Portfolio Optimization in Jump Model under Inefficiencies in the Market ()
1. Introduction
Nowadays, the essential topic in the financial engineering is the selection of the portfolios and asset pricing. If the market is efficient, then it is expected that asset prices reveal the existing information, and all investors have the same amount of information to select portfolios. However, one of the most remarkable developments of the last few decades was the most extremely thought concepts of market efficiency, the positive relationship among return and non-diversifiable risk. This was due to the strong unexpected price volatility in the markets such as stock, bond, currency and real estate markets.
[1] and [2] were among the first to emphasize that there are many market anomalies including excess volatility caused by investor overreaction and under-reaction, fashions and mispricing on their empirical studies thought on this area. [3] , [4] and [5] are among current behavioral finance articles discussed for the presence of these market anomalies. Information asymmetry has a considerable outcome on asset prices and difficulties which upsets assets are through a liquidity channel [3] . Asset mispricing and information asymmetry connection in a purely deterministic and discrete setting was first studied by [5] and [6] and later prolonged to the purely continuous random environment by [7] and [8] . So that, using this as a background asset pricing and portfolio selection should be studied in an inefficient framework by assuming that the asset has both the fundamental value and market value and there are two types of investors in the market as informed who observes both fundamental and market value and also has non-public information advantage to trade and uninformed investor observes market values and makes investment choices using public information only.
Levy processes have gained extensive interest in financial modeling as they were found to overcome many of the shortcomings associated with the Black-Scholes model and to offer a more general tool for modeling inefficiency in asset prices. The probability distributions associated with Levy processes are infinitely divisible and offer more flexibility for fitting financial data. Mispricing models for stocks under asymmetric information were first studied by [6] in a purely deterministic setting and [7] extends to the continuous random environment where he has assumed that stock prices follow geometric Brownian motion, utility function from logarithmic and the continuous mean-reverting Ornstein-Uhlenbeck (O-U) process represents the mispricing in the market. Moreover, he has derived optimal portfolios and maximum expected logarithmic utilities, including asymptotic utilities for both rational investors (uninformed and informed).
In this paper, we followed the same modeling approach using Levy jump model to predict investors wealth under inefficiencies in the market, in terms of mispricing and asymmetric information by specifying the asset price process in the large filtration of informed investor. Then we obtain its dynamics for uninformed investor using the Hitsuda representation of Gaussian processes assuming there are two distinct classes of rational investors. In this setting assuming power utility functions, the optimal portfolios, maximum expected power utilities and asymptotic utilities for investors from the terminal wealth are derived by the methods of optimization and stochastic calculus.
2. The Model
The model consists of two assets a risk-less asset B called bond earns a continuously compounded risk-free interest rate
and a risky asset S on a probability space
called stock with with total percentage appreciation rate or expected returns
at time
given as follows respectively.
(1)
(2)
The market parameters are
,
and
are Lebesgue integrable deterministic functions.
X is defined by,
(3)
to embody both a permanent component and a temporary component of price shocks represented by
and
respectively. The mean-reverting O-U process
with rate
representing the mis pricing are defined exactly as [7] and satisfies the Langevin stochastic differential equation given by:
(4)
with a unique solution
with
and
.
The stock’s Sharpe ratio or market price of risk
is square integrable
is the investment horizon. Z is a pure jump Levy process having a σ-finite Levy measure
on
with triple
where
and the Levy measure satisfies
,
N is a Poisson random measure on
that is linked to the stock. It counts the jumps of Z in the time interval
.
The returns of the stock has three components; a continuous component
, a diffusive component
which is random, and a discontinuous component
, which is also random.
The continuous component of the stock’s return
and the volatility
, are assumed to be deterministic functions with
.
The process
, the continuous random component of excess return and the over all process is assumed with finite mean and variance. By the Levy-Itô decomposition theorem [9] Z is written as
, with triplet
and
and
then the log return dynamic becomes:
(5)
And by using Itô’s formula to Equation (5) yields percentage returns:
(6)
All the investors random objects are defined on the filtered probability space
. We used index 1 and 0 for informed and uninformed investors respectively. The information flows of the informed and uninformed investors is written as follows:
where
is given by [7] . Z and X are independent, since
are independent with
3. Dynamics of Asset Price
3.1. Dynamics of Asset Price for Informed Investor
Based on information flow of informed investor from Equations (3) and (5)
where
and
, then the percentage return dynamics of informed investor is,
(7)
By using Levy characterization theorem [10] the price process is,
(8)
3.2. Dynamics of Asset Price for Uninformed Investor
Using the representation of Gaussian process from [11] , [7] gave an
-Brownian motion
and a random process
, such that
on probability space
which is independent from Brownian motion B and W, then the percentage return dynamics of uninformed investor equating with Equation (6) is given by:
(9)
with price
(10)
where
with
Remark 3.3.
is the solution of the Cauchy equation given by,
Lemma 3.4. Let
and
. The
,
,
,
, as
and
,
, as
The stock percentage return dynamic for the investors relative to the filtration with
.
