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A methodology for stochastic analysis of share prices as Markov chains with
finite states
Article  in  SpringerPlus · November 2014
DOI: 10.1186/2193-1801-3-657 · Source: PubMed
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Mettle et al. SpringerPlus 2014, 3:657
http://www.springerplus.com/content/3/1/657
a SpringerOpen JournalMETHODOLOGY Open AccessA methodology for stochastic analysis of share
prices as Markov chains with finite states
Felix Okoe Mettle1, Enoch Nii Boi Quaye1* and Ravenhill Adjetey Laryea2Abstract
Price volatilities make stock investments risky, leaving investors in critical position when uncertain decision is made.
To improve investor evaluation confidence on exchange markets, while not using time series methodology, we
specify equity price change as a stochastic process assumed to possess Markov dependency with respective state
transition probabilities matrices following the identified state pace (i.e. decrease, stable or increase). We established
that identified states communicate, and that the chains are aperiodic and ergodic thus possessing limiting
distributions. We developed a methodology for determining expected mean return time for stock price increases
and also establish criteria for improving investment decision based on highest transition probabilities, lowest mean
return time and highest limiting distributions. We further developed an R algorithm for running the methodology
introduced. The established methodology is applied to selected equities from Ghana Stock Exchange weekly
trading data.
Keywords: Markov process; Transition probability matrix; Limiting distribution; Expected mean return time;
Markov chainBackground
Stock market performance and operation has gained
recognition as a significantly viable investment field
within financial markets. We most likely find investors
seeking to know the background and historical behavior
of listed equities to assist investment decision making.
Although stock trading is noted for its likelihood of
yielding high returns, earnings of market players in
part depend on the degree of equity price fluctuations
and other market interactions. This makes earnings very
volatile, being associated with very high risks and
sometimes significant losses.
In stochastic analysis, the Markov chain specifies a
system of transitions of an entity from one state to
another. Identifying the transition as a random process,
the Markov dependency theory emphasizes “memoryless
property” i.e. the future state (next step or position) of
any process strictly depends on its current state but
not its past sequence of experiences noticed over
time. Aguilera et al. (1999) noted that daily stock
price records do not conform to usual requirements* Correspondence: enbquaye@ug.edu.gh
1Department of Statistics, University of Ghana, Accra, Ghana
Full list of author information is available at the end of the article
© 2014 Mettle et al.; licensee Springer. This is a
Attribution License (http://creativecommons.or
in any medium, provided the original work is pof constant variance assumption in conventional stat-
istical time series. It is indeed noticeable that there
may be unusual volatilities, which are unaccounted
for due to the assumption of stationary variance in stock
prices given past trends. To surmount this problem,
models classes specified under the Autoregressive
Conditional Heteroskedastic (ARCH) and its Generalized
forms (GARCH) make provisions for smoothing unusual
volatilities.
Against the characteristics of price fluctuations and
randomness which challenges application of some statistical
time series models to stock price forecasting, it is explicit
that stock price changes over time can be viewed as a
stochastic process. Aguilera et al. (1999) and Hassan
and Nath (2005) respectively employed Functional
Principal Component Analysis (FPCA) and Hidden
Markov Model (HMM) to forecast stock price trend based
on non-stationary nature of the stochastic processes which
generate the same financial prices. Zhang and Zhang
(2009) also developed a stochastic stock price forecasting
model using Markov chains.
Varied studies (Xi et al. 2012; Bulla et al. 2010; Ammann
and Verhofen 2006; and Duffie and Singleton 1993) have
researched into the application of stochastic probability ton Open Access article distributed under the terms of the Creative Commons
g/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction
roperly credited.
Mettle et al. SpringerPlus 2014, 3:657 Page 2 of 11
http://www.springerplus.com/content/3/1/657portfolio allocation. Building on existing literature, we
assume that stock price fluctuations exhibit Markov’s
dependency and time-homogeneity and we specify a three
state Markov process (i.e. price decrease, no change and
price increase) and advance the methodology for determin-
ing the mean return time for equity price increases and
their respective limiting distributions using the generated
state-transition matrices. We further replicate the case for
a two-state space i.e. decrease in price and increase in
price. Based on the methodology, we hypothesize that;
Equity with the highest state transition probability and
least mean return time will remain the best choice for
an investor.
