Jan 4, 2010 - features of this article are: first, we take the no short-sell constraint in Chinese stock .... Stocks. â Mutual funds. â Real estat...

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Abstract: This article is the term paper of the course Investments. We mainly focus on modeling long-term investment decisions of a typical utility-maximizing individual, with features of Chinese stock market in perspective. We adopt an OR based methodology with market information as input parameters to carry out the solution. Two main features of this article are: first, we take the no short-sell constraint in Chinese stock market into consideration and use an approach otherwise identical to Markowitz to work out the optimal portfolio choice; this method has critical and practical implication to Chinese investors. Second, we incorporate the benefits of multiple assets into one single well-defined utility function and use a MIQP procedure to derive the optimal allocation of funds upon each of them along the time-line.

*

Ruokun HUANG, 2006012402, [email protected], http://learn.tsinghua.edu.cn:8080/2006012402/index Yiran SHENG, 2006012400, [email protected], http://learn.tsinghua.edu.cn:8080/2006012400/index

**

Optimal Choice under Short Sell Limit with Sharpe Ratio as Criterion Among Multiple Assets Ruokun HUANG, Yiran SHENG

Contents I.

Background ........................................................................................................................ 3

II.

Introduction ....................................................................................................................... 3

III. Our Model-Overall View................................................................................................... 4 IV. Determination of Insurance value via Monte-Carlo Stimulation....................................... 7 V. Our Model-Stock Market Optimal Choice under Short Sell Limit.................................... 8 VI. Computing Process and Numerical Result ........................................................................ 9 VII. Conclusion ....................................................................................................................... 11 Appendix A - Stock to be Selected.......................................................................................... 12 Appendix B - R code ............................................................................................................... 13 Appendix C – R result ............................................................................................................. 16 Reference ................................................................................................................................. 17

2 / 17 January 4, 2010

Optimal Choice under Short Sell Limit with Sharpe Ratio as Criterion Among Multiple Assets Ruokun HUANG, Yiran SHENG

I. Background This article is for a problem described as follows: •

Suppose you are an investor – Just graduated – 200,000 annual salary – 500,000 saving

•

How to allocate your assets into: – Bank deposit – Treasures – Stocks – Mutual funds – Real estate – Others

The optimal choice of investment has long been a difficult problem for investors. The structure of people’s utility with respect to time and risk influences behaviors of various people. Along with the mainstream method used by scholars, we use well defined utility function to describe investor’s attitude toward risk and return. As we know, there are some existing theories describing the optimal investment choice, i.e. Markowitz method. However, most existing theories are based on relatively complete/perfect market, where short selling of assets is allowed. In our opinion, we must consider this short sell limit in order to avoid discrepancy, which may be large in a market with short sell limit. As a result, we are going to put forward our analysis under short sell limit on stock market.

II. Introduction In section III, we show the basic structure of our model and integrate strategy for different investment categories into one universal maximization problem. In section 3 / 17 January 4, 2010

Optimal Choice under Short Sell Limit with Sharpe Ratio as Criterion Among Multiple Assets Ruokun HUANG, Yiran SHENG

IV, we further discuss the Monte-Carlo simulation procedure of parameters used in section III. In section V, we give a detailed analysis on stock market with short sell constraint, using an approach otherwise identical to Markowitz. In section VI, we apply some data into our model and compute the results. Section VII concludes.

III.

Our Model-Overall View

To adequately characterize the long-run investment decision of a utility maximizing individual, we need to set up a discrete, multi-period model. We assume that the individual lives for M years, and that his lifetime well-being is determined by the yearend consumption of each period. Furthermore, we assume that he or she has time-additive and state-independent utilities. Thus, the maximization problem can be stated as follows: M

max{E (U k ( Dk ))} k 1

subject to feasible set of k Where Dk is his yearend consumption at kth period, Uk is his corresponding utility in kth year, and vector k is our investment strategy. For simplicity, we assume Uk has uniform structure and a discount factor r along the timeline, that is: U k e rk u( Dk )

Next, we will specify the investment strategy k . We classify our investments into four categories: risky assets, riskless asset (borrow), riskless asset (lend/save), house purchase and insurance product. Denote the money invested on each category in the kth year is jk . Notice insurance is invested only at time 0, we will simplify its purchased amount as 4 . Here we denote housing expense every year to be Hk.

