Optimal Investment and Consumption Problem with Stochastic Environments ()
1. Introduction
One key area of mathematical finance is the problem of an investor who seeks to maximize the expected utility of consumption and terminal wealth. This research work builds on the celebrated work of Merton in [1] [2] who originally studied continuous time investment and consumption problems when stock price follows a geometric Brownian motion. For Merton’s work, both the interest rate and volatility rate are constants. In real life, Interest rates and volatility rates are not constant. For example, the US 2007-2008 global financial crisis (housing bubble) made the US central banks adjust interest rates considerably. Interest rates and volatility rates, in reality, are stochastic due to uncertain events such as the Coronavirus disease 2019 (Covid19) pandemic, climate change, wars, inflation, natural disasters, fiscal policy and financial policy adjustments. In this study, a stochastic control problem of a single investor with stochastic interest rate from a bond modeled as a Ho-Lee process and a stock modeled as a Heston’s process with its volatility dynamics following a Cox-Ingersoll-Ross (CIR) process is investigated. These models are industry standard for option pricing and maximization problems. Thus, it is worth considering in solving an optimal investment and consumption problem in this mixed structure. Our main goal is to allocate initial wealth x0 between risk-free security and risky security in order to maximize the discounted expected utility of consumption and terminal wealth over a finite horizon. Bellman’s optimality principle, introduced by Bellman [3] called the Dynamic Programming Principle (DPP) will be applied in order to determine the Hamilton-Jacobi-Bellman Partial Differential equation (HJB PDE). The investor preferences are modeled as a Constant Relative Risk Aversion (CRRA) function. Our major contribution is that we have extended Merton’s work in [1] [2] problems with a unique mixture of consumption, stochastic interest rate and stochastic volatility rate simultaneously. So far, many researchers have studied such a control problem by considering constant interest and constant volatility. Some have considered one stochastic parameter in their analysis. However, such assumptions are unrealistic and not practical in the real financial world. In addition, we have also linked probability theory to PDE mathematics.
2. Links to the Literature
The problem of optimal investment and consumption has attracted a number of extensions. For instance, a paper by Benth [4] analyzed Merton’s portfolio optimization problem with stochastic volatility of Ornstein Uhlenbeck type. Yi and Guan [5] treated consumption and investment problem with volatility being constant. Zariphopoulou [6] explored consumption and investment problem with an interest rate, mean rate of return, and dispersion coefficient being constant. A paper by Sandjo et al. [7] considered constant expected return and stochastic volatility. Wang et al. [8] researched on optimal portfolio and consumption rule with a short interest rate driven by the CIR model under HARA utility function. Harrison and Kreps [9] and Harrison and Pliska [10] applied a different approach called martingale methods to solve an optimization problem. Cox and Huang [11] discussed optimal consumption and portfolio policy when asset prices follow a diffusion process. Jinzhu and Rong [12] considered a Cox-Ingersoll-Ross (CIR) model to describe the stochastic interest rate and stochastic volatility of the stock. Noh and Kim [13] Studied optimal portfolio model with stochastic volatility and stochastic interest rate with an assumption that the risky asset prices follow geometric Brownian motion. Pang [14] investigated the interest rate which varies according to a Markov diffusion process and that risky asset price obeyed a logarithmic Brownian motion. Korn and Kraft [15] obtained optimal policy for the case of a Ho-Lee model and Vasicek model for interest rates. Kraft [16] examined optimal portfolios and considered only Heston’s stochastic volatility model. Liu [17] considered the stock portfolio selection problem when stock return volatility is stochastic via Heston’s model. Zariphopoulou [18] studied optimization models in a market where assets are modeled as a diffusion process with coefficients changing with time according to correlated diffusion factors. Zariphopoulou [19] considered an optimization problem with bond price deterministic and the stock price modeled as a diffusion process such that coefficients of the stock price diffusion are arbitrary nonlinear functions of the underlying process. The paper by Fleming [20] investigated on consumption model with stochastic volatility and constant interest rate. Fouque et al. [21] considered a portfolio optimization problem with stochastic volatility and constant interest rate. In recent years jump-diffusion models, as well as Levy process models, have become popular in financial research. This is due to the shortcomings of the simple Brownian motion model developed in Black and Scholes [22].
In this study, Merton [1] [2] are extended in a unique way by studying the stochastic control problem for an agent who faces consumption, stochastic interest rates and stochastic volatility rates simultaneously. So far, many researchers have studied such models by considering either stochastic interest or stochastic volatility rates separately. However, such an assumption is unrealistic and not practical in the real financial world. Therefore, introducing stochastic interest and stochastic volatility rates simultaneously makes our model more realistic and practical although such stochastic control problems led to complex or sophisticated HJB PDE.