We explore model performance using weekly historical
data from the Ghana Stock Exchange (GSE); we set up
the respective transition probability matrix for selected
stocks to test the model efficiency and use.
Review of theoretical framework
Definition of the Markov process
The stochastic process {X (t), tϵT} is said to exhibit
Markov dependence if for a finite (or countable infinite)
set of points (t0, t1, … , tn, t), t0 < t1 < t2 <… < tn < t where t,
trϵT (r = 0, 1, 2, …, n).
PðXðtÞ ≤ xjXðtnÞ ¼ xn; Xðtn−1Þ ¼ xn−1 ; …;XðtnÞ ¼ x0Þ
¼ P½XðtÞ ≤ xjXðtnÞ ¼ xn ¼ F ½Xn; x ; tn; t
ð1Þ
From the property given by equation (1), the following
relation suffices Z
FðXn; x; tn; tÞ ¼ Fðy; x; τ; tÞdFðXn; y; tn; τÞ ð2Þ
y∈S
where tn < τ < t and S is the state space of the process {X (t)}.
When the stochastic process has discrete state and
parameter space, (2) takes the following form: for n >
n1 > n2 >… > nk and n, nrϵT (r = 1, 2, …, k)
PðXn ¼ jjXn1 ¼ i1; Xn2 ¼ i2; …; Xnk ¼ ikÞ ð3Þ
¼ P X ðnk ;nÞn ¼ jjXn1 ¼ i1Þ ¼ Pij
A stochastic process with discrete state and parameter
spaces which exhibits Markov dependency as in (3) is
known as a Markov Process.
From the Markov property, for nk < r < n we get
Pðnk ;nÞij ¼ PðXn ¼ jjXnk ¼ iÞX
¼ PðXn ¼ jjXr ¼ mÞPðXr ¼ mjXnk ¼ iÞ
Xm∈S
¼ Pðnk ;rÞPðr;nÞij mj
m∈S
ð4Þequations (2) and (4) are known as the Chapman-
Kolmogorov equations for the process.
n-step transition probability matrix and n-step transition
probabilities
If P is the transition probability matrix of a Markov chain
{Xn, n = 0, 1, 2, …} with state space S, then the elements of
Pn (P raised to the power n), PðnÞij i; jS are the n-step transi-
tion probabilities where P (n)ij is the probability that the
process will be in state j at the nth step starting from state i.
The above statement can clearly be shown from the
Chapman-Kolmogorov equation (4) as follows; for a
given r and s, write
PðsþrÞ
X
¼ PðrÞPðSÞij ik kj
k∈s
Set r = 1, s = 1 in the above equation to get
ð Þ XP 2ij ¼ PjkPkj
k∈s
Clearly, P (2)ij is the (i, j)th element for the matrix prod-
uct P × P = P2. Now suppose P (r)ij (r = 3, 4, …, n) is the
(i, j)th of Pr then by the Kolmogorov equation, the
X
Pðrþ1Þ ¼ PðrÞij ik Pkj
k∈S
which again can be seen as the (i, j)th element of the
matrix product PrP = Pr+1. Hence by induction, P (n)ij is
the (i, j)th element of Pn n = 2, 3, ….
To specify the model, the underlying assumption is
stated about the identified n-step transition probability
(stating without proof ).
The transition probability matrix is accessible with
existing state communication. Further, there exists recur-
rence and transience of states. States are also assumed to
be irreducible and belong to one class with the same
period which we take on the value 1. Thus the states are
aperiodic.
Limiting distribution of a Markov chain
If P is the transition probability matrix of an aperiodic,
irreducible, finite state Markov chain, then
2 3
α
limPt ¼ π ¼ 664α577 ð5Þt→∞ ⋮
α
XWhere α = [α1, α2, …, αm] with 0 < αj < 1 andm
αj ¼ 1. See Bhat (1984). The chain with this property
j¼1
Mettle et al. SpringerPlus 2014, 3:657 Page 3 of 11
http://www.springerplus.com/content/3/1/657is said to be ergodic and has a limiting distribution π.
The transition probability matrix P of such a chain is
primitive.
