research article mathematical model of stock prices via a

Research ArticleMathematical Model of Stock Prices via a Fractional BrownianMotion Model with Adaptive Parameters

Tidarut Areerak

School of Mathematics Institute of Science Suranaree University of Technology Nakhon Ratchasima 30000 Thailand

Correspondence should be addressed to Tidarut Areerak tidarutsutacth

Received 13 February 2014 Accepted 30 March 2014 Published 7 April 2014

Academic Editors F Sartoretto and C Zhang

Copyright copy 2014 Tidarut Areerak This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The paper presents a mathematical model of stock prices using a fractional Brownian motion model with adaptive parameters(FBMAP) The accuracy index of the proposed model is compared with the Brownian motion model with adaptive parameters(BMAP)The parameters in bothmodels are adapted at any timeTheADVANC Info Service Public Company Limited (ADVANC)and Land and Houses Public Company Limited (LH) closed prices are concerned in the paper The Brownian motion model withadaptive parameters (BMAP) and fractional Brownian motion model with adaptive parameters (FBMAP) are applied to identifyADVANC and LH closed prices The simulation results show that the FBMAP is more suitable for forecasting the ADVANC andLH closed price than the BMAP

1 Introduction

The ideas of using a Brownian motion process to explain thebehavior of the risky asset prices were presented by Black etal [1ndash3] The stock prices presented in the paper are also thetype in the risky asset pricesTherefore the Brownianmotionis usually used to model a stock price However Brownianmotion process has the independent increments propertyThis means that the present price must not affect the futureprice In fact the present stock price may influence the priceat some time in the future Hence Brownian motion processis not suitable to explain the stock price Another processa fractional Brownian motion process exhibits a long rangedependent propertyTherefore a fractional Brownianmotionprocess can be used to describe the behavior of stock priceinstead of Brownian motion process

The rate of return and volatility in general asset pricingmodel are usually the constant parameters Actually the rateof return and volatility in the model are not constant at anytime In the paper these parameters are updated dependingon time by using the new information

The ADVANC Info Service Public Company Limited(ADVANC) and Land and Houses Public Company Limited(LH) stock prices are considered in the paper These twostocks are chosen from different stock exchange of Thailand

(SET) industry groups The ADVANC and LH prices areselected from technology group (TECH) and property andconstruction group (PROPCON) respectively

The ADVANC and LH stock price models are studied byusing fractional Brownian motion process to explain uncer-tainly behavior instead of Brownian motion process TheBrownian motion model with adaptive parameters (BMAP)and the fractional Brownian motion model with adaptiveparameters (FBMAP) are presented to model the ADVANCand LH stock prices

The paper is organized as follows Preliminaries on afractional Brownian motion are given in Section 2 Theestimation of the rate of return and volatility is shown inSection 3 In Section 4 the BMAP and the FBMAP areexplained Finally Section 5 concludes the work in the paper

2 Preliminaries on a Fractional BrownianMotion Process

In the general accepted model the randomness of stockprice is modelled by Brownian motion process A stock priceprocess (119878

119905 119905 ge 0) is represented by the stochastic differential

equation (SDE) as shown in

119889119878119905= 119878119905(120583119889119905 + 120590119889119882

119905) (1)

Hindawi Publishing CorporationISRN Applied MathematicsVolume 2014 Article ID 791418 6 pageshttpdxdoiorg1011552014791418

2 ISRN Applied Mathematics

Note that the parameters 120583 and 120590 are the rate of returnand the volatility respectively The process (119882

119905 119905 ge 0) in

(1) is a standard Brownian motion process The stochasticdifferential equation (1) is driven by the Brownian motionprocess (119882

119905 119905 ge 0) In the real world 120583 and 120590 in (1) are not

constant at any time Hence these parameters in the paperare the adaptable parameters based on time In the papermodel (1) is called a Brownian motion model with adaptiveparameters (BMAP)

In practice the dynamics of stock price have a longmem-ory (long range dependence) The BMAP model in (1) is notsuitable to describe the dynamics of stock price Thereforethe fractional Brownian motion process is considered in thepaper The fractional Brownian motion process (119861119867

