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This paper considers a portfolio optimization problem with delay. The finance market is consisted of one risk-free asset and one risk asset which price process is modeled by Cox-Ingersoll-Ross stochastic volatility model. In addition, considering the history information related to investment performance, the dynamic of wealth is modeled by stochastic delay differential equation. The investor’s objective is to maximize her expected utility for a linear combination of the terminal wealth and the average performance. By applying stochastic dynamic programming approach, we provide the corresponding Hamilton-Jacobin-Bellman equation and verification theorem, and the closed-form expressions of optimal strategy and optimal value function for CRRA utility are derived. Finally, a numerical example is provided to show our results.

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 [

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 [

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.

Let ( Ω , F , ℙ ) be a probability space equipped with a filtration F = ( F t ) 0 ≤ t ≤ T satisfying the usual conditions, i.e., ( F t ) 0 ≤ t ≤ T is right-continuous and ℙ -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 ( Ω , F , F , ℙ ) . 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 S t is described by following stochastic volatility model, i.e.,

d S t S t = [ ( r + η V t ) ] d t + V t d B t S , (2.1)

where η are real constants, { B t S , t ≥ 0 } is a one-dimension standard Brow- nian motion; V t is the time varying instantaneous standard deviation of the return on the risk asset. We assume that the instantaneous variance V t follows the CIR process:

d V t = κ ( θ − V t ) d t + σ V t ( ρ d B t S + 1 − ρ 2 d B t V ) , (2.2)

where { B t V , t ≥ 0 } is a one-dimension Brownian motion defined on the filtered probability space ( Ω , F , F , ℙ ) . Parameter θ > 0 describes the long-term mean of the variance, κ ∈ ( 0,1 ) is the reversion parameter of the instantaneous va- riance process, i.e., κ describes the degree of mean reversion, and ρ is the correlation coefficient between Brownian motions B t S and B t V , which is assumed to be negative to capture the asymmetric effect.

Starting from an initial wealth x 0 , an investor invests her wealth in the financial market. Suppose that the investor invests L t and K t dollar in the risk-free asset and the risky asset at time t, respectively. Then X t = L t + K t 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 | I t | be the total dollar amount transferred from one asset to the other asset up to time t ( t ≥ 0 ) , that is, I t ≥ 0 means to transfer I t dollar from the risk-free asset account to the risky asset account, I t < 0 means to transfer − I t dollar from the risky asset account to the risk-free asset account. In addition, define two delay variables by

Y t = ∫ − δ 0 e λ s X ( t + s ) d s , (2.3)

Z t = X ( t − δ ) , ∀ t ∈ [ 0 , T ] , (2.4)

where λ ≥ 0 is a constant, δ > 0 is a delay parameter. From a view of economic point, delay variable Y t and Z t reflect the average and pointwise performance information of the wealth process in the past period [ t − δ , t ] , respectively. With slight abuse of notation, we do not distinguish X t and X ( t ) . The same is true for other variables.

L t changes with the risk-free interest rate r, the dynamic of L t is described by

d L t = ( r L t − I t ) d t . (2.5)

Generally, K t changes with risky asset’s price. In addition, the historic performance affects the investor’s investment decision, further, K t is affected by the historic performance, so we formulate the dynamic of K t as following stochastic differential equation:

d K t = [ K t ( r + ν Y t + μ Z t + η V t ) + I t ] d t + V t K t d B t S , (2.6)

where ν , μ , η are real constants, { B t S , t ≥ 0 } is a one-dimension standard Brownian motion, Y t and Z t are given by (2.3) and (2.4). V t satisfies (2.2).