Recurrence and transience of state
Let Xt be a Markov Chain with state space S, then the
probability of the first transition to state j at the tth step
starting from state i is
f ðtÞij ¼ P½Xt ¼ j; Xr ≠ j; r ¼ 1; 2; 3; …; t−1jX0 ¼ i
ð6Þ
Thus the probability that the chain ever returns to
state j is
X∞
f ij ¼ f ijðtÞ
t¼1
X∞
and μ ¼ tf ðtÞij ij is the expected value of first passage
t¼1
time. Further, if i = j, then;
f ðtÞii ¼ P½Xt ¼ i; Xr ≠ i; r ¼ 1; 2; 3; …; t−1jX0 ¼ i
ð7Þ
X∞
and μii ¼ μi ¼ tf ðtÞii is the mean recurrence time of
t¼1
state i if state i is recurrent.
A state i is said to be recurrent (persistent) if and only
if, starting from state i, eventual return to this state is
certain. Thus state i is recurrent if and only if
X∞
f  ðtÞii ¼ f ii ¼ 1 ð8Þ
t¼1
A state i is said to be transient if and only if, starting
from state i, there is a positive probability that the
process may not eventually return to this state. This
means f *ii < 1
Model specification
Defining the problem (Equity price changes as a
three-state Markov process)
Let Yt be the equity price at time t where t = 0, 1, 2,…, n
(t is measured in weekly time intervals). Further, we de-
fine dt = Yt − Yt−1 which measures the change in equity
price at time t. Considering each closing week’s price as
discrete time unit for which we define a random variable
Xt to indicate the state of equity closing price at time t, a
vector spanned by 0, 1, 2<8 0 if dt < 0 decrease inequityprice fromt−1 to t
Xt ¼ : 1 if dt ¼ 0 nochange inequityprice fromtimet−1 to t2 if dt > 0 increase inequityprice fromtimet−1 to t
Next, we define an indicator vector

¼ 1 if Xt ¼ iIi;t f or i ¼ 0; 1; 2 and t ¼ 1; 2; …; n0 if Xt ≠ i
ð9Þ
Then clearly for the outcome of Xt we have
Xn
ni ¼ Ii;t f or i ¼ 0; 1; 2 ð10Þ
t¼1
X2
where n ¼ ni . Hence estimates of the probability
i¼0
that the equity price reduce, did not change and increased
can be obtained respectively by
¼ n0 ¼ n1 ¼ nP̂0 ; P̂1 and P̂ 22 ð11Þn n n
For the stochastic process Xt obtained above for t =
1, 2, …, n we can obtained estimates of the transition
probabilities Pij = Pr (Xt = j|Xt−1 = i) for j = 0, 1, 2 by
defining
<>
8
δði;jÞ 1 if Xt ¼ i and Xtþ1 ¼ jt ¼ >> f or t ¼ 1; 2; …; n−1: 0 otherwise
and i; j ¼ 0; 1; …; k
where k + 1 is the number of states of the chain.
Xn−1
n ði;jÞ
nij
ij ¼ ¼ δt f or i; j ¼ 0; 1; 2: Then P̂ij ¼t 1 ni
f or i; j ¼ 0; 1;…; k
ð12aÞ
Therefore, an estimate for the transition matrix for
k = 2 is
2 3
4 P̂00 P̂01 P̂02P̂ ¼ P̂ 510 P̂11 P̂12 ð12bÞ
P̂20 P̂21 P̂22
Mettle et al. SpringerPlus 2014, 3:657 Page 4 of 11
http://www.springerplus.com/content/3/1/657Suppose the data in Additional file 1 is uploaded as
.csv, then R code for computing estimates in (12b)
can be found in Additional file 2 (three-state Markov
Chain function column).For a two-state Markov process