119905 119905 ge 0)

withHurst index119867 is a centeredGaussian process If119867 = 05then (119861119867

119905 119905 ge 0) is a standard Brownian motion process If

119867 = 05 then (119861119867119905 119905 ge 0) is neither a semimartingale nor a

Markov process For119867 = 05 case the (119861119867119905 119905 ge 0) is the long

memory process The (119861119867119905 119905 ge 0) is represented in (2) by

Mandelbrot and Van Ness [4] Consider the following

119861119867

119905=

1

Γ (1 + 120572)[119885119905+ 119861119905] (2)

The function Γ(sdot) is the gamma functionThe process (119885119905 119905 ge

0) is defined by 119885119905= int0

minusinfin[(119905 minus 119904)

120572minus (minus119904)

120572]119889119882119904 The process

(119861119905 119905 ge 0) is described by 119861

119905= int119905

0(119905 minus 119904)

120572119889119882119904 The parameter

120572 = 119867 minus 12 where 119867 isin (0 1) and (119882119905 119905 ge 0) is a standard

Brownian motion processThe rate of return and volatility are not constant at any

time Hence the paper also proposes the new approach of theasset pricingmodel In this case the driving process of model(1) is replaced by a fractional Brownian motion processThe rate of return and volatility are adaptive parameters Inthis case the model can be represented by the stochasticdifferential equation (SDE) as shown in

119889119878119905= 119878119905(120583119889119905 + 120590119889119861

119867

119905) (3)

The parameters 120583 and 120590 in (3) are the rate of return and thevolatility respectively The 120583 and 120590 are adaptive parametersthe same as the previousmodelThe (119861119867

119905 119905 ge 0) is a fractional

Brownian motion process In the paper model (3) is called afractional Brownian motion model with adaptive parameters(FBMAP)

Alos et al [5] have proposed to use the process (119861119905 119905 ge

0) instead of (119861119867119905 119905 ge 0) since (119885

119905 119905 ge 0) has absolutely

continuous trajectory So the process (119861119905 119905 ge 0) has long

range dependence Hence the model (3) can be consideredas shown in

119889119878119905= 119878119905(120583119889119905 + 120590119889119861

119905) (4)

An approximate approach to stochastic differential equa-tion perturbed by fractional Brownian motion was proposedby Thao [6] The process (119861120576

119905 119905 ge 0) is introduced For every

120576 gt 0 the process (119861120576119905 119905 ge 0) is defined by

119861120576

119905= int119905

0

(119905 minus 119904 + 120576)120572119889119882119904 (5)

The process (119861120576119905 119905 ge 0) is a semimartingale Therefore this

process can be written as in

119861120576

119905= 120572int

119905

0

120593120576

119904119889119904 + 120576

120572119882119905 (6)

where 120593120576119905= int119905

0(119905 minus 119904 + 120576)

120572minus1119889119882119904

The process (119861120576119905 119905 ge 0) converges to (119861

119905 119905 ge 0) in 1198712(Ω)

when 120576 approaches to 0 This convergence is uniform withrespect to 119905 isin [0 119879] Hence the model (4) can be consideredas shown in

119889119878119905= 119878119905(120583119889119905 + 120590119889119861

120576

119905) (7)

3 The Estimation of the Rate ofReturn and Volatility

In this paper the ADVANC and LH closed prices areidentified by two asset pricing models In the BMAP thedriving process is Brownian motion On the other handthe driving process is fractional Brownian motion in theFBMAPThe parameters 120583 and 120590 in both models are adaptiveparameters at any time The ADVANC and LH simulatedstock prices are compared with these empirical prices TheADVANC and LH empirical prices can be obtained fromhttpwwwsetorththindexhtml The data of ADVANCand LH empirical prices from July 9 2010 to July 8 2013 areused in the paperThese data are divided into two joint sets fortwo purposes The first set (July 9 2010ndashJuly 5 2013) is usedto estimate the drift rate and volatility The second set (July 82011ndashJuly 8 2013) is used for model validations

