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1582 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 2 ⎛ ⎞ ⎛ ⎞ K σ ln ⎜ ⎟−⎜r − ⎟T S0 ⎝ 2 ⎠ a = ⎝ ⎠ σ T And the integral interval is divided to two parts ( −∞,a] ∪ [a, +∞ ) ⎛ ⎞ e E S K S e K e dx ⎛ 2 ⎞ 2 +∞ σ σ Tx+ ⎜ r− 2 ⎟ T x + 1 − −rT ⎝ ⎠ 2 ( − ) = ⎜ ⎟ T ∫ 0 − ⎜ ⎟ a 2π where ⎝ = I + I 1 2 2 2 x ⎛ ⎞ +∞ − +σ Tx+ r− T rT S σ − 0 2 ⎜ 2 ⎟ ⎝ ⎠ ∫ I1 = e e dx 2π a 2 +∞ x rT K − − 2 ∫ I2 =−e e dx 2π For I 1 , a 2 2 x ⎛ ⎞ +∞ − +σ Tx+ r− T rT S σ − 0 2 ⎜ 2 ⎟ ⎝ ⎠ ∫ I1 = e e dx 2π = a 2 2 σ +∞ x − T σ Tx− 0 2 2 S e ∫ e dx 2π a ( x−σ T) ⎛ ⎞ e exp dx 2 2 ⎛ ⎞ r− T 2 S σ +∞ ⎜ 0 2 ⎟ ⎝ ⎠ σ T = ⎜ ⎟ ∫ − + 2π a 2 2 ⎜ ⎟ Let y= x−σ T then ⎝ 2 2 σ T +∞ y − 0 2 2 S I1 = e ∫ e dy 2π a−σ T ( ( )) ( ( )) = −Φ −σ rT Se 0 1 a T = − −σ For I 2 , rT Se 0 N a T 2 +∞ x rT K − − 2 I2 =−e ∫ e dx 2π ( ( ))( ) −rT ( a) = −Φ − −rT e 1 a K =−Ke Φ − a B. Basic Model The basic notion follows [12]. Consider a portfolio X = x x x ′ of stocks1, 2 n consisting of quantities ( ) 1, 2 n with the return vector R= ( r1, r2 r n ) ′ . We assume that for each stock there are m put and m call options that mature in one year. The m strike prices of the put and call options for one particular stock are located at equidistant points between 70% and 130% of the stock's current price. C R , R are denoted the corresponding calls and ik Pik puts returns in the portfolio with stock price S , k = 1, 2m means the k -th strike price based on the i th i stock,call price C ik , and put price P ik , whose exercise ⎠ ⎠ prices are K ik , β and ik γ denote the (decision) variables on ik the numbers of the corresponding calls and puts option. 0 S denotes the initial price of stock ,which then can be i expressed as S 0 ir at the end of the period. Using the i payoff functions of call and put options, we can explicitly express the returns of options as: C 1 0 R ik = max { 0, Si ri − Kik} = max { 0, aik + bikri } Cik with K S ik 0 aik =− , bik = P C ik ik Similarly, the return of a put option is P 1 0 R ik = max { 0, i Kik − Si ri } Pik = max{ 0, aik + bikri } S K with 0 ik aik =− , bik = Pik Pik where P ik , C will be calculated from Black–Scholes ik formula. Within this investment framework, the value a portfolio at the expired time the investor wishes to maximize can thus be formulated as: n m ⎧ C P⎫ maxV = ∑⎨xr i i + ∑ βik Rik + γik Rik ⎬ i= 1 ⎩ k= 1 ⎭ Constrains concluded in this paper will be developed based on [13], whose model also contained in optioned portfolio. The risk-return preferences of the investor are specified as mean–variance efficiency with additional shortfall constraints expressing the downside risk preferences. −1 ( I − L) wxβγ = C r and a QV ( ≥ Bα ( )) ≥1− α where the meanings of the parameters are explained as: w βγ is the share vector of stocks, call options and put x − 1 options. Set L = C rr′ and I being the matrix with 1 in the c diagonal and 0 else. Let C be the covariance matrix of the (discrete) returns, r ( r1, r2 rp ) = the vector of expected returns and e the p -dimensional vector filled with 1 in each component of the instruments. The steps of calculating the parameters are follows: (1) Covariance matrix of the (discrete) returns C is estimated from history data. r = r r r is also (2) Expected returns vector ( 1, 2 p ) estimated from history data. (3) a = eC ′ − r, b= rC ′ − r, c= eC ′ − e, d = bc− a ⎛b− ari ⎞ ⎛crj − a ⎞ ra = ⎜ ⎟ , rc = ⎜ ⎟ ⎝ d ⎠i= 1,2 p ⎝ d ⎠j= 1,2 p −1 (4) ( ) 1 1 1 2 I − L w = C r xβγ a The following portfolio optimization problem corresponds to this model: © 2013 ACADEMY PUBLISHER
JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 1583 max ⎧ n m C P ∑⎨xr i i + ∑ ( βik Rik + γik Rik ) ⎩ −1 ⎧⎪ ( I − L) wxβγ = C ra ⎨ ⎪⎩ QV ( ≥ B( α )) ≥1−α i= 1 k= 1 st . C. Parameter Uncertainty Most of the parameter such as the expected returns and covariance are estimated from noisy data. Hence, these estimates are no accurate. As a result, if the model amplifies any estimation errors, the portfolios yielding will extremely perform badly in out-of-sample tests. So it needs to solve this problem. And the robust optimization is a good choice. Generally speaking, robust optimization aims to find solutions to a given optimization problems with uncertain parameters which could achieve good objective values for all or most of realizations of the uncertain parameters. We will assume that the estimate covariance is reasonably accurate such that there is no uncertainty about it. This assumption is justified since the estimation error in expectation by far outweighs the estimation error in covariance, see e.g. [14]. So, in decision-making uncertainty is unknown. There are many factors that affect the decision-making, including human psychology state, external information input, which is usually difficult to be derived in terms of probabilistic or stochastic measurement. The well known B–S model has a number of assumptions such as the riskless interest rate and the volatility are constant, which hardly catch human psychology state and external information input although, B–S model has been improved. Now, it needs to introduce robust optimization and portfolio selection [15]. The robust counterpart of an uncertain mathematical program is a deterministic worst case formulation in which model parameters are assumed to be uncertain, but symmetrically distributed over a bounded interval known as an uncertainty set U. The structure and scale of U is specified by the modeler, typically based on statistical estimates. Structure refers to the geometry or shape of the constraint set U, such as ellipsoidal or polyhedral. Scale refers to the magnitude of the deviations of the uncertain parameters from their nominal values; it can be thought of as the size of the structure defining U. A general form of the robust counterpart to an uncertain LP is given as T max ⎡min c x ⎤ ⎣ ( ) Subject to Ax≤b, ∀( A,b,c) ∈ U There are two forms for transfer the robust into a set, linear or Ellipsoidal. (1) Linear interval In the robust optimization framework, the true value a is not certain which is given by the following equation i ⎦ − ^ ai = ai+ a iηi, ∀ i where a − i is an estimate for a i , and a ^ i is the maximum distance that a i deviated from a − i and ηi is a random ⎫ ⎬ ⎭ variable which is bounded by and symmetrically distributed within the interval[-1,1]. That is, the true value ai is symmetrically distributed with respect to i on − ^ − ^ ⎡ ⎤ the interval ai− a i,ai+ ai ⎢ ⎣ ⎥ ⎦ . (2) Ellipsoidal uncertainty sets are given by 2 ⎧ − ⎛ ⎞ ⎫ ⎜ai − ai ⎟ ⎪ 2 a: ⎝ ⎠ ⎪ ⎨ ∑ ^ 2 ≤Ω ⎬ ⎪ a ⎪ i ⎪⎩ ⎪⎭ where Ω is a user defined parameter and adjusts the trade-off between robustness and optimality. Next, the problem is how to transfer the uncertain set to a series equations or in-equations. Let J be the number of parameters. For Soyster’s and Ben-Tal and Nemirovski’s model[16], or ∑ i i a ∑ i ^ i − i − a a η = i Bertsimas and Sim (2004) relaxed this condition by defining a new parameter Γ (the budget of uncertainty) as the number of uncertain parameters that take their worst − ^ case value ai − ai .Therefore ηi ≤Γ,such that Γ ∈⎡⎣ 0, J ⎤⎦ , then the optimal problem can be rewritten as − ^ ⎛ ⎞ max⎜∑ai wi + min∑aiη iwi⎟ ⎝ ηi ⎠ S.t w = 1 ∑ ∑ i i η ≤Γ wi ≥0, −1≤ηi ≤1, ∀ i It also can be rewritten as − ^ ⎛ ⎞ max⎜∑ai wi −max∑ ai ηiwi ⎟ ⎝ ηi ⎠ S.t w = 1 ∑ ∑ i i η ≤Γ wi ≥0,0≤ηi ≤1, ∀ i However, this problem is not well-defined. Because it is difficult to obtain a different optimal solution for each return realization, there are multiple ways to specify the linear set. A nature choice is to construct an ellipsoidal uncertainty set T − Θ = r : r − μ Σ 1 r − μ ≤ δ 2 r { ( ) ( ) } ~ According to EI Ghaoui et al [17],when r has finite second-order moments, then, we can choice δ p = 1− p for p ∈[0,1 ) and δ = +∞ for p = 1 , it means the following probabilistic guarantee for any portfolio w : = J J a − © 2013 ACADEMY PUBLISHER
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Journal of Computers ISSN 1796-203X
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Corn Moisture Measurement using a C
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Call for Papers and Special Issues
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(Contents Continued from Back Cover
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