furthermore, many scholars study the optimal investment and/or consumption problems under the SV models. For instance, Chacko and Viceira  , Fleming and Hernndez-Hernndez  , Liu  and Zariphopoulou  consider an optimal investment and consumption pro- blems under SV models, and they derive explicitly the optimal strategies and optimal value functions in some situations by applying Hamilton-Jacobin- Bellman (HJB) technique; Kraft  , Taksar and Zeng  investigate a port- folio optimization problem under SV models. Moreover, Ferland and Watier  consider a mean-variance portfolio optimization problem with the CIR interest rate in a continuous-time framework, and they derive the mean-variance effi- cient portfolio by solving backward stochastic differential equations. Li & Wu  and Noh & Kim  consider portfolio optimization problems with an SV asset price process and a stochastic interest rate to maximize the expected utility of the terminal wealth. Li et al.  consider the optimal investment and rein- surance problem under Heston’s SV model. In addition,  ,  and  consider the uncertain portfolio selection.
However, in the literature above-mentioned, the past history information of risky asset price is not considered. That is, the price process of risky asset is supposed to follow a geometric Brownian motion with constant drift and cons- tant/stochastic volatility, the future movement of risky asset price is only based on the current information and is independent of the past historic information. However, there is growing evidence to demonstrate that the past price of risky asset influence its future price (See Akgiray  , Dibeh  , and Sheinkman & LeBaron  ). In other word, investors tend to make their investment decision based on the historic performance of risky asset or their portfolios in the real finance world. Specifically, if a stock price increases a lot recently, then there may be more investors would like to invest more money in this stock, which will push the stock price even higher. On the contrary, if the stock price decreases greatly, more investors tend to sell the stock and invest in other assets, which will drive the price to go down further. The dependence of asset price on the past states is called delay mathematically and a stochastic differential delay equation (SDDE) gives a mathematical formulation for such phenomena. Elsanousi and Larssen  investigate a class of optimal consumption problems where the wealth is given by a stochastic differential delay equation with a parameter, and they obtain the closed-form expressions for the optimal strategies and the value functions in two cases of deterministic parameters and random parameters. Chang et al.  study an investment and consumption problem of Merton’s type modeled by a stochastic system with delay, and they derive the closed-form expressions for the optimal strategies and the value functions in some situations by adopting stochastic control theory. Moreover, Mao  studies delay geo- metric Brownian motion in financial option valuation. Lee et al.  study a delayed geometric Brownian model with a stochastic volatility by the martingale method, they extend the geometric Brownian model by adding a stochastic volatility term, which is driven by a hidden process of fast mean reverting dif- fusion, to the delayed Black-Scholes model. A & Li  consider the optimal excess-loss-of reinsurance and investment problem with delay under the Hes- ton’s SV model. Shen  study the mean-variance portfolio selection in a random environment with unbounded coefficients.
To our best knowledge, there is little work in the literature on portfolio optimization problem when some delay factors (e.g. (2.3)-(2.4) in this paper) are added to the CIR stochastic volatility model. In this paper, we consider a new revised portfolio optimization problem in which we formulate the wealth dynamic as a stochastic differential delay equation with volatility driven by CIR model. By applying the stochastic dynamic programming approach, the corres- ponding HJB equation and a verification theorem are provided. The closed-form expressions for optimal strategy and optimal value function for CRRA utility model are derived.
The rest of this paper is organized as follows. In Section 2, the model and assumptions are described. In Section 3, the rigorous mathematical formulation of our problem is presented. HJB equation is given, and the verification theorem is proved. The closed-form expressions of optimal strategy and optimal value function for CRRA utility model are derived. In Section 4, some numerical experiment is presented to show our results. Section 5 concludes this paper and states some prospects.
2. Problem Formulation
Let be a probability space equipped with a filtration satisfying the usual conditions, i.e., is right-continuous and -com- plete, where T is a positive finite constant representing the time horizon. And all stochastic processes introduced below are supposed to be well-defined and adapted processes in the filtered complete probability space . In addition, there are no transaction costs or taxes in the financial market and trading takes place continuously.
Consider a financial market consisting of one risk-free asset and one risky asset. The risk-free asset, e.g., a bank account or a bond, can achieve a constant interest rate r. The price of risky asset is described by following stochastic volatility model, i.e.,
where are real constants, is a one-dimension standard Brow- nian motion; is the time varying instantaneous standard deviation of the return on the risk asset. We assume that the instantaneous variance follows the CIR process:
where is a one-dimension Brownian motion defined on the filtered probability space . Parameter describes the long-term mean of the variance, is the reversion parameter of the instantaneous va- riance process, i.e., describes the degree of mean reversion, and is the correlation coefficient between Brownian motions and , which is assumed to be negative to capture the asymmetric effect.