The rate of return and volatility contained in the BMAPand the FBMAP are adaptive parameters based on timeTherefore these parameters are not constant In this sectionthe rate of return and volatility of ADVANC and LH stockprices are estimated

The ADVANC and LH closed prices from July 9 2010 toJuly 5 2013 are used to estimate 120583

119895and 120590

119895by using (8) and

(9) respectively [7] Consider the following

120583119895=252

119872

119872

sum119894=1

119877119894 (8)

120590119895= radic

252

119872 minus 1

119872

sum119894=1

(119877119894minus )2

(9)

where 119877119894is the return of stock price which can be computed

by 119877119894= (119878119894+1minus 119878119894)119878119894 is the average of return 119877

119894 and

119872 is the number of returns The parameters 120583119895and 120590

119895are

estimated by the set of data as shown in Figure 1 In this figurethe data from July 9 2010 to July 8 2011 are used to estimatethe initial 120583

0and 1205900The data from July 9 2010 to July 11 2011

are used to estimate 1205831and 120590

1 The stock market is closed on

the weekend Therefore the closed prices on July 9 2011 andJuly 10 2011 are not available The data from July 9 2010 toJuly 12 2011 are used to estimate 120583

2and 120590

2 and so on Using

the same procedure the data from July 9 2010 to July 5 2013are used to estimate 120583

482and 120590

482

ISRN Applied Mathematics 3

July 11 2011

July 12 2011

+

+

+

+

+

+

1205830 1205900

1205831 1205901

1205832 1205902

120583482 120590482

Stock priceat the time

Estimators

Estimators

Estimators

Estimators

July 5 2013

July 9 2010ndashJuly 8 2011




eq (8)-eq (9)

eq (8)-eq (9)

eq (8)-eq (9)

eq (8)-eq (9)

Figure 1 The flowchart to estimate the parameters 120583119895and 120590

119895

From the estimation results the parameters 120583119895and 120590

119895of

ADVANC and LH closed prices are shown in Figures 2 and3 respectively

4 Stock Prices Mathematical Models

41 Brownian Motion Model with Adaptive Parameters(BMAP) The BMAP can be considered by the SDE asshown in (1) The rate of return 120583 and the volatility 120590 areadaptive parameters and can be estimated using the flowchartin Figure 1 In the paper the Euler discretization methodis applied to solve the SDE The solution of discretizedform of the SDE (1) is denoted by 119878

119895 Therefore the Euler

discretization form of (1) can be written in

119878119895+1= 119878119895+ 120583119895119878119895Δ119905 + 120590

119895119878119895Δ119882119895 (10)

where 119895 is time index (119895 = 0 119873) 119873 is the number ofdatasets Δ119905 is a sampling time and 120583

119895and 120590

119895are estimated

in the previous section For the paper119873 is equal to 484 andΔ119905 is set to 1252 The initial value 119878

0is equal to stock price at

July 8 2011 The term Δ119882119895can be approximated by

Δ119882119895= 119885119895radicΔ119905 (11)

The random variable 119885119895is the standard normally distributed

random variable with mean = 0 and variance = 1 It isgenerated by method of Box and Muller [8]

The ADVANC and LH stock prices calculated fromthe BMAP are simulated by MATLAB programming Thesesimulated data are compared with the second set data ofempirical prices for a model validation The average relativepercentage error (ARPE) as given in (12) is the accuracy indexin the paper Consider the following

ARPE = 1119873

119873

sum119894=1

1003816100381610038161003816119883119894 minus 1198841198941003816100381610038161003816

119883119894

times 100 (12)

where 119873 is the number of datasets 119883119894is the empirical price

(market price) and 119884119894is the model price (simulated price)

For the simulation results by the BMAP Figure 4 showsthe empirical prices compared with the prices simulated bythe BMAP for a given path of Brownian motion process Inthe paper the date period for simulation is between July 82011 and July 8 2013

42 Fractional BrownianMotionModel with Adaptive Param-eters (FBMAP) The FBMAP can be described by SDE (7)The rate of return 120583 and the volatility 120590 in this model arevariable parameters depending on time The 120583 and 120590 can beestimated using the block diagram as shown in Figure 1 TheSDE (7) is solved by using the Euler discretization methodTherefore the Euler discretization form of (7) can be writtenin