However, since the strike event for insurance happens on an unpredictable date in some future time and may not be exactly the same yearend consumption realization 4 / 17 January 4, 2010

Optimal Choice under Short Sell Limit with Sharpe Ratio as Criterion Among Multiple Assets Ruokun HUANG, Yiran SHENG

date. Therefore we need to make a little modification in the model, introducing a continuous random variable T called survive time which is the interval starts from time 0 until the strike event happens. We assume T follows some distribution F(t). Once the strike event happens, the person’s annual income drops from IH to IL. The expected utility gain from insurance is, L is the lump sum payment once the strike event happens: M

V E (e rT u (3 L)) e ru u (3 L) Pr[T du ] 0

With this factor included, the full model is: M M rk max ( k , H k ) E e u ( Dk ) e ru u 4 L Pr T du u House k 1 0

3

3

j 1

j 1

Dk I L ( I H I L ) T k jk j ,k 1 (1 R j ) s 4 T k H k 1, T k where T k 0, T k s.t.

Dk D

jM 0 ,j=1,2,3

Where IH and IL are the given annual income, R1 is the random return on risky portfolio, R2 r is the risk free (borrow) rate, R3 r is the risk free (lend/save) rate, s is the yearly spread payment of insurance. D is the survival level annual consumption. Now suppose E(u( D)) aE ( D) b var( D) , the above equation becomes, assuming different category of assets are independent with each other: 5 / 17 January 4, 2010

Optimal Choice under Short Sell Limit with Sharpe Ratio as Criterion Among Multiple Assets Ruokun HUANG, Yiran SHENG

M rk max ( k , H k ) e aE ( Dk ) b var( Dk ) V u House k 1 3

3

j 1

j 1

E ( Dk ) I L ( I H I L )(1 F (k )) jk j ,k 1 (1 E ( R j )) s4 (1 F (k ))

var( Dk ) ( I H I L )2 F (k )(1 F (k )) 1,k 12 var( R2 ) s 24 2 F (k )(1 F (k )) The value of V can be derived from Monte-Carlo stimulation, we will discuss this later. Next, we are going to discuss the term Hk. In order to make our planning applicable for computer, we set up a binary vector B , where

0 , n o t b u y h o u s e a t t h i s y e a r Bk 1 , b u y h o u s e ( p a y i n i t i a l ) t h i s y e a r We can easily reach the result that:

Ip Ap H Ap 0

Ip Ap

B , Ip=Initial payment, Ap=Annual payment Ap Ip

We denote the transformation matrix as P Note that the optimization problem is just a quadratic problem like:

1 max xT Σx cT x 2 s.t.Ax b with x 1,1 ,1,2

1,M ,2,1 2, M ,3,1 3, M , H1

H M ,4

T

This problem is equivalent to:

1 max y T Σy d T y 2 s.t.Dx e with y 1,1 ,1,2

1,M ,2,1 2, M ,3,1

3, M , B1

BM ,4

T

6 / 17 January 4, 2010

Optimal Choice under Short Sell Limit with Sharpe Ratio as Criterion Among Multiple Assets Ruokun HUANG, Yiran SHENG

1 T T and d c 0 0 0

0 0 0 1 0 0 0 P 0 0 0 1

This is always applicable since P 0 .

IV.

Determination

of

Insurance

value

via

Monte-Carlo Stimulation In order to specify the mechanism of insurance product in a sense of its contribution to life-time utilities, we introduce a latent variable model. Suppose Y is a latent variable follows standard normal distribution. When Y falls below a critical value, the strike event in insurance contract happens, id est: Pr[T t ] F (t ) Pr[Y y] ( y)

Therefore there is a one-to-one mapping between Y and T:

T F 1 ((Y )) Since T is a rv in nature, we need to carry out a Monte-Carlo Stimulation to determine the expected utility gain from insurance products. Throughout this session we assume F(t)=1-e-ht; where h is a constant hazard rate, its mathematical meaning is the strike event conditional probability density at any time. T

log(1 (Y )) h

The following table gives input parameters for the stimulation: Parameters

Input value

Hazard rate: h

0.06

Lump sum payments: L

30000

Spread payment: s

500

For h =0.06, the probability that the strike event will not happen within 30 years 7 / 17 January 4, 2010