During the investment time horizon [ 0, T ] , the investor continuously invests her wealth in the risk-free asset and the risky asset. Let π t be the proportion of the investor’s wealth invested in the risky asset at time t. The remaining propor- tion 1 − π t is invested in the risk-free asset. Then K t = π t X t and L t = X t − K t = ( 1 − π t ) X t . The process π = { π t } t ∈ [ 0, T ] is called an investment strategy. We assume that short-selling and borrowing are prohibited, i.e., 0 ≤ π t ≤ 1 , and the investment strategies satisfy the self-financial condition, that is, d X t = d K t + d L t . Then the dynamic of the wealth { X t , t ≥ 0 } under the in- vestment strategy { π t } t ∈ [ 0, T ) is given by the following stochastic differential delay equation (SDDE):

d X t = X t [ ( π t ( ν Y t + μ Z t + η V t ) + r ) d t + π t V t d B t S ] , ∀ t ∈ [ 0 , T ] . (2.7)

where Y t and Z t are given by (2.3)-(2.4) and V t satisfies (2.2), respectively. We further assume that X t = x 0 > 0 , t ∈ [ − δ ,0 ] , which can be interpreted that the investor is endowed with the initial wealth x 0 at time − δ and do not start investment until time 0. Then the initial value of the delay variable Y t is

y 0 : = Y 0 = x 0 ( 1 − e − λ δ ) λ .

Definition 2.1. (ADMISSIBLE STRATEGY) For any fixed t ∈ [ 0, T ] , a stra- tegy π t is said to be admissible if it satisfies the following conditions:

i) π t is F t -measurable for any t ∈ [ 0, T ] ,

ii) For any t ∈ [ 0, T ] , | π t X t | ≤ k | X t + μ Y t | , here k ≥ 0 is a constant.

Let Π denote the set of all admissible strategies.

In this subsection, we formulate the portfolio optimization problem with delay.

Definition 2.2. A utility function U : [ 0, ∞ ) × [ 0, ∞ ) → R { ∞ } is a two varia- bles function i.e. u = U ( x , y ) , ( x , y ) ∈ ℝ 2 . u is strictly increasing, strictly con- cave, twice continuously differentiable with respect to the first variable x , and satisfies l i m x → ∞ u x ( x , y ) = 0 and l i m x → 0 u x ( x , y ) = ∞ .

We consider an optimization problem of the investor who starts with an initial wealth x 0 and initial historic information of y 0 and z 0 . The investor wants to select a investment strategy π ∈ Π so as to maximize the expected utility E [ U ( X T , Y T ) ] . Here, we consider an expected utility of a combination of the terminal wealth X T and the average performance information of the wealth process in the past period [ T − δ , t ] , i.e., Y T . That is, U ( X T , Y T ) is the terminal utility function which depends on both the terminal wealth X T and delay variable Y T such that

E t , x , y , v [ U ( X T , Y T ) ] < ∞ ,

for all x , y and v , where E t , x , y , v [ ⋅ ] = E [ ⋅ | X t = x , Y t = y , V t = v ] .

In mathematical terms, the portfolio optimization problem on a finite time horizon [ 0, T ] can be modeled as the following optimization problem.

Problem 2.3. (Portfolio optimization problem with delay)

s u p π ∈ Π E [ U ( X T , Y T ) ] (2.8)

s . t . d X t = X t [ ( π t ( ν Y t + μ Z t + η V t ) + r ) d t + π t V t d Z t S ] , (2.9)

d V t = κ ( θ − V t ) d t + σ V t ( ρ d Z t S + 1 − ρ 2 d Z t V ) . (2.10)

For convenience, we first provide a notation. Let A 0 ⊂ R 3 be an open set and A = [ 0, T ] × A 0 . Denote

C 1,2,1,2 ( A ) = { ϕ ( t , x , y , z ) | ϕ ( t , ⋅ , ⋅ , ⋅ ) and ϕ ( ⋅ , ⋅ , ⋅ , z ) areoncecontinuouslydifferentiable on [ 0, T ] and ϕ ( ⋅ , x , ⋅ , ⋅ ) and ϕ ( t , ⋅ , y , ⋅ , ) aretwicecontinuouslydifferentiableon A 0 } .