(11)
4. Wealth and Portfolio Dynamics of Investors
Before going to the main results we introduce some preliminary concepts on portfolio and wealth process.
Definition 4.1. Portfolio process
A portfolio
, is an
-adapted process satisfying
almost surely (12)
where
is a function of
and
is kept as the background and it is assumed that
is a function of time t and
is the proportion of an investor’s wealth invested in the stock at time t. The remainder
is invested in the bond or money market.
Definition 4.2. Wealth process
The wealth process for the investors is
, where
is the value of the portfolio (stock and bond) at time t.
is the proportion of the wealth invested in the stock.
is the initial capital which is
. For the sake simplicity, it will be denoted by
Let
be the wealth process of the uninformed investors at time t as a result of investing
in the stock.
Let
is the number of stocks in the portfolio at the time t. Then
where
is the value of the portfolio just before time t. Now by assuming all portfolios are self-financing,
(13)
By substituting the asset return dynamics Equation (11) in (13) and using Stochastic Exponential the wealth process
has dynamics.
(14)
where
is market price of risk or Sharpe ratio.
The discounted value
is given by:
Remark 4.3. The interest rate
is set to be zero in the following discussion then the terminal wealth
is equivalent to to the discounted terminal wealth
.
5. Optimal Portfolio and Power Utility Maximization
It is assumed that each investor has a utility function
that satisfies the Inanda condition. A function
satisfies Inanda Condition, if it is strictly increasing, strictly concave, continuously differentiable
with:
,
. We assumed power utility function which is defined by
with constant relative risk aversion (RRA),
.
Particularly it shown that
,
, when
,
.
By defining
and taking the expectation of
with
and by Ito’s Isometry
.
The expectation of
is given by:
, the following result is written for the expected power utility for investors from the terminal wealth.
Theorem 5.1. The expected power utility for investors from the terminal wealth
is given by:
where
,
and
.
The admissible set for the investors is denoted by
where
A process
is called admissible if it a member of
. It is assumed that
is the discounted wealth process of
, and
is predictable if it is measurable with respect to the predictable sigma-algebra of left continuous with right limit function on
.
The optimal portfolio for the investors
such that
which means
.
From theorem (5.1)
is independent of
. So
is maximized if and only if
maximized which means,
is maximized on the admissible set
.
By using similar approach to the optimization method used by [12] [13] [14] , and yields the same optimal as the HJB approach.
The major result as unique admissible optimal portfolio and the maximum expected power utility are presented in the following theorem.
Theorem 5.2. Assume that the first and second differentiation of H(.) with respect
exist.
1) Let
and
with
is strictly concave on R, with unique maximum optimal portfolio
for each investor that satisfies
. Further more,
and
then there is unique admissible optimal portfolio
2) For jth investor the maximum expected power utility from terminal wealth for investors with initial wealth
is given by
where
is the optimal portfolio of investors.
Remark 5.1. The optimal portfolio
is random and becomes deterministic when there is no information asymmetry. Since the drift term is random the dynamics of the return are random in the presence of information asymmetry in which optimal demand for the risky asset is random. If there is no information asymmetry investors perceive both fundamental and market values of the asset. Hence, there is no mispricing, which means
, and therefore the return dynamics are the same,which yielding a common deterministic continuous optimal.
We analyzed theorem (5.1) in terms of continuous and jump component of the the portfolio
by defining for
as
and
respectively and
and
is the maximum expected power utility on an optimal
portfolio
with investment horizon T which is assumed to be adapted to their filtration.Using this notation Theorem (5.1) 1) states for both investors,there is unique optimal portfolio
, such that
, where
is the continuous optimal portfolio for an asset with Geometric Brownian motion dynamics
and
is the excess asset holding resulting from the jumps for jth investor. Further more, 2) establishes that the maximum expected power utility from the terminal wealth for each investors, having
initial wealth is given by
which means,
, where
is the maximum expected power utility from the terminal wealth for the continuous part with optimal portfolio
and
is the excess utility from the jump.
The value
depends on the growth rate of H and optimality is achieved if H is twice differentiable. That is, if
finitely exists. If
, then
when ever
.
If
, then
exists if
. Hence, Theorem (5.1) is true if no short-selling
or borrowing
from the bank account is allowed, and
.