Optimal Choice under Short Sell Limit with Sharpe Ratio as Criterion Among Multiple Assets Ruokun HUANG, Yiran SHENG

is about 17%. After 10000 stimulation runs over Y, we have V = 0.6629u(4L)

V. Our Model-Stock Market Optimal Choice under Short Sell Limit Firstly, in classical model, where short selling of assets is allowed, the optimal problem has simple solution as follows:

w

Σ 1e E rM rf H

Where E rM

A C

D A C rf C

,

2

A 1T Σ-1e B eT Σ-1e C 1T Σ-1 1 D BC A2

H B 2 Arf Crf 2 Actually under Markowitz’s world, it is the solution of the maximizing Sharpe Ratio. Here in our analysis, we also use Sharpe Ratio as a criterion of portfolio evaluation. However, we assume that short sell is limited, as in Chinese stock market. The mathematical description of our analysis is as follows: E rw rf w

max w

1T w 1, subject to wi 0, i There is no simple solution to this non-linear planning problem as that of Markowitz. Hereby we use software R (R Development Core Team, 2008), package 8 / 17 January 4, 2010

Optimal Choice under Short Sell Limit with Sharpe Ratio as Criterion Among Multiple Assets Ruokun HUANG, Yiran SHENG

Rdonlp2 to solve this problem. Secondly, there are too many stocks in the market, specifically more than 1600 in A share market. From a general view these stocks are to some extent homogeneous, since they are often highly related. In our analysis, for the convenience of coumputing, we choose several stocks from the same industry, id est 深发展 A (000001) from Financial Institution industry, et cetera. Hereby we choose 143 stocks from the 1600+ stocks in A share market, listed in Appendix A (page 12).

VI.

Computing Process and Numerical Result

The data we choose are as follows: Stock: 143 A stocks as in Appendix A (page 12). Utility discount rate: 3% Risk free rate (Borrow): 6.5% Risk free rate (Lend/save):2.5% House initial payment: $1,800,000 House annual payment: $150,000 (other input data are in Appendix B-R code)

The numerical result is: Choose these stocks as a mutual fund. 变量 No.

股票号码

股票名称

比例

2

000002

万科 A

0.142771581

13

000400

许继电气

0.0105711138

22

000515

攀渝钛业

0.0230618999

25

000538

云南白药

0.2125265671

35

000661

长春高新

0.0240573083

40

000792

盐湖钾肥

0.2222784446

46

000816

江淮动力

0.0147827086

53

000895

双汇发展

0.0642883098

75

600038

哈飞股份

0.010492248

83

600096

云天化

0.070783097 9 / 17

January 4, 2010

Optimal Choice under Short Sell Limit with Sharpe Ratio as Criterion Among Multiple Assets Ruokun HUANG, Yiran SHENG

113

600519

贵州茅台

0.1458883638

139

600875

东方电气

0.058498358

Invest in stock market by buying following pieces of mutual fund: Year

Amount

Year

Amount

1

0.02688647

16

0.00929332

2

0.02524552

17

0.00901866

3

0.02462977

18

0.00875212

4

0.02402905

19

0.00849345

5

0.02344297

20

0.00824243

6

-0.00029566

21

0.00799883

7

0.01273575

22

0.00776243

8

0.01236226

23

0.00753302

9

0.01146496

24

0.00731038

10

0.01112612

25

0.00709433

11

0.0107973

26

0.00688466

12

0.01047819

27

0.00668119

13

0.01016851

28

0.00648373

14

0.009867986

29

0.00629211

15

0.009576343

30

0

The borrowing behavior is: Borrow 25.41707 pieces= $25,417.07 at year 6, zero otherwise. The saving behavior is: (zero otherwise) Year

Amount

1

$

689,222.7

2

$

895,705.3

3

$1,107,349

4

$1,324,284

5

$1,546,643

Buy house at year 6. Besides, the mutual fund has the frontier like:

10 / 17 January 4, 2010

Optimal Choice under Short Sell Limit with Sharpe Ratio as Criterion Among Multiple Assets

0.10 0.00

0.05

pltreturn

0.15

0.20

Ruokun HUANG, Yiran SHENG

0.0

0.2

0.4

0.6

0.8

pltsigma2

The ‘straight line’ in the graph above is the original frontier without short sell limit, which appears to be much larger than our frontier. Note that that frontier is a parabolic curve at a proper scale.