First, we show an important Itô’s formula. Let f ∈ C 1,2,1,2 ( ℝ 4 ) and define

G ( t ) = f ( t , x , y , v ) ,

where

x = x ( X t ) , y = y ( X t ) = ∫ − δ 0 e λ s X ( t + s ) d s , z = z ( X t ) = X ( t − δ ) , v = V t , π = π ( t ) .

Lemma 2.4. (Itô’s formula)

d G ( t ) = L π f d t + f x π x v d B t S + f v σ v ( ρ d B t S + 1 − ρ 2 d B t V ) + f y [ x − e − λ δ z − λ y ] d t , (2.11)

where

L π f = L π f ( t , x , y , v ) = f t + x [ π ( ν y + μ z + η v ) + r ] f x + κ ( θ − v ) f v + 1 2 π 2 x 2 v f x x + π σ ρ x v f V x + 1 2 σ 2 v f v v . (2.12)

Proof. For each t ∈ [ 0, T ] , using the Leibnitz formula, by (2.3)-(2.4) we have

d d t ( y ( X t ) ) = d d t [ ∫ − δ 0 e λ s X ( t + s ) d s ] = d d t [ ∫ t − δ t e λ ( u − t ) X ( u ) d u ] ( u = t + s ) = ζ ( t ) − e − λ δ X ( t − h ) − λ ∫ t − δ t e λ ( u − t ) X ( u ) d u = x ( ζ t ) − e − λ δ z ( X t ) − λ y ( X t ) .

Since G ( t ) = f ( t , x , y , v ) , by the classical Itô’s formula we can obtain

d G ( t ) = { f t + x [ π ( ν y + μ z + η v ) + r ] f x + κ ( θ − v ) f v + 1 2 π 2 x 2 v f x x + π σ ρ x v f v x + 1 2 σ 2 v f v v } d t + f x π x v d B t S + f v σ v ( ρ d B t S + 1 − ρ 2 d B t V ) + f y [ x − e − λ δ z − λ y ] d t ≡ L π f d t + f x π x v d B t S + f v σ v ( ρ d B t S + 1 − ρ 2 d B t V ) + f y [ x − e − λ δ z − λ y ] d t .

where L π f is the form of (2.12).

Remark 2.5. Lemma 2.4 yields the following useful formula for d y :

d y = ( x − e − λ δ z − λ y ) d t . (2.13)

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.

For an admissible strategy π , define the value function

J ( t , x , y , v ) = E [ U ( X T , Y T ) | X t = x , Y t = y , V t = v ] . (3.1)

Then the optimal value function is

J π ( t , x , y , v ) = s u p π ∈ Π J ( t , x , y , v ) , (3.2)

with boundary condition J π ( T , x , y , v ) = U ( x , y ) .

Using the Itô’s formula in Lemma 2.4, we can show the following HJB equa- tion. Assume that J ( t , x , y , v ) ∈ C 1,2,1,2 ( [ 0, T ] × ℝ × ℝ × ℝ ) , then the value func- tion J ( t , x , y , v ) solves the following HJB equation

s u p π ∈ Π { L π J ( t , x , y , v ) + J y ( x − e − λ δ z − λ y ) } = 0 (3.3)

with boundary condition

J ( T , x , y , v ) = U ( x , y ) , (3.4)

where

L π J ( t , x , y , v ) = J t + x [ π ( ν y + μ z + η v ) + r ] J x + κ ( θ − v ) J v + 1 2 π 2 x 2 v J x x + π σ ρ x v J v x + 1 2 σ 2 v J v v . (3.5)

Though we have known that the value function J ( t , x , y , v ) 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 X t be a strong solution of (2.7), Y t and Z t are given by (2.3) and (2.4), and J ( t , x , y , v ) ∈ C 1,2,1,2 ( [ 0, T ] × ℝ × ℝ × ℝ ) is a solution of HJB Equation (3.3) with boundary condition (3.4) such that

E { ∫ 0 T ( v J v ) 2 d t } < ∞ , E { ∫ 0 T ( π t v x J x ) 2 d t } < ∞ , ∀ π ∈ Π . (3.6)

Then we have

J ( t , x , y , v ) ≥ s u p π ∈ Π E t , x , y , v [ U ( X T , Y T ) ] .