The outline of this paper is as follows: Section 1 Introduction. Section 2 Literature review. Section 3 Description of the financial market model. In section 4, the wealth model is determined. Section 5 Optimization criterion description. In section 6, the HJB PDE for the value function is derived. Section 7, we investigate the value function, optimal investment and consumption policies. In Section 8, numerical examples and simulations are provided. Here, the effect of market parameters on the optimal investment and consumption policies are illustrated. In Section 9, the conclusion and suggested possible future research work are stated.
3. Financial Market Model
Let
be a filtered complete probability space with filtration
satisfying the usual conditions such as
being right continuous complete filtration and
-complete. Let all stochastic processes be well defined and adapted in the filtered complete probability space
.
Consider a financial market of a single investor with a portfolio consisting of one risk-free security (e.g. a money market account or bond)
and one risky security (e.g. a stock or stock index)
.
Let the price dynamics of the risk-free security
evolve as follows:
(1)
with stochastic interest rate
following a Ho-Lee model given by:
(2)
where
is the expected instantaneous change in the interest rate,
is a constant volatility factor and
is a one-dimensional wiener process on a filtered probability space
. Assumed that
can be written as
, where
and
are constants.
Let the price dynamics of the risky security a stock (or share)
, follow a Heston’s model given by:
(3)
where
is the appreciation factor,
is the volatility of the risky price and
is a Wiener process on a filtered probability space
. Note that
is risky stock price,
is risk-free interest rate,
is the expected returns parameter of risky asset and
is the volatility of the volatility
of risky asset.
In addition, let
follow a Cox-Ingersoll-Ross (CIR) model given by:
(4)
where
,
, and
are constants. Also note that
for all
.
is a wiener process on a filtered probability space
.
4. The Wealth Model
Consider an investor with an initial amount of money
and a time horizon of interest T. Over the time interval
, the investor changes his portfolio dynamically. Let
denote the rate of continuous consumption. Let
denote the wealth to be invested in the risky asset S. Then the amount invested in the risk-free security is given by
. Note that the pair
is an investment and consumption strategy.
Lemma 1 The net wealth for an investor who faces intermediate consumption, stochastic interest rate and stochastic volatility rate evolve as follows:
(5)
Proof. In lemma 1, we prove the net wealth model for our financial market. Note that net wealth with intermediate consumption is defined by:
(6)
Substituting 1 and 3 into 6 gives:
(7)
Rearranging 7 gives us that:
(8)
5. The Optimization Criterion
Suppose the set of all admissible strategies is denoted by
.
Definition 5.1 An investment and consumption strategy pair
is said to be admissible if the following conditions are satisfied.
1) The pair
is progressively
-measurable and
,
, for all
.
2)
.
3) For all admissible pair
, the wealth process 5 with
has a path wise unique solution.
Remark 1 The investor’s objective is to maximize the net expected discounted utility of consumption plus the expected discounted utility of terminal wealth.
In this study, the power utility function which belongs to the CRRA class is used. The investor’s objective is to maximize the expected discounted utility of consumption plus the expected discounted utility of terminal wealth formulated mathematically as follows:
(9)
subject to the budget constraint
(10)
(11)
and
(12)
Definition 5.2 The value function is defined as
(13)
with boundary conditions
(14)
where
for all t, with T being the date of death,
is the value at time T of a trading strategy. The parameter
is the subjective discount rate and
determines the relative importance of the intermediate consumption.
denotes the conditional expectation operator.
and
are consumption and bequest functions respectively.
Remark 2 Note that
and
are such that
is twice differentiable with
and
.
Remark 3 When
, the expected utility only depends on the terminal wealth and the problem is reduced to an investment problem without intermediate consumption.
Definition 5.3 Let
be the initial wealth, the investor’s optimal investment and consumption problem is to maximize the expected discounted utility over the set of all admissible strategies
such that:
(15)
and
(16)
for all
,
.
6. The Hamilton-Jacobi-Bellman PDE
Bellman’s optimality principle, introduced by Bellman [3] called the Dynamic Programming Principle (DPP) will be applied in order to determine the Hamilton-Jacobi-Bellman Partial Differential equation (HJB PDE). By applying DPP, the fully HJB PDE associated with the stochastic control problem 13 is the non-linear second order PDE given as follows:
(17)
where
,
,
,
,
,
,
and
denotes partial derivatives.
Remark 4 For the sake of closed form solutions from the HJB PDE 17, we assume the following:
1) The correlation coefficient
of
,
and
.
2) Interest rate for risk-free security and risky security are equal.
Remark 5 The reduction of the initial nonlinear HJB PDE to a linear PDE is useful for obtaining both the value function and the optimal policies.