We maintain the above defined terms and set

¼ 0 if dt ≤ 0 no increase in equity price from t−1 to tXt 1 if dt > 0 increase in equity price from time t−1 to t
further set i, j = 0, 1, (for k = 1) and apply (9), (10), (11),
(12a), and (12b) sequentially, we obtain 
P̂ ¼ P̂00 P̂01
P̂10 P̂11
without loss of generality, suppose Xt has state space
s = {0, 1} and transition probability matrix
 
¼ 1−θ θP ; 0 < α; β < 1 ð13Þ
β 1−β
Then, f (1)00 = 1 − θ and for n ≥ 2, we have;
Mettle et al. SpringerPlus 2014, 3:657 Page 5 of 11
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Table 1 Summary statistics on the weekly trading price change over the study period
Number of weekly price change Weekly price change summary
Decrease No change Increase Mean SD Max Min Skew. Kurt. Count
ALW 15 77 12 0.00 0.01 0.01 −0.04 −2.30 14.23 104
AYRTN 8 89 7 0.00 0.00 0.01 −0.01 −0.10 4.39 104
BOPP 26 45 33 0.01 0.13 0.44 −0.62 −1.80 11.79 104
CAL 27 40 37 0.00 0.03 0.12 −0.07 1.69 7.26 104
EBG 30 44 30 0.00 0.19 0.50 −1.60 −5.65 51.32 104
EGL 21 46 37 0.01 0.06 0.39 −0.25 1.05 15.79 104
ETI 18 59 27 0.00 0.01 0.04 −0.04 −0.71 5.62 104
FML 22 38 44 0.03 0.12 0.85 −0.19 4.08 25.22 104
GCB 25 37 42 0.02 0.13 0.79 −0.41 1.68 12.51 104
GGBL 5 51 48 0.04 0.10 0.73 −0.20 3.84 22.70 104
GLD 7 79 18 0.04 0.34 3.13 −0.72 7.22 64.43 104
GOIL 16 53 35 0.00 0.03 0.12 −0.23 −2.99 23.63 104
HFC 8 75 21 0.01 0.03 0.27 −0.08 5.76 47.80 104
MLC 7 75 22 0.00 0.01 0.05 −0.03 1.04 5.42 104
PBC 13 81 10 0.00 0.01 0.04 −0.02 1.53 11.02 104
PZC 22 55 27 0.01 0.38 3.02 −1.00 4.78 39.15 104
SCB 38 40 26 0.14 1.30 9.54 −4.19 4.45 30.38 104
SCBPREF 11 87 6 0.00 0.01 0.01 −0.03 −2.35 9.67 104
SIC 19 67 18 0.00 0.02 0.16 −0.06 5.55 50.15 104
SOGEGH 12 89 3 0.00 0.02 0.01 −0.18 −6.56 50.60 104
SWL 9 84 11 0.01 0.45 3.15 −2.00 2.28 27.27 104
TBL 21 62 21 0.02 0.62 2.99 −3.00 0.16 12.33 104
TLW 16 56 32 0.23 0.97 6.56 −1.97 3.77 19.38 104
TOTAL 16 66 22 0.01 0.08 0.52 −0.16 4.89 29.16 104
TRANSOL 12 63 29 0.04 0.18 1.26 −0.50 4.24 24.87 104
UNIL 3 76 25 0.03 0.23 1.79 −0.77 5.23 38.66 104
UTB 12 87 5 0.00 0.01 0.02 −0.02 −1.30 6.18 104f ðtÞ00 ¼ P½Xt ¼ 0;Xr ≠ 0; r ¼ 1; 2; 3; …; t−1jX0 ¼ 0
¼ P½Xt ¼ 0;Xr ¼ 1; r ¼ 1; 2; 3; …; t−1jX0 ¼ 0
By the Markov property and the definition of conditional
probability, we have
 Yt−1  
f ðnÞ00 ¼ P Xt ¼ 0jXt−1 ¼ 1 P½Xr ¼ 1jXr−1 ¼ 1 P½X1 ¼ 1jX0 ¼ 0
r¼2
¼ βð1−βÞt−2θ ¼ θβð1−βÞt−2 t ≥ 2
ð14Þ
X∞
solving μ ðtÞ0 ¼ μ00 ¼ tf 00 to obtain the respective
t¼1
mean recurrence time. Thus,X∞
μ ¼ μ ¼ tf ðtÞ
X∞
0 00 00 ¼ 1−θ þ tθβð1−βÞt−2
t¼1  t¼2
¼ θ þ β ð15Þ
β
Similarly, we have
f ðtÞ01 ¼ θð1−θÞt−1; t ≥ 1 μ01 ¼ 1
f ðtÞ10 ¼ βð1−βÞt−1; t ≥ 1 μ10 ¼ 1
f ð1Þ11 ¼ ð1−βÞ ¼
 
; t 1 g θ þ βμ11 ¼ μ1 ¼
f ðtÞ t−2 t ≥ 2 θ11 ¼ θβð1−αÞ ;
ð16Þ
With the corresponding R algorithm shown in
Additional file 2 (two-state Markov Chain function column).