VII. Conclusion a) Short sell limit plays an important role in Chinese stock market From the graph we showed above, we can see that with limit of short sell, the feasible set shrinks much compared to situation without short sell constraint. So, we must consider short sell limit in Chinese stock market. b) Stock market as an asset of investment should be careful considered In our optimal solution, mutual fund is invested at relatively a very low level. This implies one should better choose other investment assets other than stock. c) House purchase is the core of life due to large amount of utility it brings All the financing method in our analysis, including borrowing and saving, contribute to the ‘final target’ of buying a house. The optimal choice indicates one 11 / 17 January 4, 2010

Optimal Choice under Short Sell Limit with Sharpe Ratio as Criterion Among Multiple Assets Ruokun HUANG, Yiran SHENG

should save money for 5 years, and as long as he has enough or near to enough money, he purchases a house with small amount of loan. This solution is very close to modern Chinese young people. d) Further work to be done The overall utility assumption also has its limitations. The state independent assumption tends to be invalid since the insurance is introduced in this model. The utility of house can be decided by more factors. Besides, the period of house payment and/or payment structure should be flexible.

Appendix A - Stock to be Selected No

股票号码 股票名称

No

股票号码 股票名称

No

股票号码 股票名称

1

000001

深发展 A

51

000860

顺鑫农业

101

600299

蓝星新材

2

000002

万科 A

52

000878

云南铜业

102

600300

维维股份

3

000009

中国宝安

53

000895

双汇发展

103

600305

恒顺醋业

4

000012

南玻 A

54

000911

南宁糖业

104

600313

ST 中农

5

000031

中粮地产

55

000931

中关村

105

600315

上海家化

6

000034

ST 深泰

56

000933

神火股份

106

600320

振华港机

7

000040

深鸿基

57

000936

华西村

107

600333

长春燃气

8

000043

中航地产

58

000938

紫光股份

108

600350

山东高速

9

000049

德赛电池

59

000951

中国重汽

109

600356

恒丰纸业

10

000060

中金岭南

60

000962

东方钽业

110

600382

广东明珠

11

000061

农产品

61

000968

煤气化

111

600416

湘电股份

12

000089

深圳机场

62

000972

新中基

112

600418

江淮汽车

13

000400

许继电气

63

000988

华工科技

113

600519

贵州茅台

14

000401

冀东水泥

64

000990

诚志股份

114

600528

中铁二局

15

000402

金融街

65

000998

隆平高科

115

600530

交大昂立

16

000410

沈阳机床

66

000999

三九医药

116

600559

老白干酒

17

000420

吉林化纤

67

600001

邯郸钢铁

117

600585

海螺水泥

18

000423

东阿阿胶

68

600007

中国国贸

118

600587

新华医疗

19

000425

徐工科技

69

600008

首创股份

119

600597

光明乳业

20

000426

富龙热电

70

600011

华能国际

120

600598

北大荒

21

000509

SST 华塑

71

600026

中海发展

121

600611

大众交通

22

000515

攀渝钛业

72

600028

中国石化

122

600631

百联股份

23

000518

四环生物

73

600036

招商银行

123

600633

*ST 白猫

24

000527

美的电器

74

600037

歌华有线

124

600655

豫园商城

25

000538

云南白药

75

600038

哈飞股份

125

600661

交大南洋 12 / 17

January 4, 2010

Optimal Choice under Short Sell Limit with Sharpe Ratio as Criterion Among Multiple Assets Ruokun HUANG, Yiran SHENG

