Persistence Process of Stock Price Movements Based On Markov Chain Analysis
3.2 Method
In forecasting the stock price, the accuracy of the prediction obtained is not good enough because it cannot predict the long-term predictions. It happened because the changes in the stock price are very volatile from time to time so that the forecasting of the stock price is best done to short-term period prediction. The investors then have other alternative method to take the decision whether to buy the shares or not by looking at the probability of the stock price movements and its persistence in the future period. The method used to estimate the probability of the stock price movements is time series model and Discrete Markov Chain Analysis (DMCA). The time series model is used to predict the stock price index while DMCA is used to predict the probability of the its movements. Time series model used in this work is the Moving Average smoothing of order three because the data behaves like those which have stationary property.
Markov Chain is a stochastics process Xt with t0,1,2, which has the following properties [4]:
i. The value of Xt is finite or countable
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Kesuma. Z.M., et al., “Application of Fuzzy Logic to Diagnose Severity of Coronary Heart Disease: Case Study in dr. Zainoel Abidin General Hospital, Banda Aceh Indonesia”
ISBN:978-602-99849-2-7
ii. Transition probability from state "i" (at time t) to state "j" (at time t1) is Pij that is
Xt jXt i Xt it X i X i
PXt jXt i
PijP 1
,
1 1,
,
1 1,
0 0 1 ,
iii. Conditional probability of Xt1 given the past states X0,X1,,Xt1 and the present state Xt
is only depend on the present state (Markov property)
Pij represents the probability that the process, when in state
i
, next make transition into state j
0
, 1 , 0 , 1
; 0 , , 0
j ij
ij i j P i
P
The transition probability from state i to state j is represented to the form of Markov Chain or the transition probability matrix. The transition chain was shown in Figure 1
Figure 1.Markov Chain
Figure 1 illustrates the Markov Chain with three states in which the arrows indicate the possibility of a transition from one state to another. The other representation of Markov Chain is the transition probability matrix P which denote one-step transition probabilities Pij
2 1 0
12 11 10
02 01 00
i i
i P P
P
P P P
P P P P
For an irreducible ergodic Markov Chain ijn
n P
lim exist and is independent of i. Suppose ,
0 ,
lim
Pijn j
j n
then j is the unique nonnegative solution of
0
, 0 ,
i
n ij i
j P j
with
0
1
j
j .
Pierre Vallois and Charles Tapiero [6] in their paper construct the persistence index to claim process and insurance. Those index indicate the pattern of claim process to insurance client based on the pattern of change in the provision for claims using Markov Chain. The persistence index formulated in their work was constructed to Markov Chain with two states only. Based on those idea, we will formulated the persistence index of stock price movements to Markov Chain with more than two states. The formulation of the index is given in Proposition 1
Proposisi 1.SupposeXt,t0is a sequence of random variables which represent the states of the stock price movements that has been classified from the Difference of Price (DoP) at time t. Suppose that the transition probability of stock price movements is written in matrix P. Then, the persistence index of the stock price movements is
i j
i ji ii
i n 1P P , n 1, n 1
with n is the size of the states.
P01
P12
P00 0
2 1
1 0 6 Kesuma. Z.M., et al., “Application of Fuzzy Logic to Diagnose Severity of Coronary Heart Disease: Case Study
in dr. Zainoel Abidin General Hospital, Banda Aceh Indonesia”
ISBN:978-602-99849-2-7
Copyright © 2015by Advances on Science and Technology 0
5 10 15 20 25 30
25-08-14 25-10-14 25-12-14 25-02-15 25-04-15 25-06-15 25-08-15
Adj Close Prediction
If i 0, then
1
n P P j i
ji
ii . This process is a persistence process, where the state probability of the stock price at the next period will be equal with the previous period is greater than all of the other states.
If i 0, then
1
n P P j i
ji
ii . This process is not a persistence process, where the state probability of the stock price at the next period will be equal with the previous period is smaller than all of the other state.
Proof.
Suppose that there are n possible states in which the initial state is i, than the stock would have a persistence property if Pii Pjiwith jifor all j, hence
0 )
1 (
0 )
1 (
, 0
2 1 2 1
ji ji iii j i
j i j ii
i j ii i
j ii i j ii
P P
n
P P
P P n
i j P
P P
P P P
n n
4. Results and Discussion
The prediction of the stock price using Moving Average smoothing of order three (MA(3)) is shown by Figure 2,
Figure 2.Stock Price Prediction Using MA(3)
Figure 2 shows that the prediction of the stock price using MA(3) is closed enough with the actual price. It is because the data have a stationary properties so that MA(3) method is appropriately applied to this kind of data. Following are the prediction of the transition probability of stock price movements using DMCA.