2.1. Investment Strategy and Wealth Processes
Starting from an initial wealth , an investor invests her wealth in the financial market. Suppose that the investor invests and dollar in the risk-free asset and the risky asset at time t, respectively. Then denotes the total wealth at time t. In addition, the investor is free to transfer money from the risk-free asset account to the risky asset account and conversely. Let be the total dollar amount transferred from one asset to the other asset up to time , that is, means to transfer dollar from the risk-free asset account to the risky asset account, means to transfer dollar from the risky asset account to the risk-free asset account. In addition, define two delay variables by
where is a constant, is a delay parameter. From a view of economic point, delay variable and reflect the average and pointwise performance information of the wealth process in the past period , respectively. With slight abuse of notation, we do not distinguish and . The same is true for other variables.
changes with the risk-free interest rate r, the dynamic of is described by
Generally, changes with risky asset’s price. In addition, the historic performance affects the investor’s investment decision, further, is affected by the historic performance, so we formulate the dynamic of as following stochastic differential equation:
where are real constants, is a one-dimension standard Brownian motion, and are given by (2.3) and (2.4). satisfies (2.2).
During the investment time horizon , the investor continuously invests her wealth in the risk-free asset and the risky asset. Let be the proportion of the investor’s wealth invested in the risky asset at time t. The remaining propor- tion is invested in the risk-free asset. Then and . The process is called an investment strategy. We assume that short-selling and borrowing are prohibited, i.e., , and the investment strategies satisfy the self-financial condition, that is, . Then the dynamic of the wealth under the in- vestment strategy is given by the following stochastic differential delay equation (SDDE):
where and are given by (2.3)-(2.4) and satisfies (2.2), respectively. We further assume that , , which can be interpreted that the investor is endowed with the initial wealth at time and do not start investment until time 0. Then the initial value of the delay variable is
Definition 2.1. (ADMISSIBLE STRATEGY) For any fixed , a stra- tegy is said to be admissible if it satisfies the following conditions:
i) is -measurable for any ,
ii) For any , here is a constant.
Let denote the set of all admissible strategies.
2.2. Portfolio Optimization Problem
In this subsection, we formulate the portfolio optimization problem with delay.
Definition 2.2. A utility function is a two varia- bles function i.e. . is strictly increasing, strictly con- cave, twice continuously differentiable with respect to the first variable , and satisfies and .
We consider an optimization problem of the investor who starts with an initial wealth and initial historic information of and . The investor wants to select a investment strategy so as to maximize the expected utility . Here, we consider an expected utility of a combination of the terminal wealth and the average performance information of the wealth process in the past period , i.e., . That is, is the terminal utility function which depends on both the terminal wealth and delay variable such that
for all and , where .
In mathematical terms, the portfolio optimization problem on a finite time horizon can be modeled as the following optimization problem.
Problem 2.3. (Portfolio optimization problem with delay)
For convenience, we first provide a notation. Let be an open set and . Denote
First, we show an important Itô’s formula. Let and define
Lemma 2.4. (Itô’s formula)
Proof. For each , using the Leibnitz formula, by (2.3)-(2.4) we have
Since , by the classical Itô’s formula we can obtain
where is the form of (2.12).
Remark 2.5. Lemma 2.4 yields the following useful formula for :
3. Optimal Investment Strategy
In this section, the HJB equation and a verification of Problem 2.3 are showed. Moreover, the closed-form expression of optimal investment strategy and value function are derived for CRRA utility function. By applying the dynamic pro- gramming approach, the portfolio optimization problem is equivalent to the problem of finding a solution to the HJB equation.
3.1. HJB Equation and Verification Theorem
For an admissible strategy , define the value function
Then the optimal value function is
with boundary condition .
Using the Itô’s formula in Lemma 2.4, we can show the following HJB equa- tion. Assume that , then the value func- tion solves the following HJB equation
with boundary condition
Though we have known that the value function is the solution of the HJB equation, we need to prove a verification theorem to ensure that a solution to the HJB equation is actually equal to the value function.
Theorem 3.1. (Verification Theorem) Let be a strong solution of (2.7), and are given by (2.3) and (2.4), and is a solution of HJB Equation (3.3) with boundary condition (3.4) such that
Then we have
If , then is the optimal strategy of Problem 2.3 and
Proof. Let be a solution of the HJB equation (3.3) with boundary condition (3.4). For any given admissible strategy and for any , we must have
On the other hand, applying Itô’s formula (2.11) to , we have
Integrating it from s to T, and using (3.8), we obtain
Therefore, by virtue of boundary condition (3.4), we have
Using the condition (3.6), it is easy to see that
are square integrable martingale whose expectation vanish. Therefore, by taking expectations on both sides of (3.9), we derive
Because it holds for all , we must get
On the other hand, if we take as defined in (3.7), and if , then all the above inequalities can be replaced by equalities. In other words, we have
Now the proof is complete.