26

000552

靖远煤电

76

600062

双鹤药业

126

600663

陆家嘴

27

000554

泰山石油

77

600064

南京高科

127

600701

工大高新

28

000585

东北电气

78

600066

宇通客车

128

600710

常林股份

29

000591

桐君阁

79

600073

上海梅林

129

600723

西单商场

30

000597

东北制药

80

600085

同仁堂

130

600737

中粮屯河

31

000598

蓝星清洗

81

600087

长航油运

131

600754

锦江股份

32

000619

海螺型材

82

600090

啤酒花

132

600756

浪潮软件

33

000629

攀钢钢钒

83

600096

云天化

133

600775

南京熊猫

34

000630

铜陵有色

84

600100

同方股份

134

600798

宁波海运

35

000661

长春高新

85

600106

重庆路桥

135

600806

昆明机床

36

000682

东方电子

86

600109

国金证券

136

600835

上海机电

37

000729

燕京啤酒

87

600111

包钢稀土

137

600849

上海医药

38

000758

中色股份

88

600115

东方航空

138

600867

通化东宝

39

000768

西飞国际

89

600135

乐凯胶片

139

600875

东方电气

40

000792

盐湖钾肥

90

600138

中青旅

140

600879

火箭股份

41

000798

中水渔业

91

600150

中国船舶

141

600887

伊利股份

42

000799

酒鬼酒

92

600162

香江控股

142

600889

南京化纤

43

000802

北京旅游

93

600169

太原重工

143

600895

张江高科

44

000807

云铝股份

94

600186

莲花味精

45

000811

烟台冰轮

95

600188

兖州煤业

46

000816

江淮动力

96

600192

长城电工

47

000830

鲁西化工

97

600195

中牧股份

48

000833

贵糖股份

98

600197

伊力特

49

000856

唐山陶瓷

99

600229

青岛碱业

50

000858

五粮液

100

600283

钱江水利

Appendix B - R code library("Rcplex") library("Matrix") library("xlsReadWrite") library("Rdonlp2") #Read libraries #Parameters, ‘1’=$1000 MonteCarloNumber=10000

#Times of experiment in Monte-Carlo

InitialSaving=500

#Initial saving

years=30

#Total Years

HouseInitial=1800

#House Initial payment

HouseAnnual=150

#House annual payment

HouseY=10

#House payment years

r=0.03

#Utility Discount rate 13 / 17

January 4, 2010

Optimal Choice under Short Sell Limit with Sharpe Ratio as Criterion Among Multiple Assets Ruokun HUANG, Yiran SHENG

rbottom=0.025

#Risk free (lend,save) rate

rtop=0.065

#Risk free (Borrow) rate

L=30

#Insurance lamp-sum payment

HouseUtil=3500

#utility of House, in money sense

h=0.06

#hazard rate

s=.5

#spread of Insurance

Dbottom=10

#lowest living expense

Ih=200

#Annual Salary

Il=10

#Annual salary after losing job

B=3

#risk averse

rhouse=0.0

#House price growth rate

yy=rnorm(MonteCarloNumber,0,1) tt=-log(1-pnorm(yy))/h vv=exp(-r*tt) V=mean(vv)

#Calculate V using Monte Carlo

kstart=ceiling(mean(tt)) Fk<-function(x){ tst=1-exp(-x*h) tst }#Define Fk #start to optimize stock fn <- function(x){ -(expectret%*%x - rbottom/12)/ sqrt(+t(x)%*%sigma%*%x)

#Min fn.}

inmat=read.xls("s2ok.xls",sheet=1,type="double") n=ncol(inmat) sigma=cov(inmat);expectret=apply(inmat,MARGIN=2,FUN=mean) p <- rep((1/n),n)

#initial value

par.l <- rep(0,n); par.u <- rep(1,n)

#Parameter range

lin.u <- 1; lin.l <- 1 A <- t(rep(1,n)) ret <- donlp2(p, fn, par.lower=par.l, par.upper=par.u, A=A, lin.u=lin.u, lin.l=lin.l, name="stockopl") rstock<-expectret%*%(ret$par) sigma2stock=t(ret$par)%*%sigma%*%(ret$par)*12 sigma2stock=as.numeric(sigma2stock) rstock=as.numeric(rstock) #vector=(stock1,stock2...stock30,borrow1,borrow2..borrow30,save1...sa ve30,B1...B30,Insurance) tempm=diag(HouseInitial,years) for (i in 1:years){ for (j in (i+1):(i+HouseY)){ if (j<=years ) tempm[j,i]=HouseAnnual } tempm[i,i]=HouseInitial*exp(i*rhouse)} transform_mat_q=as.matrix(bdiag(diag(1,3*years),tempm,1)) 14 / 17 January 4, 2010

