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1584 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 ~ T T { r∈Θ } r P w r ≥minw r ≥ p The optimal problem reduces to a convex second-order cone program[18]. T 2 T max w μ − δ Σ 1/ w 1 = 1, l ≤ w ≤ u w { } 2 According to the Central Limit Theorem, it is ^ concluded that the sample mean μ is approximately normally distributed. That is ,it follows: ^ ⎛ Σ ⎞ μ ~ N ⎜ μ, ⎟ ⎝ n ⎠ Similarly, the ellipsoidal uncertainty set for the mean μ can be expressed as Θ μ −1 ⎪⎧ ^ ⎛ Σ ^ ⎛ ⎞ ⎞ ⎛ ⎞ ⎪⎫ 2 = ⎨μ : ⎜ μ − μ ⎟⎜ ⎟ ⎜ μ − μ ⎟ ≤ κ ⎬ ⎪⎩ ⎝ ⎠⎝ n ⎠ ⎝ ⎠ ⎪⎭ where κ = q / 1− q for some q ∈[0,1) The problem reduces to ⎪⎧ 1/ 2 ^ ⎛ Σ ^ 1/ 2 ⎪⎫ T ⎞ T max⎨w μ−κ ⎜ ⎟ w −δ Σ w w 1 = 1, l ≤ w ≤ u⎬ w ⎪⎩ ⎝ n ⎠ 2 2 ⎪⎭ See[19] the problem is finally reduced to ⎪⎧ ^ ^ 1 / 2 ⎪⎫ T 1 / 2 T max⎨w μ − κ Ω w − δ Σ w w 1 = 1, l ≤ w ≤ u⎬ w 2 ⎪⎩ 2 ⎪⎭ where Σ 1 Σ T Σ Ω = − 11 n T Σ n n 1 1 n In this paper, firstly, we consider the uncertain set for return mean. We define r′ is the estimation of real value r , the uncertainly set I as { : i ′ i i i ′ i } I = r r −s ≤r ≤ r + s for mean μ . According to Anna[20], the robust counterpart: rx ≥ r ∑ min i i p can be transferred to the following form: ⎧∑rx ′ i i −∑sm i i ≥rp ⎪ ⎨ mi ≥ xi ⎪ ⎩ si ≥ 0 The robust counterpart of objective function is n m ⎛ ⎧ C P ⎫⎞ max ⎜∑⎨xr i i + ∑ { βik Rik + γik Rik} ⎬⎟ x, βγ , ⎝ i= 1 ⎩ k= 1 ⎭⎠ let x αβγ is the share of stock and options. The goal is to determine the solution of above problem under the constraints. Ⅲ.MONTE CARLO SIMULATION AND EMPIRICAL EXAMPLE A. Monte Carlo Method and the Simulation Process Comparing with other numerical methods, Monte Carlo simulation has two major advantages: first, more flexible, easy to implement and improvement; secondly, the simulation of estimation error and convergence speed in solving the problem has strong independence of dimension. European option because of its execution time is fixed, not to be executed in advance, therefore it only need to calculate the earnings of the option of each sample path at expiration date, which is available by Matlab programming. [21-23] discuss the application of the simulation methods in various area. Monte Carlo method can overcome the obstacle and we use it further to improve the accuracy of simulated price with the enhancement of reduction variate technique for more complex options whose payoff function is dependent on the underlying asset path and sum of asset is more than one. Now, we illustrate the key steps in Monte Carlo. It is saw that to draw samples of the terminal stock price S( T ) it suffices to have a mechanism for drawing samples from the standard normal distribution. For now we simply assume the ability to produce a sequence Z1, Z2 of independent standard normal random variables. Given a mechanism for generating the Z i , we rT can estimate E⎡ − e ( ST − K ) + ⎤ using the following ⎣ ⎦ algorithm: For i = 1, 2n generate Z i ⎛⎡ 1 ⎤ ⎞ Si T = S0 exp⎜⎢ r− σ + 2 ⎥ T σ TZi⎟ ⎝⎣ ⎦ ⎠ set ( ) 2 C = e S −K −rT set ( ) + ^ i set n = ( + + + ) T C C C C n 1 2 n / For any n ≥ 1, the estimator C n is unbiased, in the sense that its expectation is the target quantity: ^ ⎛ ⎞ − rT + E⎜Cn ⎟= C = E⎡ ⎣e ( ST −K) ⎤ ⎝ ⎠ ⎦ The estimator is strongly consistent meaning that as n →∞. In this paper, we suppose z = z() t is a random process, the change in a very small time interval Δt is expressed as Δ z . If Δz satisfies that Δ z = ε Δ t where ε ∼ N ( 0,1) . For different time interval Δ t , Δz are independent, then call z = z() t follows Wiener process. Suppose the stock price follows ds = μsdt + σ sdz , where dz is the Standard Brown motion. In the practical ^ © 2013 ACADEMY PUBLISHER
JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 1585 application, more accurate simulation not starts from S, but log-price ln S . Monte Carlo simulation steps: (1) To generate sample paths for underlying asset, given the initial value i i i i S = t 1 S + t μiS Δ t t + + σiS Δ t tεt (2)To calculate option price of each sample path. (3) To average option price for each sample path. IV.EMPIRICAL EXAMPLE In order to illustrate the features and applications of this model, we make a numerical example. For simple, we only consider two stocks. And there is a call and a put option based on each stock. Suppose that the investment horizon is T = 1year, including 20 trading days in each month, so there is 240 trading days in total. To divide T by daily, that is Δ t =1day, equals to 1 year. The 240 price of each stock is supposed to follow log-normal distribution, then the price of stock i,( i = 1,2) in t + 1day is: i i i i S S μ S t σ S tε ε ∼ N 0,1 = + Δ + Δ , ( ) + t t 1 t i t i t t i i i Generate a path S 0 , S 1 S 240 for stock i by Monte Carlo method. Each stock will only correspond to a European call option and a European put option, asset specific parameters are as follows: μ1 = 11%, σ1 = 26.86%; μ2 = 8.05%, σ2 = 16.3% Other parameters are as shown in the following. The kind of option, underlying, market price, option price, time and strike price respectively are: For call option C 1 whose underling is S with initial 1 value S 1 ( 0 ) =15.59, the option premium C 1 ( 0 ) =2.17, the strike price is K 11 =14.5, the expired time is 6 month. For call option P 1 whose underling is S 1 with initial P =1.87, value S ( ) =15.59, the option premium ( ) 1 0 1 0 the strike price is K 12 =16.5, the expired time is 12 month. For call option C whose underling is 2 S 1 with initial value ( ) 2 0 C 2 0 =1.32, the strike price is K =13, the expired time is 3 month. 21 For call option P 2 whose underling is S 2 with initial P =1.48 , S =13.71, the option premium ( ) value S ( ) =13.71, the option premium ( ) 2 0 2 0 the strike price is K =15, the expired time is 9 month. 22 If the option j based on stock i is exercised on the l ( ≤ 240) day, the option value is i i i i ( Srr 01 2 rl − Kij) max 0, ,and in the rest of investment horizon, that is, in the following 240 − l days,the value is treated as risk free asset,so the total value of the option in the investment horizon is: i i i i r( 240 −l) / 365 max ( 0, Srr 0 1 2rl − Kij) e , r= 5% is the risk free interest rate. The European call option price before expiration day, for example, on the v− th day is i i i i −r l−v i max 0, Srr r − K e − c , v≤ l ( ) ( l ij) 0 1 2 0 i where C is the option current price (option premium) 0 based on stock i. If v> l , then on the v− th day, the call option price is r ⎛ ⎞ 365 i i i i ⎜e −1⎟max( 0, S0r1r2 rl −Kij) ⎝ ⎠ Suppose that the portfolio assets real returns are μ = μ , μ , r 1 , r 1 , r 2 , r 2 , and the means of samples ( 1 2 c p c p ) return are μ ( μ′ ) 1 1 2 2 1 , μ ′ 2 , r ′ c , r ′ p , r ′ c , r ′ p ′ = with investment share x = ( x , x , w , w , w , w ) . We construct the model: 1 2 1 2 3 4 μ x μ x r w r w r w r w 1 1 2 1 max 1 1+ 2 2+ c 1+ p 2+ c 3+ p 4 ⎧ 1 1 2 1 μ′ 1 x1+ μ ′ 2 x2 + r ′ c w1+ r ′ p w2 + r ′ c w3+ r ′ p w4 ≥0.001 ⎪ −1 ⎪ ( I − L) wxβγ = C μa ⎪ −1 −1 −1 2 a= eC ′ μ, b= μ′ C μ, c= eC ′ e, d = bc−a ⎪ ⎪ ⎛b−aμ ⎞ ⎛cμ−a⎞ −1 ⎪ μa = ⎜ ⎟, μc = ⎜ ⎟, L= C μcμ′ ⎨ ⎝ d ⎠ ⎝ d ⎠ ⎪x1+ x2 + w1+ w2 + w3+ w4 = 1 ⎪ ⎪μ′ x + μ ′ x + r ′ w + r ′ w + r ′ w + r ′ w −∑ sm ≥r ⎪ ⎪mi ≥ xi ⎪ ⎩si ≥ 0 1 1 2 1 1 1 2 2 c 1 p 2 c 3 p 4 i i p where C is the covariance matrix between the assets , the minimum return for an investor is 2%. By solving the above model, we obtain the optimal portfolio is (0.4,-0.04,-0.1,-0.3,-0.16), the objective is0.00682456. If it is set s i = 0 , that is there is no robust of return mean, the result is 0.0070175439. It is easy to understand that under robust, investment is more conservative. Because the advantage of combining option in portfolio is option could hedging with risk. In order to test it, we change the variance from small to large, for example, supposeσ 1 = 30%; σ 2 = 25% , we find that the objective is 0.0052984, if there is without options, the objective is 0.000215. That is, options in portfolio could hedge risks. V. CONCLUSION This paper extents the general portfolio model in two aspects. The first is to combined option in the portfolio could hedge the risk, and the options can also considered as an asset in the portfolio, extending the general model.And we use Monte Carlo method to simulate the © 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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(Contents Continued from Back Cover
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