Portfolio Optimization in Jump Model under Inefficiencies in the Market

This paper evaluates the use of modeling approach that depends on Levy jump model to predict investors wealth under inefficiencies in the market, in terms of mispricing and asymmetric information where the traded stock or risky asset price is considered to be as a function of a Levy jump process (i.e. the driving Levy process has Brownian component) 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.


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.

The Model
The market parameters are t µ , t σ and t r are Lebesgue integrable deterministic functions.
X is defined by, to embody both a permanent component and a temporary component of price shocks represented by t W and t U respectively.The mean-reverting O-U process ( ) with rate λ representing the mis pricing are defined exactly as [7] and satisfies the Langevin stochastic differential equation given by: with a unique solution 1 1 e 2 The stock's Sharpe ratio or market price of risk θ is square integrable 0 T > is the investment horizon.Z is a pure jump Levy process having a σ-finite Levy measure ν on 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 ,d ,d ( ) then the log return dynamic becomes: And by using Itô's formula to Equation (5) yields percentage returns: .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: is given by [7].Z and X are independent, since , , Z W U are independent with

Dynamics of Asset Price for Informed Investor
Based on information flow of informed investor from Equations ( 3) and ( 5) , then the percentage return dynamics of informed investor is, ( ) ( ) d , where .
By using Levy characterization theorem [10] the price process is, ( )

Dynamics of Asset Price for Uninformed Investor
Using the representation of Gaussian process from [11], [7] gave an 0  -Brownian motion 0 B and a random process 0 ν , 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: ( ) ( ) d , where with price ( ) where is the solution of the Cauchy equation given by, The stock percentage return dynamic for the investors relative to the filtration with

Wealth and Portfolio Dynamics of Investors
Before going to the main results we introduce some preliminary concepts on portfolio and wealth process.
where Π is a function of ( ) t ω and ω is kept as the background and it is assumed that j Π is a function of time t and is the value of the portfolio (stock and bond) at time t.t Π is the proportion of the wealth invested in the stock.
For the sake simplicity, it will be denoted by ( ) By substituting the asset return dynamics Equation ( 11) in (13) The interest rate t r is set to be zero in the following discussion then the terminal wealth j T W is equivalent to to the discounted terminal wealth j t W  .

Optimal Portfolio and Power Utility Maximization
It is assumed that each investor has a utility function Condition, if it is strictly increasing, strictly concave, continuously differentiable with: ( ) ( ) We assumed power utility function which is defined by ( ) Particularly it shown that ( ) ) and taking the expectation of ( ) and by Ito's Isometry The expectation of ( ) , 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 j t W  is given by: The admissible set for the investors is denoted by It is assumed that j W  is the discounted wealth process of j W , and Π is predictable if it is measurable with respect to the predictable sigma-algebra of left continuous with right limit function on [ ] 0,T × Ω .The optimal portfolio for the investors ( ) From theorem (5.1) 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 Π exist.
1) Let ( ) g Π is strictly concave on R, with unique maximum optimal portfolio * j Π for each investor that satisfies then there is unique admissible optimal portfolio ( ) 2) For j th investor the maximum expected power utility from terminal wealth for investors with initial wealth 0 x > is given by ( ) is the optimal portfolio of Journal of Mathematical Finance investors.
Remark 5.1.The optimal portfolio * j Π 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 0 U = , and therefore the return dynamics are the same,which yielding a common deterministic continuous optimal.
We analyzed theorem (5.1) is the maximum expected power utility on an optimal portfolio Π 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 ( ) Π is the excess asset holding resulting from the jumps for j th investor.Further more, 2) establishes that the maximum expected power utility from the terminal wealth for each investors, having 0 x > initial wealth is given by x is the maximum expected power utility from the terminal wealth for the continuous part with optimal portfolio * c Π and ( ) x is the excess utility from the jump.
The value ( ) x depends on the growth rate of H and optimality is ) ( ) is true if no short-selling ( ) from the bank account is allowed, and

Holding of Excess Stock by Investors
Define the continuous optimal for informed and uninformed investors by: By the Mean Value Theorem there exists ( ) Substituting Equation (15) in Equation ( 17) the optimal portfolio that maximizes the expected power utility from terminal wealth is written as follows, ( ) ( ) From Equation (18) the holding of excess stock by investors over the continuous optimal γ which is strictly due to the presence of jumps is given by, , The holding of excess stock by investors over the continuous optimal γ can be negative or positive depending on ( ) that is determined numerically.

Asymptotic Utilities of Investors
Let ( ) j T U x be the maximum expected power utility of the uninformed investor resulting from an optimal portfolio * j t Π .The risk-adjusted stock's Sharpe ratio for the investor is From [7] we have the following result.1) The optimal portfolio and maximum expected logarithmic utility from terminal wealth for the investors in a purely continuous market are given by * 2 2) As T → ∞ 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 T → ∞ the maximum expected power utility from the terminal wealth for investors is given by ( ) 2) As T → ∞ , the asymptotic maximum expected power utility from the terminal wealth for investors is given by ( ) 3) The relative asymptotic maximum expected basis power utility for the informed investor is:

H γ and Optimal Portfolios
To approximate

( )
H γ let as use Taylor expansion and drive some useful formulas.Define .Define the i th instantaneous centralized moments of returns for the stock dynamics (2) by: So now we have the following from the quadratic approximation as a result.
1) The optimal portfolio for each investor resulting from the quadratic approximation of H, where 2) The jump component of the maximum expected power utility for investors resulting from the quadratic approximation of H, where ( ) 3) The maximum expected power utility from the terminal wealth for the investor resulting from the quadratic approximation of H, where ( ) From theorem (5.1) the optimal utility due to the jump for the investor is given by: By substituting Equation (24) in (23) where ( ) ( ) ( ) Theorem 5.7.The total asymptotic excess optimal utility of the informed investor over the uninformed,as T → ∞ is given by: ( )

Π
is the proportion of an investor's wealth invested in the stock at time t.The remainder 1 j t − Π is invested in the bond or money market.Definition 4.2.Wealth process The wealth process for the investors is [ ]

H
′′ Π < and from Theorem (5.1) the optimal portfolio * t Π of investors is: γ exists on [ ] 0,1 .The following definition is useful which is connected to the Levy measure that will be used for the computation of approximation.
The model consists of two assets a risk-less asset B called bond earns a continuously compounded risk-free interest rate t in terms of continuous and jump component of