Further let

π * = − ( ν y + μ z + η v ) J x + σ ρ v J x v x v J x v . (3.7)

If π * ∈ Π , then π * is the optimal strategy of Problem 2.3 and

J ( t , x , y , v ) = E t , x , y , v [ U ( X T , Y T ) | π = π * ] .

Proof. Let J ( t , x , y , v ) be a solution of the HJB equation (3.3) with boundary condition (3.4). For any given admissible strategy π ∈ Π and for any ( t , x , y , v ) ∈ [ 0, T ] × ℝ × ℝ × ℝ , we must have

{ L π J ( t , x , y , v ) + J y ( x − e − λ δ z − λ y ) } ≤ 0. (3.8)

On the other hand, applying Itô’s formula (2.11) to J ( t , x , y , v ) , we have

d [ J ( t , x , y , v ) ] = [ L π J ( t , x , y , v ) + J y ( x − e − λ δ z − λ y ) ] d t + π t v x J x d B t S + f v σ v ( ρ d B t S + 1 − ρ 2 d B t V ) .

Integrating it from s to T, and using (3.8), we obtain

J ( T , X T , Y T , V T ) − J ( s , X s , Y s , V s ) = ∫ s T { L π J ( t , x , y , v ) + J y ( x − e − λ δ z − λ y ) } d t + ∫ s T π t v x J x d B t S + ∫ s T f v σ v ( ρ d B t S + 1 − ρ 2 d B t V ) ≤ ∫ s T π t v x J x d B t S + ∫ s T f v σ v ( ρ d B t S + 1 − ρ 2 d B t V ) .

Therefore, by virtue of boundary condition (3.4), we have

J ( s , X s , Y s , V s ) ≥ U ( X T , Y T ) − ∫ s T π t v x J x d B t S − ∫ s T f v σ v ( ρ d B t S + 1 − ρ 2 d B t V ) . (3.9)

Using the condition (3.6), it is easy to see that

∫ s T f v v d B t S , ∫ s T f v v d B t V and ∫ s T π t v x J x d B t S

are square integrable martingale whose expectation vanish. Therefore, by taking expectations on both sides of (3.9), we derive

J ( t , x , y , v ) ≥ E t , x , y , v [ U ( X T , Y T ) ] .

Because it holds for all π ∈ Π , we must get

J ( t , x , y , v ) ≥ s u p π ∈ Π E t , x , y , v [ U ( X T , Y T ) ] .

On the other hand, if we take π = π * as defined in (3.7), and if π * ∈ Π , then all the above inequalities can be replaced by equalities. In other words, we have

J ( t , x , y , v ) = E t , x , y , v [ U ( X T , Y T ) | π = π * ] .

Now the proof is complete.

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

U ( x , y ) = 1 1 − γ ( x + μ e λ δ y ) 1 − γ , (3.10)

where γ > 0 , γ ≠ 1 , and γ is the investor’s relative risk aversion coefficient. In fact, Chang et al. [

The maximum in HJB Equation (3.3) is obtained when

∂ ∂ π ( L π J ( t , x , y , v ) + J Y ( x − e − λ δ z − λ y ) ) = 0 ,

furthermore,

π * = − ( ν y + μ z + η v ) J x + σ ρ v J v x v x J x x . (3.11)

We try to seek a function J satisfying the HJB equation (3.3) in the form

J ( t , x , y , v ) = ( x + μ e λ δ y ) 1 − γ 1 − γ e x p { γ h ( T − t ) + γ H ( T − t ) v } . (3.12)

Define

u ≡ x + μ e λ δ y , H ˜ ( t ) = exp { γ h ( T − t ) + γ H ( T − t ) v } .

