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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
July 9 2010ndashJuly 8 2011
July 9 2010ndashJuly 11 2011
July 9 2010ndashJuly 4 2013
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)
4 ISRN Applied Mathematics
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
ISRN Applied Mathematics 5
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
DateJuly 8 2011 July 8 2013
(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
6 ISRN Applied Mathematics
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
DateJuly 8 2011 July 8 2013
(b) For LH closed prices (ARPE = 68788)
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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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
July 9 2010ndashJuly 8 2011
July 9 2010ndashJuly 11 2011
July 9 2010ndashJuly 4 2013
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)
4 ISRN Applied Mathematics
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
ISRN Applied Mathematics 5
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
DateJuly 8 2011 July 8 2013
(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
6 ISRN Applied Mathematics
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
DateJuly 8 2011 July 8 2013
(b) For LH closed prices (ARPE = 68788)
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
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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
July 9 2010ndashJuly 8 2011
July 9 2010ndashJuly 11 2011
July 9 2010ndashJuly 4 2013
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)
4 ISRN Applied Mathematics
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
ISRN Applied Mathematics 5
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
DateJuly 8 2011 July 8 2013
(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
6 ISRN Applied Mathematics
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
DateJuly 8 2011 July 8 2013
(b) For LH closed prices (ARPE = 68788)
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
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 ISRN Applied Mathematics
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
ISRN Applied Mathematics 5
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
DateJuly 8 2011 July 8 2013
(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
6 ISRN Applied Mathematics
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
DateJuly 8 2011 July 8 2013
(b) For LH closed prices (ARPE = 68788)
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
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
ISRN Applied Mathematics 5
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
DateJuly 8 2011 July 8 2013
(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
6 ISRN Applied Mathematics
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
DateJuly 8 2011 July 8 2013
(b) For LH closed prices (ARPE = 68788)
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
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 ISRN Applied Mathematics
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
DateJuly 8 2011 July 8 2013
(b) For LH closed prices (ARPE = 68788)
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
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Recommended