119878120576

119895+1= 119878120576

119895+ 120583119895119878120576

119895Δ119905 + 120590

119895119878120576

119895[120572120593119895Δ119905 + Δ119882

119895120576120572] (13)


0 100 200 300 400 500

06

055

05

045

04

035

03

025

120583j

j

(a) ADVANC

0 100 200 300 400 500

j

120583j

04

035

03

025

02

015

01

005

0

minus005

(b) LH

Figure 2 The historical drift rate 120583119895for closed prices prediction from July 11 2011 to July 8 2013

0 100 200 300 400 500

j

03

0295

029

0285

028

0275

027

0265

026

120590j

(a) ADVANC

0 100 200 300 400 500

j

120590j

041

039

038

037

036

035

04

(b) LH

Figure 3 The historical volatility 120590119895for closed prices prediction from July 11 2011 to July 8 2013

In (13) the 119878120576119895is the discretized solution of the SDE (7) 119895

Δ119905 and 119873 have the same meaning as the BMAP case Theparameter120572 = 119867minus05 where119867 isHurst index and119867 isin (0 1)The estimation of this parameter is shown in Section 421The parameters 120583

119895and 120590

119895are calculated the same as those

of the BMAP case In the paper119873 Δ119905 and 120576 are set equal to484 0005 and 1252 respectivelyThe initial value 119878120576

0is equal

to stock price at July 8 2011 The term Δ119882119895can be generated

by (11) The term 120593119895in (13) can be calculated by [9]

120593119895= radic119895Δ119905

119873

119873minus1

sum119896=0

(119905 minus119896119895Δ119905

119873+ 120576)

120572minus1

119885119896 (14)

The random variable 119885119896in (14) is the standard normally

distributed random variable with mean = 0 and variance =1 It is generated by Box and Muller method

The MATLAB programming is also used to calculatethe ADVANC and LH closed prices in FBMAP For modelvalidation these simulated data are compared with theempirical prices on July 8 2011ndashJuly 8 2013 The accuracyindex in this case uses the average relative percentage error(ARPE) as calculated by (12)

421 Parameter Estimation The parameters 120572 for ADVANCand LH stock prices are the unknown values Therefore theestimation of these parameters is proposed in this sectionThe parameter 120572 is calculated by 120572 = 119867minus05 In this equation119867 is Hurst index and 0 lt 119867 lt 1 Hence minus05 lt 120572 lt 05Firstly 120572 is varied from minus05 to 05 with step size equal to01 However the parameter 120572 cannot be equal to minus05 or 05Therefore minus049 and 049 are used instead of minus05 and 05respectively The 10000 sample paths of Brownian motion


350

300

250

200

150

100

AD

VAN

C sto

ck p

rices

ADVANC empirical pricesADVANC prices simulated by BMAP

DateJuly 8 2011 July 8 2013

(a) For ADVANC closed prices (ARPE = 23268)

5

10

15

20

LH st

ock

pric

es

LH empirical pricesLH prices simulated by BMAP


(b) For LH closed prices (ARPE = 18483)

Figure 4 The simulation results using the BMAP

Table 1 ADVANC

120572 minus049 minus04 minus03 minus02 minus01 0 01 02 03 04 049Average of ARPE 12383 14713 21128 32555 42932 21496 11455 38315 times 10

1437474 times 10

210 Inf InfSD of ARPE 35199 54755 93025 18166 30492 10474 74464 28687 times 1016 Inf Inf Inf

Table 2 LH

120572 minus049 minus04 minus03 minus02 minus01 0 01 02 03 04 049Average of ARPE 13362 17722 28059 44745 59309 29370 11819 50349 times 1041 80212 times 10268 Inf InfSD of ARPE 33415 62432 13703 31977 59243 15935 74936 40179 times 1043 Inf Inf Inf

process are considered In each path the average relativepercentage error (ARPE) is computed using every value 120572The simulation results forADVANCandLH stock priceswith120572 varied from minus049 to 049 are addressed in Tables 1 and 2respectively