Model 1: Markov Chain with 3 States
The states defined in Model 1 are '0': jump down Xt 0,
'1'
: stable X
t 0
, and' 2 '
: jump up X
t 0
. The result of Model 1 is106
Kesuma. Z.M., et al., “Application of Fuzzy Logic to Diagnose Severity of Coronary Heart Disease: Case Study in dr. Zainoel Abidin General Hospital, Banda Aceh Indonesia”
ISBN:978-602-99849-2-7
667 . 0 027 . 0 306 . 0
667 . 0 0 333 . 0
246 . 0 0 754 . 0 P
Then we obtain the long-run probability of each state by solving the equations
1 027
. 0
667 . 0 667 . 0 246 . 0 306
. 0 333 . 0 754 . 0
2
0 2
1
2 1
0 2
2 1
0 0
jj
The solution of those equations is obtained by using software Matlab and given in the vector
Based on the transition probability matrix, the persistence properties for each state are shown in Table 1
Table 1.Persistence Properties of Model 1
Initial State Persistence Index Persistence Properties
Jump Down 02(0.754)(0.3330.306)0.869 Persistence Stable 12(0)(00.027)0.027 Not Persistence Jump Up 2 2(0.667)(0.2460.667)0.421 Persistence
Table 1 shows that the initial state ‘jump down’ gives the greatest persistence index, it means that if the stock price is in state ‘jump down’ then the probability to tend to fall is greater than the probability to stay stable or to go up. It is also applied to the initial state ‘jump up’. Meanwhile, it is not happened if the initial state is ‘stable’. It means that if the initial state is ‘stable’, then the probability to stay stable is very small than to change to another state.
Model 2: Markov Chain with 5 States
The states defined in Model 2 are '0': drastically down Xt 0.2,
'1'
: jump down0.2Xt 0,
' 2 '
: stable Xt 0, '3': jump up 0Xt 0.2, and' 4 '
: drastically up Xt 0.2. The result of Model 2 is
421 . 0 474 . 0 0 105 . 0 0
098 . 0
0 02 . 0
0
522 . 0 033 . 0 315 . 0 033 . 0
667 . 0 0 333 . 0 0
294 . 0 0 539 . 0 147 . 0
075 . 0 0 375 . 0 55 . 0 P
Then we obtain the long-run probability of each state by solving the equations
4 3
1 4
4 3
2 1
0 3
3 2
4 3
2 1
0 1
3 1
0 0
421 . 0 098 . 0 02 . 0
474 . 0 522 . 0 667 . 0 294 . 0 075 . 0
033 . 0
105 . 0 315 . 0 333 . 0 539 . 0 375 . 0
033 . 0 147 . 0 55 . 0
] 4334 . 0 0117 . 0 5549 . 0
[
1 0 8 Kesuma. Z.M., et al., “Application of Fuzzy Logic to Diagnose Severity of Coronary Heart Disease: Case Study
in dr. Zainoel Abidin General Hospital, Banda Aceh Indonesia”
ISBN:978-602-99849-2-7
Copyright © 2015by Advances on Science and Technology
1
4
0
j
j
The solution of those equations is given in the vector below
] 0745 . 0 3591 . 0 0119 . 0 3981 . 0 1564 . 0
[
Table 2 shows the persistence properties of Model 2
Table 2.Persistence Properties of Model 2
Initial State Persistence Index Persistence
Properties
Drastically Down 04(0.55)(0.1470.033)2.02 Persistence Jump Down 1 4(0.539)(0.3750.3330.3150.105)1.028 Persistence
Stable 24(0)0.0330.033 Not Persistence
Jump Up 340.522 0.0750.2940.6670.4740.578 Persistence Drastically Up 4 40.421 0.020.0981.566 Persistence
Table 2 shows that the greatest persistence index is when the initial state of the stock price movement is in the state ‘drastically down’. Furthermore, the state will also be persistence if the initial state of the stock price is in states ‘jump down’, ‘jump up’, and ‘drastically up’. Similar to the Model 1, Model 2 also gives the result that the process will not be persistence if the initial state of the stock price movement is in state ‘stable’. It means that whenever the price is in ‘stable’ condition, the probability to remain ‘stable’ is smaller than to move to the other state.
5. Conclusion
Discrete-Time Markov Chain Analysis (DMCA) is one of some methods which is used by the investors in making a decision whether to buy shares or not. This method is more effective than using another method that predicts the stock price index because the DMCA predict the transition probability of the stock price movements which is less volatile. In addition, we can observe the tendency of the pattern of stock price movements by looking its persistence in the next period. In this paper have been formulated the persistence index of the stock price movements which was proposed in Proposition 1. This index can indicate the behavior of the shares if it is in a certain initial states.
The results of the application of DMCA and persistence process show that, in the period time 25 August 2014 until 25 August 2015, the stock price has a tendency to persist like the previous state except when the initial state is stable. It means that if the initial state is either down or up, then it will also be down or up in the next period with greater probability. This analysis gives a general result about persistence process. In addition, we can classify the level of the persistence based on certain criteria. This study can be conducted as a continuation of the discussion which has been done in this paper.
References
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Kesuma. Z.M., et al., “Application of Fuzzy Logic to Diagnose Severity of Coronary Heart Disease: Case Study in dr. Zainoel Abidin General Hospital, Banda Aceh Indonesia”
ISBN:978-602-99849-2-7
[2] Dharmawan, Y.V., “Forecasting Loss Based on Stochastic Processes,” Bachelor thesis, Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, 2012.
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[7] Xu, S.Y., “Stock Price Forecasting Using Information from Yahoo Finance and Google Trend,”
https://www.econ.berkeley.edu/sites/default/files/Selene Yue Xu.pdf Retrieved 02 September, 2015.
[8] Zhang, D., Zhang, X., “Study on Forecasting the Stock Market Trend Based on Stochastic Analysis Method,” International Journal of Business and Management Vol. 4, No. 6, 163-170 (2009).
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1 1 0 Kesuma. Z.M., et al., “Application of Fuzzy Logic to Diagnose Severity of Coronary Heart Disease: Case Study
in dr. Zainoel Abidin General Hospital, Banda Aceh Indonesia”
ISBN:978-602-99849-2-7
Copyright © 2015by Advances on Science and Technology