Definition 6.1 Applying the first-order maximizing conditions to 17, we obtain the following candidate optimizers:
(18)
and
(19)
Substituting the candidate optimizers 18 and 19 into the PDE 17 we get the following after simplification:
(20)
At this stage, we can apply power transformation and change of variable techniques to reduce PDE 20 to a linear PDE with well-defined solutions.
7. The Value Function and Optimal Policies
Solving PDE 20 by applying power transformation and change of variable techniques results in a linear PDE with well-defined solutions and thus, the value function and optimal policies can be established.
Taking a trial solution for PDE 20 take the form
(21)
The partial derivatives for 21 are given by:
(22)
Note that the optimal consumption strategy becomes
(23)
Substituting 22 and 23 into 20 gives the following after simplification:
(24)
Eliminating the dependence on x gives a PDE of the form:
(25)
Again we assume PDE 25 is of the form:
(26)
The partial derivatives for 26 are given by:
(27)
Substituting 27 into PDE 25, we obtain:
(28)
Eliminating
and simplifying further, 28 result to a nonlinear second-order PDE in
given by:
(29)
Remark 6 Note that PDE 29 is still complex and cannot be solved directly since there exists the term
.
Inspired by the paper of Liu [17], we further assume a solution to 29 as stated below:
Lemma 2 Assume
given by
(30)
is the solution to 29. Then we can prove that
can be written as:
(31)
with the boundary condition
.
Proof. In lemma 2, we seek to convert PDE 29 to PDE 31. Define the differential operator
on any function
as
(32)
Then equation 29 can also be written as
(33)
where
(34)
Notice that, on the other hand, we find
(35)
Note also that
(36)
Therefore,
(37)
where
(38)
Implying 29 can be converted to the following:
(39)
with the boundary condition
.
PDE 29 has been converted to PDE 31. PDE 31 has well defined solutions.
Thus, solving 20 or 29 is equivalent to solving 31.
Theorem 1 Suppose
is continuously differentiable and twice continuously differentiable for all
and
, then the solution of the HJB PDE 20 is given by
(40)
with
,
and
given by:
(41)
(42)
and
(43)
In addition, the pair
given by
(44)
and
(45)
are the optimal investment and consumption policies when interest rates of a risk-free security follow a Ho-Lee model and stock price dynamics evolve as a Heston’s model.
Proof. In theorem 1, assume we can fit a solution
given by:
(46)
with the boundary condition
.
From equation 46, we have the following partial derivatives:
(47)
Substituting 47 into 31 gives:
(48)
Canceling the term
on both sides of 48 gives:
(49)
Rewriting equation 49 to collect like terms in r and
gives:
(50)
After eliminating
and r, we can split Equation (50) into three ODE’s as follows:
(51)
(52)
and
(53)
Rewriting Equations (51)-(53), we get
(54)
(55)
and
(56)
Rewriting Equation alone (54), we obtain:
(57)
Let
denote the discriminant of the quadratic equation given by:
(58)
Implying
(59)
Let the discriminant
have distinct real solutions, that is
, then we obtain the following condition for
necessary for numerical analysis:
(60)
Considering condition 60, if we integrate both sides of 57 with respect to t, we obtain:
(61)
where
and
are two distinct real solutions for 57 given by:
(62)
Solving 61 with terminal conditions
, we obtain:
(63)
The solutions to the Equations (55) and (56) are obtained directly as follows:
(64)
and
(65)
Therefore, the value function is represented as follows:
(66)
where
,
and
are given in 63, 64 and 65 respectively.
In addition, the optimal feedback portfolio functions are given as follows:
(67)
and
(68)
where
(69)
with
,
and
determined in 63, 64 and 65.
8. Numerical Examples and Simulations
In this section, we determine how parameters affect investment
and consumption
controls.
When
, then problem becomes an investment and consumption problem with the following optimal policies:
(70)
and
(71)
When
, then the problem becomes an investment and consumption problem giving the following optimal policies:
(72)
and
(73)
When
, then the problem becomes an investment and consumption problem having the following optimal policies:
(74)
and
(75)
When
, the problem becomes an asset allocation problem without consumption. In such a case, optimal policies can be investigated further in another study.
8.1. Effects of Wealth
on Optimal Investment
and Consumption
Here, we assess the effects of Wealth on investment and consumption. Note that
(76)
and
(77)
Figure 1 and Figure 2 show the effect of the wealth
on the investment
and consumption
. The curve results analysis indicates that wealth
affects investment and consumption rates in a positive way when
and
. In summary, we can confidently state that optimal investment
and consumption
increase with the accumulation of the wealth
. This agrees with practical investments and our intuition.
Figure 1. The effects of wealth
on optimal investment
when
;
;
;
;
;
;
;
;
;
;
;
and
.
Figure 2. The effects of wealth
on consumption
when
;
;
;
;
;
;
;
;
;
;
;
and
.