It can be seen that the average and standard deviation ofARPE in case of 120572 = minus049 are the minimum in both stockprices (ADVANC and LH) Therefore 120572 = minus049 is chosenfor ADVANC and LH cases

422 Model Validation For the simulation results usingthe FBMAP Figure 5 shows the empirical prices comparedwith the prices simulated by the FBMAP (7) with the samescenario of Brownianmotion of Figure 4 In Figure 5 the dateperiod to simulate the stock prices using FBMAP is betweenJuly 8 2011 and July 8 2013

43 Comparison of Accuracy Index between BMAP andFBMAP For a given standard Brownian motion samplepath Figures 4 and 5 show that the average relative percentage

Table 3 The average and standard deviation of ARPE using theBMAP and FBMAP

Stock name Model Average ofARPE

Standard deviationof ARPE

ADVANC BMAP 21496 10474FBMAP 12383 35199

LH BMAP 29370 15935FBMAP 13362 33415

error (ARPE) of the FBMAP is smaller than those of theBMAP in case of ADVANC and LH

In general the 10000 scenarios or sample paths ofBrownian motion process are considered In each path theARPE is computed frombothmodelsThe comparison resultsbetween the BMAPand the FBMAP can be seen fromTable 3It can be seen that the average ARPE of the FBMAP is lessthan the average ARPE of the BMAP Moreover the standarddeviation of ARPE from the FBMAP is smaller comparedwith the BMAPThe simulation results show that the FBMAP


Date

AD

VAN

C sto

ck p

rices

ADVANC empirical pricesADVANC prices simulated by FBMAP

300

280

260

240

220

200

180

160

140

120

100July 8 2011 July 8 2013

(a) For ADVANC closed prices (ARPE = 69208)LH

stoc

k pr

ices

LH empirical pricesLH prices simulated by FBMAP

14

13

12

11

10

9

8

7

6

5



Figure 5 The simulation results using the FBMAP

can provide the small ARPE comparedwith theBMAP in caseof ADVANC and LH

5 Conclusion

Two asset pricing models are presented in the paper One isthe Brownian motion model with adaptive parameters calledBMAP and another one is the fractional Brownian motionmodel with adaptive parameters called FBMAP The rate ofreturn and volatility in both models are adaptive at any timeThe driven process in the BMAP is Brownian motion whilethe driven process in the FBMAP is a fractional Brownianmotion The BMAP and the FBMAP are applied to simulatetheADVANCand LH stock pricesThe simulated prices fromboth models are compared with the empirical prices Theaccuracy index ARPE is used in the paper From the 10000scenarios of simulated prices of each model the average andstandard deviation of ARPE from both models show that theFBMAP provides a better appropriateness with the datasetthan the BMAP in case of ADVANC and LH Therefore theFBMAP is suitable to predict the ADVANC and LH closedprices in the future

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] F Black and M Scholes ldquoThe pricing of options and corporateliabilitiesrdquo Journal of Political Economy vol 81 no 3 pp 637ndash654 1973

[2] F Black and M Scholes ldquoTaxes and the pricing of optionsrdquoJournal of Finance vol 31 no 2 pp 319ndash332 1976

[3] R C Merton ldquoTheory of rational option pricingrdquo The RandJournal of Economics vol 4 pp 141ndash183 1973

[4] B B Mandelbrot and J W van Ness ldquoFractional Brownianmotions fractional noises and applicationsrdquo SIAM Review vol10 pp 422ndash437 1968

[5] E Alos O Mazet and D Nualart ldquoStochastic calculus withrespect to fractional Brownian motion with Hurst parameterlesser than 12rdquo Stochastic Processes and their Applications vol86 no 1 pp 121ndash139 2000

[6] T HThao ldquoAn approximate approach to fractional analysis forfinancerdquo Nonlinear Analysis Real World Applications vol 7 no1 pp 124ndash132 2006

[7] P Wilmott Paul Wilmott on Quantitative Finance John Wileyamp Sons Chichester UK 2006

[8] R Seydel Tools for Computational Finance Springer BerlinGermany 2002

[9] T H Thao and T T Nguyen ldquoFractal Langevin equationrdquoVietnam Journal of Mathematics vol 30 no 1 pp 89ndash96 2002