It is easy to verify that

J t = u 1 − γ 1 − γ ( − γ h ′ − γ H ′ v ) H ˜ ( t ) , J y = μ e λ δ H ˜ ( t ) u − γ , J x = H ˜ ( t ) u − γ , J v = γ 1 − γ H H ˜ ( t ) u 1 − γ , J x x = ( − γ ) H ˜ ( t ) u − γ − 1 , J x v = γ H H ˜ ( t ) u − γ , J v v = u 1 − γ 1 − γ ( γ H ) 2 H ˜ ( t ) . (3.13)

Plugging them into (3.11), we can derive

π * = ( ν y + μ z + η v + γ σ ρ v H ) u γ v x . (3.14)

Substituting (3.13)-(3.14) into (3.3) yields

γ 1 − γ ( h ′ + v H ′ ) u 1 − γ = x ( ( ν y + μ z + η v ) ( ν y + μ z + η v + γ σ ρ v H ) u γ v x + r ) u − γ + u − γ μ e λ δ ( x − e − λ δ z − λ y ) + u 1 − γ 1 − γ γ κ ( θ − v ) H + 1 2 u − γ − 1 ( ν y + μ z + η v + γ σ ρ v H ) 2 u 2 γ v + u − γ γ σ ρ v x H ( ν y + μ z + η v + γ σ ρ v H ) u γ v x + 1 2 u 1 − γ 1 − γ σ 2 γ 2 v H 2 . (3.15)

Let Q = ν y + η v . From (3.15) we have

γ 1 − γ ( h ′ + v H ′ ) u 1 − γ = ( Q + μ z ) ( Q + γ σ ρ v H + μ z ) γ v u 1 − γ + r x u − γ + μ e λ δ ( x − λ y ) u − γ − μ z u − γ + γ 1 − γ κ ( θ − v ) H u 1 − γ − 1 2 ( Q + γ σ ρ v H + μ z ) 2 γ v u 1 − γ + 1 2 γ 2 σ 2 v H 2 1 − γ u 1 − γ + ( Q + μ z + γ σ ρ v H ) ρ σ H u 1 − γ , (3.16)

or equivalently,

γ 1 − γ ( h ′ + v H ′ ) u 1 − γ = Q ( Q + γ σ ρ v H ) γ V u 1 − γ + γ 1 − γ κ ( θ − v ) H u 1 − γ + 1 2 γ 2 σ 2 v H 2 1 − γ u 1 − γ + ( Q + γ σ ρ v H ) ρ σ H u 1 − γ + [ ( r + μ e λ δ ) x − λ μ e λ δ y ] u − γ − 1 2 ( Q + γ σ ρ v H ) 2 γ v u 1 − γ + ( μ z Q γ v + 1 2 μ 2 z 2 γ v + μ ρ σ z H ) u 1 − γ − μ z u − γ . (3.17)

Since (3.12) has a solution that does not depend on z, we have the following condition

( μ z Q γ v + 1 2 μ 2 z 2 γ v + μ ρ σ z H ) u 1 − γ − μ z u − γ = 0. (3.18)

By (3.18), (3.17) becomes

γ 1 − γ ( h ′ + v H ′ ) u 1 − γ = Q ( Q + γ σ ρ v H ) γ v u 1 − γ + γ 1 − γ κ ( θ − v ) H u 1 − γ + 1 2 γ 2 σ 2 v H 2 1 − γ u 1 − γ + ( Q + γ σ ρ v H ) ρ σ H u 1 − γ − 1 2 ( Q + γ σ ρ v H ) 2 γ v u 1 − γ + [ ( r + μ e λ δ ) x − λ μ e λ δ y ] u − γ .