8.2. Effects of the Expected Returns Parameter of Risky Asset k on Optimal Investment
and Consumption
In Figure 3 and Figure 4, the optimal investment
and optimal consumption
increases with respect to the increase in expected returns of risky asset k when
and
. Note that in 3, the product
is considered as the appreciation rate of the stock implying the more the investor wishes to invest in the stock for more wealth and consumption. This agrees with practical investments and our intuition.
Figure 3. The effects of the expected returns parameter of risky asset k on optimal investment
when
;
;
;
;
;
;
;
;
;
;
and
.
Figure 4. The effects of the expected returns parameter of risky asset k on consumption
when
;
;
;
;
;
;
;
;
;
;
and
.
8.3. Effects of Risk Aversion Factor
on Optimal Investment
and Consumption
In Figure 5 and Figure 6, the optimal investment
and consumption
increase with larger values of risk aversion factor
as this lead to smaller relative risk aversion
for the investor. The investor becomes vigorous in investing in the stock resulting in more wealth and thus more consumption.
Figure 5. The effects of risk aversion factor
on optimal investment
when
;
;
;
;
;
;
;
;
;
;
and
.
Figure 6. The effects of risk aversion factor
on optimal consumption
when
;
;
;
;
;
;
;
;
;
;
and
.
8.4. Effects of Weight for Intermediate Consumption
on Optimal Consumption
In Figure 7,
is increasing as weight for intermediate consumption
increase. When
gets larger, optimal consumption amount will also increase for
and
. In conclusion, the amount to consume increases for larger values of
.
Figure 7. The effects of weight for intermediate consumption
on optimal consumption
when
;
;
;
;
;
;
;
;
;
and
.
8.5. Effects of Risk-Free Interest Rate r on Optimal Investment
and Consumption
Figure 8. The effects of risk-free interest rate r on optimal investment
when
;
;
;
;
;
;
;
;
;
;
and
.
8.6. Effects of Risk-Free Interest Rate r on Optimal Consumption
In Figure 8, the optimal investment
decreases as interest rate r increases when
and vise verse for
. In this case, the investor will reduce the investment amount in the stock in order to avoid the risks and invest more in risk-free assets since income is increasing in this asset. In Figure 9, consumption
increases as interest rate r increases when
. Again the investor will reduce the investment amount in the stock in order to avoid the risks and invest more in risk-free assets since income is increasing in these assets. Net wealth still increases resulting in more consumption.
8.7. Effects of Volatility of Risky Security
on Optimal Investment
and Consumption
In Figure 10, the optimal investment policy
increases as the volatility of risky security
increases when
but decreases when
. Implying the value of correlation is key when making an investment decision in this financial market setup. In Figure 11, the consumption policy
increases as
Figure 9. The effects of risk-free interest rate r on optimal consumption
when
;
;
;
;
;
;
;
;
;
;
and
.
Figure 10. The effects of volatility of risky security
on optimal investment
when
;
;
;
;
;
;
;
;
;
;
and
.
Figure 11. The effects of volatility of risky security
on optimal consumption
when
;
;
;
;
;
;
;
;
;
;
and
.
the volatility of risky security
increases. Higher risky investments yield higher returns implying more consumption. This agrees with practical investments and our intuition.
9. Conclusion
This research work builds on the celebrated work of Merton [1] [2] who originally studied continuous-time investment and consumption problems. We investigate an optimal investment and consumption problem for a single investor with a portfolio consisting of one risk-free security (e.g. a money market account or bond)
and one risky security (e.g. a stock or stock index)
. The interest rate dynamics of risk-free security follow a Ho-Lee model. In addition, the risky asset price follows Heston’s model with its volatility evolving as the CIR model. Our main goal is to allocate initial wealth
between risk-free security and risky security to maximize the discounted expected utility of consumption and terminal wealth over a finite horizon. By applying the Dynamic Programming Principle (DPP), we obtain the HJB PDE. Upon solving the HJB PDE, we derive the closed-form solutions of optimal investment and consumption strategies for the power utility case. The impact and economic implications of market parameters on optimal investment and consumption strategies showed that the wealth
, the weight for intermediate consumption
, the risk aversion factor
and the expected returns parameter of risky asset k affect the optimal investment
and optimal consumption
is a positive way regardless of the value of the correlation coefficient
. In addition, an increase in risk-free interest rate r and volatility of risky security
led to an increase in net wealth resulting in more consumption. Also the value of
is key for optimal investment in this financial market setup. The future research will focus on extending our study to include other utility functions. We will also introduce multiple risky securities resulting in more sophisticated nonlinear second-order partial differential equations.
Acknowledgements
Special thanks go to the anonymous reviewers for their helpful comments on our earlier submitted version of this paper. This work was fully supported financially by Mulungushi University.