5.1. Holding of Excess Stock by Investors
Define the continuous optimal for informed and uninformed investors by:
Let
with
,
then
and from Theorem (5.1) the optimal portfolio
of investors is:
(15)
where
, so
(16)
By the Mean Value Theorem there exists
such that
(17)
Substituting Equation (15) in Equation (17) the optimal portfolio that maximizes the expected power utility from terminal wealth is written as follows,
(18)
From Equation (18) the holding of excess stock by investors over the continuous optimal
which is strictly due to the presence of jumps is given by,
(19)
The holding of excess stock by investors over the continuous optimal
can be negative or positive depending on
that is determined numerically.
5.2. Asymptotic Utilities of Investors
Let
be the maximum expected power utility of the uninformed investor resulting from an optimal portfolio
. The risk-adjusted stock’s Sharpe ratio
for the investor is
where
and
.
From [7] we have the following result.
Theorem 5.3. ( [7] Theorem 2.1) Let
be the initial wealth of the investor then:
1) The optimal portfolio and maximum expected logarithmic utility from terminal wealth for the investors in a purely continuous market are given by
and
2) As
the asymptotic maximum expected utility is:
3) The excess asymptotic maximum expected utility of the informed investor is:
.
Now we have maximum expected power utility and asymptotic maximum expected power utility as a result for the jump case which contains the result of [7] when there is no jump and the utility is logarithmic.
Theorem 5.4.
1) As
the maximum expected power utility from the terminal wealth for investors is given by
.
2) As
, the asymptotic maximum expected power utility from the terminal wealth for investors is given by
(20)
3) The relative asymptotic maximum expected basis power utility for the informed investor is:
.
5.3. Utilities under Quadratic Approximation of
and Optimal Portfolios
To approximate
let as use Taylor expansion and drive some useful formulas. Define
for
and assume
, there exists
which ensures that
, the nth derivative of
exists on
.
The following definition is useful which is connected to the Levy measure that will be used for the computation of approximation.
Definition 5.5. Let
. Define the ith instantaneous centralized moments of returns for the stock dynamics (2) by:
(21)
is well defined because
and
By defining,
,
and
Then,
(22)
So now we have the following from the quadratic approximation as a result.
Theorem 5.5.
1) The optimal portfolio for each investor resulting from the quadratic approximation of H, where
is defined on
is given by,
2) The jump component of the maximum expected power utility for investors resulting from the quadratic approximation of H, where
is defined on
and
is the risk adjusted Sharpe ratio, is
3) The maximum expected power utility from the terminal wealth for the investor resulting from the quadratic approximation of H, where
is defined on
and
is the risk adjusted Sharpe ratio, is
From theorem (5.1) the optimal utility due to the jump for the investor is given by:
(23)
Since
, from lemma (3.4)
then
.
Hence
(24)
By substituting Equation (24) in (23)
(25)
The following result is asymptotic power optimal utilities due to jump for each investor.
Theorem 5.6.
1) Let
be the initial wealth of the investor. By assuming
and
is defined on
then the asymptotic optimal power utility for the investor due to jump,as
is given by:
where
,
.
2) Let
be the initial wealth of the investor.By assuming
and
is defined on
then the excess asymptotic optimal utility of the informed investor over the uninformed, due to jump,as
is given by:
By lemma (3.4) as
and by theorem (5.4) we have,
where
,
.
Theorem 5.7. The total asymptotic excess optimal utility of the informed investor over the uninformed,as
is given by:
where
and
is the intial wealth of investors
.
Theorem 5.8. As
the maximum expected power utility from the terminal wealth for investors with initial wealth
and
is given by:
(26)
Using theorem (5.1) and theorem (5.6) it is the direct substitution. From theorem (5.4), since
is continuous, then by mean value theorem,
Thus,
and
as
So the explicit formula for
by solving Equation (20) and Equation (26) is written as follows.
(27)
6. Conclusions
We derived the optimal portfolios, maximum expected power utilities and asymptotic utilities for each investor from the terminal wealth when there are inefficiencies in the market. The traded stock or risky asset price is modeled as a Levy jump process by specifying the asset price process in the large filtration of informed investor, and then obtaining its dynamics for uninformed investor using the Hitsuda representation of Gaussian processes assuming there are two distinct classes of rational investors in a financial market.
Using quadratic approximation of the portfolios, we show that extra assets are detained by an investor if and only if the ratio of the first and second instantaneous Central moments of return is larger than the continuous optimal portfolio of that investor. We also show that jumps minimize the excess long run utility for the informed investor relative to that of the uninformed investor, which implies that jump risk may be good for market efficiency as an unintended reducer of information asymmetry.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgements
We are grateful to the authors for their numerous and valuable contributions to this work.