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Note that the parameters 120583 and 120590 are the rate of returnand the volatility respectively The process (119882

119905 119905 ge 0) in

(1) is a standard Brownian motion process The stochasticdifferential equation (1) is driven by the Brownian motionprocess (119882

119905 119905 ge 0) In the real world 120583 and 120590 in (1) are not

constant at any time Hence these parameters in the paperare the adaptable parameters based on time In the papermodel (1) is called a Brownian motion model with adaptiveparameters (BMAP)

In practice the dynamics of stock price have a longmem-ory (long range dependence) The BMAP model in (1) is notsuitable to describe the dynamics of stock price Thereforethe fractional Brownian motion process is considered in thepaper The fractional Brownian motion process (119861119867

119905 119905 ge 0)

withHurst index119867 is a centeredGaussian process If119867 = 05then (119861119867

119905 119905 ge 0) is a standard Brownian motion process If

119867 = 05 then (119861119867119905 119905 ge 0) is neither a semimartingale nor a

Markov process For119867 = 05 case the (119861119867119905 119905 ge 0) is the long

memory process The (119861119867119905 119905 ge 0) is represented in (2) by

Mandelbrot and Van Ness [4] Consider the following

119861119867

119905=

1

Γ (1 + 120572)[119885119905+ 119861119905] (2)

The function Γ(sdot) is the gamma functionThe process (119885119905 119905 ge

0) is defined by 119885119905= int0

minusinfin[(119905 minus 119904)

120572minus (minus119904)

120572]119889119882119904 The process

(119861119905 119905 ge 0) is described by 119861

119905= int119905

0(119905 minus 119904)

120572119889119882119904 The parameter

120572 = 119867 minus 12 where 119867 isin (0 1) and (119882119905 119905 ge 0) is a standard

Brownian motion processThe rate of return and volatility are not constant at any

time Hence the paper also proposes the new approach of theasset pricingmodel In this case the driving process of model(1) is replaced by a fractional Brownian motion processThe rate of return and volatility are adaptive parameters Inthis case the model can be represented by the stochasticdifferential equation (SDE) as shown in

119889119878119905= 119878119905(120583119889119905 + 120590119889119861

119867

119905) (3)

The parameters 120583 and 120590 in (3) are the rate of return and thevolatility respectively The 120583 and 120590 are adaptive parametersthe same as the previousmodelThe (119861119867

119905 119905 ge 0) is a fractional

Brownian motion process In the paper model (3) is called afractional Brownian motion model with adaptive parameters(FBMAP)

Alos et al [5] have proposed to use the process (119861119905 119905 ge

0) instead of (119861119867119905 119905 ge 0) since (119885

119905 119905 ge 0) has absolutely

continuous trajectory So the process (119861119905 119905 ge 0) has long

range dependence Hence the model (3) can be consideredas shown in

119889119878119905= 119878119905(120583119889119905 + 120590119889119861

119905) (4)

An approximate approach to stochastic differential equa-tion perturbed by fractional Brownian motion was proposedby Thao [6] The process (119861120576

119905 119905 ge 0) is introduced For every

120576 gt 0 the process (119861120576119905 119905 ge 0) is defined by

119861120576

119905= int119905

0

(119905 minus 119904 + 120576)120572119889119882119904 (5)

The process (119861120576119905 119905 ge 0) is a semimartingale Therefore this

process can be written as in

119861120576

119905= 120572int

119905

0

120593120576

119904119889119904 + 120576

120572119882119905 (6)

where 120593120576119905= int119905

0(119905 minus 119904 + 120576)

120572minus1119889119882119904

The process (119861120576119905 119905 ge 0) converges to (119861

119905 119905 ge 0) in 1198712(Ω)

when 120576 approaches to 0 This convergence is uniform withrespect to 119905 isin [0 119879] Hence the model (4) can be consideredas shown in

119889119878119905= 119878119905(120583119889119905 + 120590119889119861

120576

119905) (7)