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Mettle et al. SpringerPlus 2014, 3:657 Page 7 of 11
http://www.springerplus.com/content/3/1/657Generating eigen vectors for computation of
limiting distributions
After the transition probabilities are obtained for both
two-state and three-state chains, the R codes in the
lower portions of columns one and two in Additional file
2 were used to generate the respective eigen vectors for
computation of limiting distributions.
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Figure 1 A plot of mean and standard deviation of weekly price changes of equities. The plot indicates a very volatile weekly market price
fluctuation for any market participating investor. This indicates high level of risk associated with equity purchase decision. We consider that the
rational investor would basically seek to maximize purchasing decisions faced with this risk.Findings and discussions
Data structure and summary statistics
Data used for this paper are weekly trading price changes
for five randomly selected equities on the Ghana Stock
Exchange (GSE), each covering period starting from
January 2012-December 2013. We obtain the weekly
price changes using the relation dt = Yt − Yt−1 where Yt
represents the equity closing price on week t and Yt−1 is
the opening price for the immediate past week. The equi-
ties selected include Aluworks (ALW), Cal Bank (CAL),
Ecobank Ghana (EBG), Ecobank Transnational Incorporated
(ETI), and Fan Milk Ghana Limited (FML).
In all, 104 (52 weeks) observational data points
where obtained. Summary statistics on all respectiveFigure 2 t-step transition probabilities for share price increases.equities on the GSE are shown in Table 1. We
present summaries on the respective number of weekly
price decreases, no change in price and price increase.
Descriptive statistics for each equity weekly price change
is also shown.
Overall, the frequency of “no price change” was
more experienced over the study period. The lowest
and highest price changes for the trading period are
respectively −4.19 and 9.54. The estimated values of
the kurtosis and skewness are also shown. Figure 1
presents a plot of the average weekly equity price
changes of respective equities listed on the GSE over the
study period in comparison to the standard deviation of
weekly price changes.
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Table 2 Entries of the limiting distribution at for Table 3 Entries of two-state transition matrices for
respective equities selected equities
Equity Limiting distribution Equities P00 P01 P10 P11
α1 α2 α3 1 − θ θ β 1 − β
ALW 0.141509 0.745283 0.113208 ALW 0.133333 0.866667 0.142857 0.857143
CAL 0.244980 0.406396 0.348625 CAL 0.296296 0.703704 0.227848 0.772152
EBG 0.269568 0.443025 0.287407 EBG 0.433333 0.566667 0.210526 0.789474
ETI 0.168470 0.586912 0.244618 ETI 0.166667 0.833333 0.170455 0.829545
FML 0.198113 0.386792 0.415094 FML 0.380952 0.619048 0.152941 0.847059
Table 4 Expected mean return time for respective stocks
Equity μ00 μ11
ðθþβÞ θþβ
β θ
ALW 1.1555556 7.4285714
CAL 1.3837280 3.6060127
EBG 1.5488889 2.8218623
ETI 1.2009132 5.9772727
FML 1.4497355 3.2235294Empirical results on model application (three-state
Markov chain)
For the five randomly selected equities, the transition
probabilities of the equities are presented as follows.
These were obtained from equation (12a) defining
Pij ¼ nijn w.r.t. the three-state space Markov process.i
A 3 × 3 transition matrix is obtained for respective equities
as defined by (12b).
From the results of the algorithm, we select 5 equities
with which we implement the hypothesis. They include;
ALW tr2ansition probability matrix 3
4 0:133333 0:666667 0:200000P̂ ¼ 0:139241 0:759494 0:1012665
0:166667 0:750000 0:083333
CAL tra2nsition probability matrix 3
4 0:296296 0:407407 0:296296P̂ ¼ 0:261905 0:476190 0:2619055
0:189189 0:324324 0:486486
EBG tra2nsition probability matrix 3
4 0:433333 0:366667 0:200000P̂ ¼ 0:255319 0:553191 0:1914895
0:137931 0:344828 0:517241
ETI tra2nsition probability matrix 3
4 0:166667 0:611111 0:222222P̂ ¼ 0:131148 0:639344 0:2295085
0:259259 0:444444 0:296296
FML tr2ansition probability matrix 3
4 0:380952 0:523810 0:095238P̂ ¼ 0:170732 0:487805 0:3414635
0:136364 0:227273 0:636364
Clearly, P̂ ij > 0 for all i, j = 0, 1, 2 indicating irreducibility
of the chains for all equities. Hence state 0 for all the
equities is aperiodic and since periodicity is a class
property, the chains are aperiodic. These imply that
the chains are ergodic and have limiting distributions.