3.2. Solution of HJB Equation
To obtain a closed-form solution of HJB equation (3.3) with boundary condition (3.4), we assume that the investor has a utility function of the following form
where , , and is the investor’s relative risk aversion coefficient. In fact, Chang et al.  and Elsanousi  also adopt the utility function of (3.10) to study the optimal investment and consumption problem with delay.
The maximum in HJB Equation (3.3) is obtained when
We try to seek a function J satisfying the HJB equation (3.3) in the form
It is easy to verify that
Plugging them into (3.11), we can derive
Substituting (3.13)-(3.14) into (3.3) yields
Let . From (3.15) we have
Since (3.12) has a solution that does not depend on z, we have the following condition
By (3.18), (3.17) becomes
Equation (3.17) has a solution depending only on t and u if
Plugging Q and (3.19) into (3.17), we have
Equation (3.20) is equivalent to the following ordinary differential equations
From boundary condition (3.4), we have
Solving equations (3.21) with boundary condition (3.22), we derive the fol- lowing Theorem.
Theorem 3.2. (Optimal strategy of Problem 2.3) Given wealth and CRRA utility (3.10), the solution to HJB Equation (3.3) is given by
where and are time-dependent coefficients that are independent of the state variables. That is, for any ,
The optimal investment proportion in the risky asset of Problem 2.3 is given by
Remark 3.3. It is interesting that our results are similar to the results of Liu and Pan  . Liu and Pan study optimal investment strategies given an investor access not only to bond and stock markets but also to the derivatives market and the price process of the risk asset is associated with stochastic volatility and jump. Here are some comparisons between them.
i) In our results, the proportion in the risky asset depends on wealth , delay variables and , and stochastic volatility at time t. However, in Liu and Pan  the optimal strategy is a deterministic function and does not depend on wealth and stochastic volatility . Moreover, our results are consistent with the results of Liu and Pan  , when the delay variables are not considered in our model (i.e. ). The details are given in Lemma 3.4.
ii) Let the delay approach 0 then , in this case the delay variable vanishes. Assuming that (3.19) holds and , then . At this time, the dynamics of wealth (2.7) degenerates as
which is the case without delay. The corresponding problem without delay and its optimal strategy are given in Lemma 3.4.
iii) The value function of (3.23) depends only on delay vari- able and does not depend on delay variable .
3.3. A Special Case
Now we consider a special case of our model. Suppose that the dynamics of risky asset does not depend on historic performance, then our model degenerates into a CIR SV model without delay. The results of our model will be reduced to the following special case.
Proposition 3.4. (Optimal strategy without delay) Consider the following problem without delay:
The value function is given by
and HJB equation is given by
with boundary condition .
Then, for given wealth , the solution to HJB equation is given by
where and are time-dependent coefficients that are independent of the state variables. That is, for any ,
The optimal investment proportion in the risky asset is given by
Proof. Let in Theorem 4.5, we can easily obtain the results of this lemma.
Remark 3.5. Our results without considering delay (i.e. ) is similar to the results of Liu and Pan  . Their results are consistent with our results without considering delay, when there is only stock and not derivatives consi- dering in Liu and Pan’s model. In some sense, we extend the results of Liu and Pan  .
4. Numerical Experiment
In this section, we investigate the effect of delay variables, stochastic volatility and VaR constraint on the optimal strategies and the optimal value functions, and provide some numerical examples to demonstrate the effect. We set the initial wealth level in million dollars between 0 and 10. The VaR horizon period is chosen to be 1 trading day, nearly 1/360 calendar year, while the terminal year is set to be 10 calendar years. In the following numerical illustrations, unless otherwise stated, the basic parameters are given by .
4.1. Analysis of Optimal Strategy
In general, the dynamic changes of wealth must depend on both delay variables and at the same time in a similar manner, i.e., .
(1) From (3.27), for given and , we have
According to the above results, we know that i) if , then , which is the case without delay, the optimal investment strategy dose not depend on Y and Z; ii) if and , then and , which mean that the delay factors take a positive effect on the optimal investment strategy; iii) if and , then and , which mean that the delay factors take a passive effect on the optimal investment strategy.