3 The Estimation of the Rate ofReturn and Volatility

In this paper the ADVANC and LH closed prices areidentified by two asset pricing models In the BMAP thedriving process is Brownian motion On the other handthe driving process is fractional Brownian motion in theFBMAPThe parameters 120583 and 120590 in both models are adaptiveparameters at any time The ADVANC and LH simulatedstock prices are compared with these empirical prices TheADVANC and LH empirical prices can be obtained fromhttpwwwsetorththindexhtml The data of ADVANCand LH empirical prices from July 9 2010 to July 8 2013 areused in the paperThese data are divided into two joint sets fortwo purposes The first set (July 9 2010ndashJuly 5 2013) is usedto estimate the drift rate and volatility The second set (July 82011ndashJuly 8 2013) is used for model validations

The rate of return and volatility contained in the BMAPand the FBMAP are adaptive parameters based on timeTherefore these parameters are not constant In this sectionthe rate of return and volatility of ADVANC and LH stockprices are estimated

The ADVANC and LH closed prices from July 9 2010 toJuly 5 2013 are used to estimate 120583

119895and 120590

119895by using (8) and

(9) respectively [7] Consider the following

120583119895=252

119872

119872

sum119894=1

119877119894 (8)

120590119895= radic

252

119872 minus 1

119872

sum119894=1

(119877119894minus )2

(9)

where 119877119894is the return of stock price which can be computed

by 119877119894= (119878119894+1minus 119878119894)119878119894 is the average of return 119877

119894 and

119872 is the number of returns The parameters 120583119895and 120590

119895are

estimated by the set of data as shown in Figure 1 In this figurethe data from July 9 2010 to July 8 2011 are used to estimatethe initial 120583

0and 1205900The data from July 9 2010 to July 11 2011

are used to estimate 1205831and 120590

1 The stock market is closed on

the weekend Therefore the closed prices on July 9 2011 andJuly 10 2011 are not available The data from July 9 2010 toJuly 12 2011 are used to estimate 120583

2and 120590

2 and so on Using

the same procedure the data from July 9 2010 to July 5 2013are used to estimate 120583

482and 120590

482


July 11 2011

July 12 2011

+

+

+

+

+

+

1205830 1205900

1205831 1205901

1205832 1205902

120583482 120590482


Estimators

Estimators

Estimators

Estimators

July 5 2013





eq (8)-eq (9)

eq (8)-eq (9)

eq (8)-eq (9)

eq (8)-eq (9)


119895


119895of






119878119895+1= 119878119895+ 120583119895119878119895Δ119905 + 120590

119895119878119895Δ119882119895 (10)


119895and 120590

119895are estimated




Δ119882119895= 119885119895radicΔ119905 (11)




ARPE = 1119873

119873

sum119894=1

1003816100381610038161003816119883119894 minus 1198841198941003816100381610038161003816

119883119894

times 100 (12)





119878120576

119895+1= 119878120576

119895+ 120583119895119878120576

119895Δ119905 + 120590

119895119878120576

119895[120572120593119895Δ119905 + Δ119882

119895120576120572] (13)


0 100 200 300 400 500

06

055

05

045

04

035

03

025

120583j

j

(a) ADVANC

0 100 200 300 400 500

j

120583j

04

035

03

025

02

015

01

005

0

minus005

(b) LH


0 100 200 300 400 500

j

03

0295

029

0285

028

0275

027

0265

026

120590j

(a) ADVANC

0 100 200 300 400 500

j

120590j

041

039

038

037

036

035

04

(b) LH




119895and 120590



0is equal



120593119895= radic119895Δ119905

119873

119873minus1

sum119896=0

(119905 minus119896119895Δ119905

119873+ 120576)

120572minus1

119885119896 (14)






350

300

250

200

150

100

AD

VAN

C sto

ck p

rices




5

10

15

20

LH st

ock

pric

es





Table 1 ADVANC


1437474 times 10


Table 2 LH










LH BMAP 29370 15935FBMAP 13362 33415




Date

AD

VAN

C sto

ck p

rices


300

280

260

240

220

200

180

160

140

120

100July 8 2011 July 8 2013


stoc

k pr

ices


14

13

12

11

10

9

8

7

6

5





5 Conclusion




References

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










July 11 2011

July 12 2011

+

+

+

+

+

+

1205830 1205900

1205831 1205901

1205832 1205902

120583482 120590482


Estimators

Estimators

Estimators

Estimators

July 5 2013





eq (8)-eq (9)

eq (8)-eq (9)

eq (8)-eq (9)

eq (8)-eq (9)