Equation (3.17) has a solution depending only on t and u if

− λ = μ e λ δ + r , λ < 0. (3.19)

Plugging Q and (3.19) into (3.17), we have

h ′ + v H ′ = 1 − γ γ ( ν 2 y 2 2 γ v + ν η y γ + ν ρ σ y H ) + ( κ θ H + 1 − γ γ ( r + μ e λ δ ) ) + ( 1 2 σ 2 ( γ ( 1 − ρ 2 ) + ρ 2 ) H 2 + [ ( 1 − γ ) η ρ σ γ − κ ] H + 1 − γ 2 γ 2 η 2 ) v . (3.20)

Equation (3.20) is equivalent to the following ordinary differential equations

{ ν 2 y 2 2 γ v + ν η y γ + ν ρ σ y H = 0 h ′ = κ θ H + 1 − γ γ ( r + μ e λ δ ) H ′ = 1 2 σ 2 ( γ ( 1 − ρ 2 ) + ρ 2 ) H 2 + [ ( 1 − γ ) η ρ σ γ − κ ] H + 1 − γ 2 γ 2 η 2 (3.21)

From boundary condition (3.4), we have

H ( T ) = 0 , h ( T ) = 0. (3.22)

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 X t and CRRA utility (3.10), the solution to HJB Equation (3.3) is given by

#Math_178# (3.23)

where h ( ⋅ ) and H ( ⋅ ) are time-dependent coefficients that are independent of the state variables. That is, for any 0 ≤ τ ≤ T ,

H ( τ ) = exp ( k 3 τ ) − 1 2 k 3 + ( k 1 + k 3 ) ( exp ( k 3 τ ) − 1 ) ϑ , (3.24)

h ( τ ) = 2 κ θ σ 2 ln ( 2 k 3 exp ( ( k 1 + k 3 ) τ / 2 ) 2 k 3 + ( k 1 + k 3 ) ( exp ( k 3 τ ) − 1 ) ) + 1 − γ γ ( r + μ exp ( λ δ ) ) τ , (3.25)

where

ϑ = 1 − γ γ 2 η 2 , k 1 = κ − 1 − γ γ η ρ σ , k 2 = σ 2 ( γ ( 1 − ρ 2 ) + ρ 2 ) , k 3 = k 1 2 − ϑ k 2 . (3.26)

The optimal investment proportion in the risky asset of Problem 2.3 is given by

π t * = ( ν Y t + μ Z t + η V t + γ σ ρ V t H ( T − t ) ) ( X t + μ e λ δ Y t ) γ V t X t . (3.27)

Remark 3.3. It is interesting that our results are similar to the results of Liu and Pan [

i) In our results, the proportion in the risky asset depends on wealth X t , delay variables Y t and Z t , and stochastic volatility V t at time t. However, in Liu and Pan [

ii) Let the delay δ approach 0 then Y t → 0 , in this case the delay variable Y t vanishes. Assuming that (3.19) holds and λ = − r , then μ = 0 . At this time, the dynamics of wealth (2.7) degenerates as

d X t = X t [ ( π t η V t + r ) d t + π t V t d Z t S ] ,

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 J ( t , X t , Y t , V t ) of (3.23) depends only on delay vari- able Y t and does not depend on delay variable Z t .

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:

m a x π t ∈ Π E [ U ( X T ) ] = E [ X T 1 − γ 1 − γ ]

s . t . d X t = X t [ ( π t η V t + r ) d t + π t V t d B t S ] , (3.28)

d V t = κ ( θ − V t ) d t + σ V t ( ρ d B t S + 1 − ρ 2 d B t V ) . (3.29)

The value function is given by

J ( t , x , v ) = m a x π t ∈ Π E [ U ( X T ) | X t = x , V t = v ] (3.30)

and HJB equation is given by

max { J t + x ( π η v + r ) J x + κ ( θ − v ) J v + 1 2 π 2 x 2 v J x x + π σ ρ v x J v x + 1 2 σ 2 v J v v } = 0 (3.31)

with boundary condition J ( T , x , v ) = x 1 − γ 1 − γ .