In Figure 1, we draw the optimal investment strategy with t under the different delay. Let (or ) denote the case without delay, i.e., the curves denoted by “ ” in two Figures. In Figure 1(a), we set , and or respectively. Here, and mean that delay variables Y and Z take a positive effect on optimal investment strategy, that is, the curves are arranged by , and from top to bottom. In other words, the higher positive historic performance of the portfolio promote the higher proportion of an investor’s wealth invested on stocks. From another point of view, the result implies that a higher historic stock price encourages investor to invest more money to the stock. Conversely, in Figure 1(b), let , and or respectively. And and mean that delay variables Y and Z take a negative effect on optimal investment strategy, that is, the curves are arranged by , and from top to bottom.
Figure 1. Optimal strategies with time t under different delay.
Figure 1(b) shows that the higher historic performance of the portfolio gives the investors a chance to harvest their gain by cut down the proportion invested on stocks, which implies that a higher historic stock price cause investor to invest less money on the stock so as to avoid to chase the high price. This seems to be consistent with the facts.
(2) From (3.27), for given and , we have
So, the optimal investment strategy decrease w.r.t. stochastic volatility V and wealth X, respectively.
Figure 2 plot the optimal investment strategy with stochastic volatility V under the different delay. Let , . or means the case without delay, i.e., the curve denoted by “ ” in two Figures. Figure 2 shows that the optimal investment strategy decreases with increas- ing stochastic volatility V. That is to say, the higher stochastic volatility take a passive effect on the investment enthusiasm. Moreover, in Figure 2(a), we set , and or respectively. Here, and mean that delay variables Y and Z take a posi- tive effect on optimal strategy, that is, the curves are arranged by , and from top to bottom. Conversely, in Figure 2(b), let , and or respectively. and mean that delay variables Y and Z take a negative effect on optimal investment strategy, that is, the curves are arranged by , and from top to bottom. That is to say, Figure 1 and Figure 2 have the same result about the effect of delay variables on the optimal investment strategy.
In Figure 3, we demonstrates the optimal investment strategy w.r.t.
Figure 2. Optimal strategies with stochastic volatility V under different delay.
Figure 3. Optimal strategies under the different stochastic volatility.
different wealth X under the different volatility V. Let , , and respectively, and with delay variables . The curves are arranged by , , and from top to bottom. The two figure show the larger is stochastic volatility V and the smaller is the proportion of the investor’s wealth invested on stocks. This fits with the fact that market instability has negative impact on investors. Furthermore, in Figure 3(a) the curves show that the optimal investment strategy decreases with the accumulation of wealth X. This suggests that an investor has an increas- ing preference to risk aversion with the increase of wealth, thereby, an investor cut down the stockholding.
4.2. Analysis of Optimal Value Function
In this subsection, we analyze the effect of delay variable Y and stochastic volatility V on the value function. According to (3.23), the value function de- pends on and we have
Figure 4 plot the optimal value function with t when the risk aversion coefficient and . They shows that the value functions are almost the same with different delay variables.
Figure 5 plot the optimal value function with wealth X when the risk aversion coefficient and . They shows that the optimal value functions are almost the same with different delay variables. Figure 6 plot the optimal value function w.r.t. stochastic volatility V when the risk aversion coefficient and . However, it is interesting enough that the higher stochastic volatility seemed to induce the higher value of the value functions. That is, the curves are arranged by , , and from top to bottom. The two figures show that the optimal value func- tions increase with the increasing of stochastic volatility V.
Figure 4. Optimal value function w.r.t time t under the different delay.
Figure 5. Optimal value function w.r.t. wealth X under the different delay.
Figure 6. Optimal value function under different stochastic volatility V.
5. Conclusions and Prospect
This paper considers portfolio optimization problem with delay under CIR sto- chastic volatility model. Adopting the stochastic dynamic programming approach, we derive the optimal portfolio strategy in closed-form for a CRRA type utility function, and verification theorem is showed.
The results show that the historic performance of portfolio has obvious effect on the optimal strategy. Specifically, the higher positive history performance seems to induce the higher investment proportion on risky asset. On the con- trary, the higher negative historic performance of the portfolio leads to the lower proportion on risky asset. And the historic performance of portfolio has similar effect on value function. As a result, it is meaningful to put delay variables into the portfolio optimization problem.
There are several topics which deserve to be studied in the future. First, as illustrated in this paper, the portfolio optimization problem with a single risky and single risk-free asset obtains an explicit solution via the dynamic program- ming principle and the verification theorem. However, it is anticipated that explicit solutions of the similar type for the model with multiple risky assets will not be available.
This research is supported by NSFC (No. 71501050) and Startup Foundation for Doctors of ZhaoQing University (No. 611-612282).