119895


119895of






119878119895+1= 119878119895+ 120583119895119878119895Δ119905 + 120590

119895119878119895Δ119882119895 (10)


119895and 120590

119895are estimated




Δ119882119895= 119885119895radicΔ119905 (11)




ARPE = 1119873

119873

sum119894=1

1003816100381610038161003816119883119894 minus 1198841198941003816100381610038161003816

119883119894

times 100 (12)





119878120576

119895+1= 119878120576

119895+ 120583119895119878120576

119895Δ119905 + 120590

119895119878120576

119895[120572120593119895Δ119905 + Δ119882

119895120576120572] (13)


0 100 200 300 400 500

06

055

05

045

04

035

03

025

120583j

j

(a) ADVANC

0 100 200 300 400 500

j

120583j

04

035

03

025

02

015

01

005

0

minus005

(b) LH


0 100 200 300 400 500

j

03

0295

029

0285

028

0275

027

0265

026

120590j

(a) ADVANC

0 100 200 300 400 500

j

120590j

041

039

038

037

036

035

04

(b) LH




119895and 120590



0is equal



120593119895= radic119895Δ119905

119873

119873minus1

sum119896=0

(119905 minus119896119895Δ119905

119873+ 120576)

120572minus1

119885119896 (14)






350

300

250

200

150

100

AD

VAN

C sto

ck p

rices




5

10

15

20

LH st

ock

pric

es





Table 1 ADVANC


1437474 times 10


Table 2 LH










LH BMAP 29370 15935FBMAP 13362 33415




Date

AD

VAN

C sto

ck p

rices


300

280

260

240

220

200

180

160

140

120

100July 8 2011 July 8 2013


stoc

k pr

ices


14

13

12

11

10

9

8

7

6

5





5 Conclusion




References

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 100 200 300 400 500

06

055

05

045

04

035

03

025

120583j

j

(a) ADVANC

0 100 200 300 400 500

j

120583j

04

035

03

025

02

015

01

005

0

minus005

(b) LH


0 100 200 300 400 500

j

03

0295

029

0285

028

0275

027

0265

026

120590j

(a) ADVANC

0 100 200 300 400 500

j

120590j

041

039

038

037

036

035

04

(b) LH




119895and 120590



0is equal



120593119895= radic119895Δ119905

119873

119873minus1

sum119896=0

(119905 minus119896119895Δ119905

119873+ 120576)

120572minus1

119885119896 (14)






350

300

250

200

150

100

AD

VAN

C sto

ck p

rices




5

10

15

20

LH st

ock

pric

es





Table 1 ADVANC


1437474 times 10


Table 2 LH










LH BMAP 29370 15935FBMAP 13362 33415




Date

AD

VAN

C sto

ck p

rices


300

280

260

240

220

200

180

160

140

120

100July 8 2011 July 8 2013


stoc

k pr

ices


14

13

12

11

10

9

8

7

6

5





5 Conclusion




References

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










350

300

250

200

150

100

AD

VAN

C sto

ck p

rices




5

10

15

20

LH st

ock

pric

es





Table 1 ADVANC


1437474 times 10


Table 2 LH










LH BMAP 29370 15935FBMAP 13362 33415




Date

AD

VAN

C sto

ck p

rices


300

280

260

240

220

200

180

160

140

120

100July 8 2011 July 8 2013


stoc

k pr

ices


14

13

12

11

10

9

8

7

6

5





5 Conclusion




References

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Date

AD

VAN

C sto

ck p

rices


300

280

260

240

220

200

180

160

140

120

100July 8 2011 July 8 2013


stoc

k pr

ices


14

13

12

11

10

9

8

7

6

5





5 Conclusion




References

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article mathematical model of stock prices via a

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