Then, for given wealth X t , the solution to HJB equation is given by

J ( t , X t , V t ) = 1 1 − γ X t 1 − γ exp { γ h ( T − t ) + γ H ( T − t ) V t } (3.32)

where h ( ⋅ ) and H ( ⋅ ) are time-dependent coefficients that are independent of the state variables. That is, for any 0 ≤ τ ≤ T ,

H ( τ ) = exp ( k 3 τ ) − 1 2 k 3 + ( k 1 + k 3 ) ( exp ( k 3 τ ) − 1 ) ϑ , (3.33)

h ( τ ) = 2 κ θ σ 2 ln ( 2 k 3 exp ( ( k 1 + k 3 ) τ / 2 ) 2 k 3 + ( k 1 + k 3 ) ( exp ( k 3 τ ) − 1 ) ) + 1 − γ γ r τ , (3.34)

where

ϑ = 1 − γ γ 2 η 2 , k 1 = κ − 1 − γ γ η ρ σ , k 2 = σ 2 ( γ ( 1 − ρ 2 ) + ρ 2 ) , k 3 = k 1 2 − ϑ k 2 . (3.35)

The optimal investment proportion in the risky asset is given by

π ¯ t * = η + γ σ ρ H ( T − t ) γ . (3.36)

Proof. Let μ = ν = 0 in Theorem 4.5, we can easily obtain the results of this lemma.

Remark 3.5. Our results without considering delay (i.e. ν = μ = 0 ) is similar to the results of Liu and Pan [

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 r = 0.05 , α = 0.01 , η = 2 , ρ = − 0.9 , σ = 0.1 , κ = 0.2 , δ = 1 .

In general, the dynamic changes of wealth X t must depend on both delay variables Y t and Z t at the same time in a similar manner, i.e., ν μ ≥ 0 .

(1) From (3.27), for given X > 0 and V > 0 , we have

∂ π * ∂ Y = ν X + μ e λ δ γ V X { = 0 , ν = μ = 0 , > 0 , ν > 0 , μ > 0 < 0 , ν < 0 , μ < 0 ,

∂ π * ∂ Z = μ γ V X { = 0 , μ = 0 , > 0 , μ > 0 , < 0 , μ < 0 ,

According to the above results, we know that i) if ν = μ = 0 , then ∂ π * ∂ Y = ∂ π * ∂ Z = 0 , which is the case without delay, the optimal investment strategy π * dose not depend on Y and Z; ii) if ν < 0 and μ < 0 , then ∂ π * ∂ Y > 0 and ∂ π * ∂ Z > 0 , which mean that the delay factors take a positive effect on the optimal investment strategy; iii) if ν < 0 and μ < 0 , then ∂ π * ∂ Y < 0 and ∂ π * ∂ Z < 0 , which mean that the delay factors take a passive effect on the optimal investment strategy.

In

(2) From (3.27), for given Y > 0 and Z > 0 , we have

∂ π * ∂ V = − ( ν Y + μ Z ) ( X + μ e λ δ ) γ X V 2 < 0 , ∂ π * ∂ X = − ν Y + μ Z + η V + γ σ ρ H ( T − τ ) μ e λ δ γ V < 0.

So, the optimal investment strategy π * decrease w.r.t. stochastic volatility V and wealth X, respectively.

In

different wealth X under the different volatility V. Let V = 0.1 , V = 0.15 , and V = 0.2 respectively, and with delay variables Y = 2 , Z = 2 , μ = 0.01 , ν = 0.1 . The curves are arranged by V = 0.1 , V = 0.15 , and V = 0.2 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

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 t , X , Y , V and we have

J X = ( X + μ e λ δ Y ) − γ exp { γ h ( T − t ) + γ H ( T − t ) V } > 0 ,

J V = ( X + μ e λ δ Y ) 1 − γ 1 − γ γ H ( T − t ) exp { γ h ( T − t ) + γ H ( T − t ) V } > 0 ,

J Y = μ e λ δ ( X + μ e λ δ Y ) − γ > 0 ( μ > 0 ) ,

Let X = 100 , ν = 0.01 , μ = 0.01 , Z = 2 , Y = 2 or Y = 200 , and ν = μ = 0 (or Y = 0 , Z = 0 ) denotes without delay in

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).

A, C.X. and Shao, Y. (2017) Portfolio Optimization Problem with Delay under Cox-Ingersoll-Ross Mo- del. Journal of Mathematical Finance, 7, 699-717. https://doi.org/10.4236/jmf.2017.73037