Figure 2 presents the t − step transition probabilities
for share price increases based on the assumption oftime-homogeneity. This shows linear plot of transition
probabilities for P (t)22 for each selected stock as com-
puted above. It measures the probability that a share
at initial state (i. e. state 2) at inception transited to
state 2 again after t weeks. Regarding the plot of the
transition probabilities, the logical reasoning is to
choose the equity which has the highest P22.
From the plot, FML share is the best choice for the
investor since the probability that it increases from a
high price to another higher price is higher when
compared to the other selected stocks. ALW recorded
the least probability of transition within the period.
Comparing CAL to EBG, the methodology shows that
CAL shares maintain high probability of moving to
higher prices as compared to EBG shares although
the later started with high prices at inception.
Using equation (5), the limiting distributions of the
respective equities were computed. These probabilities
measure the proportions of times the equity states within
a particular state in the long run. From Table 2, ALW
equity has 14% chance of reducing and 11% chance of
increasing in the long run. It however has 75% chance of
no change in price. Similarly, in the long run, FML equity
has 20% chance of reducing, 39% chance of experiencing
no change in price and 42% chance of increasing in price.
It is easily seen that for this instance, FML equity has the
highest probability of price increase in the long run.
Empirical model application (the two-state Markov process)
Defining a two-state space Markov process following from
equation (13), we derive the state transition probabilities.
The two-state transition probability matrix entries are
indicated in Table 3 below;
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Figure 3 Mean recurrence time of selected shares.Applying equations (15) and (16) to the transition
probabilities, we obtain the respective mean return time
of the selected equities. These are shown in Table 4 below;
Mean return time is measured in weeks with μij as
defined in (15) and (16). The mean return time measures
the expected time until the equity price’s next return to
the state it was initially in at time 0. Figure 3 presents a
plot of expected return time of the selected stocks at μ11.
This determines the expected time until the next increase
in share. We expect that the choice of share should not
only have the highest transition probability, but should
relatively possess a lower mean return time. Possessing
the least mean return time for μ11 signifies the shortest
return time to a price increase.
Conclusion
The Markov Process provides a credible approach for
successfully analyzing and predicting time series data
which reflect Markov dependency. The study finds that
all states obtained communicate and are aperiodic and
ergodic hence possessing limiting distributions. It is
distinctive from Figures 1 and 2 (expected return time
and t-step state transition probabilities of equity price in-
creases i.e. Pij transition from state 2 to state 2) that the
investor gains good knowledge about the characteris-
tics of the respective equities hence improving deci-
sion making in the light return maximization. With
regards to the selected stocks, FML equity recorded the
highest state transition probabilities, highest limiting dis-
tribution but the second lowest mean return time to price
increases (i.e. 3.224 weeks).
Our suggested use of Markov chains as a tool for
improving stock trading decisions indeed aids in improving
investor knowledge and chances of higher returns given
risk minimization through best choice decision. We showedthat the proposed method of using Markov chains as
a stochastic analysis method in equity price studies
truly improves equity portfolio decisions with strong
statistical foundation. In our future work, we shall explore
the case of specifying an infinite state space for the
Markov chains model in stock investment decision making.
Additional files
Additional file 1: Weekly Price Change Data for GSE.
Additional file 2: R algorithm for respective methodologies.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
FOM introduced the idea, undertook the theoretical and methodology
development. ENBQ developed the codes and helped in the analysis and
typesetting of mathematical equations. RAL also helped in the analysis and
the general typesetting of the manuscript. All authors read and approved
the final manuscript.
Author details
1Department of Statistics, University of Ghana, Accra, Ghana. 2Department of
Banking and Finance, University of Professional Studies, Accra, Ghana.
Received: 19 August 2014 Accepted: 24 October 2014
Published: 6 November 2014
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doi:10.1186/2193-1801-3-657
Cite this article as: Mettle et al.: A methodology for stochastic analysis
of share prices as Markov chains with finite states. SpringerPlus
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