Optimal Choice under Short Sell Limit with Sharpe Ratio as Criterion Among Multiple Assets Ruokun HUANG, Yiran SHENG

term1_v1=matrix(0,1,4*years+1) for (i in 1:years){ if (i

(i

term1_v1[i+years]=+exp(-i*r)-exp(-(i+1)*r)*(1+rtop)

else term1_v1[i+years]=-exp(-i*r) }#coefficient of borrow for (i in 1:years){ if (i

a1[i,i-1]=1+rstock}

a3=diag(-1,years) for (i in 2:years){

a3[i,i-1]=1+rbottom}

a2=diag(1,years) for (i in 2:years){

a2[i,i-1]=-1-rtop}

a4=diag(-1,years); a4=a4%*%tempm a5=rep(-s,years) for (i in kstart:years){a5[i]=0} a=cbind(a1,a2,a3,a4,a5) al=t(rep(0,4*years+1)) for (i in (3*years+1):(4*years)){al[i]=-1} a=rbind(a,al) #Amat ok 15 / 17 January 4, 2010

Optimal Choice under Short Sell Limit with Sharpe Ratio as Criterion Among Multiple Assets Ruokun HUANG, Yiran SHENG

b=rep(Dbottom-Ih,years+1) for (i in kstart:years){b[i]=Dbottom-Il} b[years+1]=-1 b[1]=b[1]-InitialSaving #bvec ok. Q=Q*(-2)*B myvtype=rep("C",4*years+1) for (i in (3*years+1):(4*years)){myvtype[i]="B"} ub=rep(Inf,4*years+1) for (i in (4*years-HouseY+1):(4*years)){ub[i]=0} myresult

Appendix C – R result The optimal choice of stocks under short sell limit, whose parameter is not zero: 变量 No.

股票号码

股票名称

比例

2

000002

万科 A

0.142771581

13

000400

许继电气

0.0105711138

22

000515

攀渝钛业

0.0230618999

25

000538

云南白药

0.2125265671

35

000661

长春高新

0.0240573083

40

000792

盐湖钾肥

0.2222784446

46

000816

江淮动力

0.0147827086

53

000895

双汇发展

0.0642883098

75

600038

哈飞股份

0.010492248

83

600096

云天化

0.070783097

113

600519

贵州茅台

0.1458883638

139

600875

东方电气

0.058498358

Final MIQP result is: (total years=30) [1] 2.688647e-02 2.524552e-02 2.462977e-02 2.402905e-02 2.344297e-02 [6] -2.956562e-04 1.273575e-02 1.236226e-02 1.146496e-02 1.112612e-02 [11] 1.079730e-02 1.047819e-02 1.016851e-02 9.867986e-03 9.576343e-03 [16] 9.293319e-03 9.018660e-03 8.752118e-03 8.493454e-03 8.242434e-03 [21] 7.998834e-03 7.762432e-03 7.533018e-03 7.310384e-03 7.094329e-03 [26] 6.884660e-03 6.681188e-03 6.483729e-03 6.292105e-03 0.000000e+00 [31] 0.000000e+00 -3.409720e-04 0.000000e+00 0.000000e+00 0.000000e+00 [36] 2.541707e+01 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [41] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [46] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [51] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00

16 / 17 January 4, 2010

Optimal Choice under Short Sell Limit with Sharpe Ratio as Criterion Among Multiple Assets Ruokun HUANG, Yiran SHENG [56] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [61] 6.892227e+02 8.957053e+02 1.107349e+03 1.324284e+03 1.546643e+03 [66] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [71] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [76] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [81] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [86] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [91] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [96] 1.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [101] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [106] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [111] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [116] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [121] 1.500728e+00

Reference Huang, Chi-fu and Litzenberger, Robert H. 1989. Valuation of Comlex Securities with Preference Restrictions. Foundations for Financial Economics. s.l. : Prentice Hall, 1989, 6. Hull, John. 2009. Options, Futures and Other Derivatives. s.l. : Pearson Education Ltd., 2009. ISBN 0-13-500994-4. P. Spellucci, TAMURA Ryuichi. Rdonlp2. R package. [Online] http://arumat.net/Rdonlp2/. R Development Core Team. 2008. R: A Language and Environment for Statistical Computing. [Online] R Foundation for Statistical Computing, 2008. Vienna, Austria. http://www.R-project.org. 3-900051-07-0. Schmidt, Wolfgang. 2008. Monte-Carlo Variance Reduction-A new universal method for multivariate problems. 2008. working paper. Sharpe, W. 1964. Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance. 1964, 19, pp. 425-442.

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