The document analyzes historical share prices for Standard Charted PLC to determine if they follow a Geometric Brownian Motion model. Tests for normality and independence were conducted on the returns. The returns were found to not be normally distributed and independent based on p-values from Shapiro-Wilk, Kolmogorov-Smirnov, and chi-squared tests. As the log returns are not a Brownian motion, the analysis concludes that the share prices do not follow a Geometric Brownian Motion.

1. pg. 0
FINANCIAL
REPORT ON
STOCK PRICE
CHANGES FOR
STANDARD
CHARTED PLC
Thomas Allan Cox - 13000015
ABSTRACT
In thisreportI shall be analysinghistorical share prices
for StandardChartedPLC.I will describe the behaviour
of the share price as well asfindingoutwhetherthe
data followsthe GeometricBrownian Model.
London Metropolitan University
Financial MathematicsandStatistics –MA5040

2. pg. 1
Introduction Page No.
Stock Market 2
Geometric BrownianModel 2
StandardCharted PLC 2
Analysis
Time Series Graph 2
Theoryof methods 3
Results andAnalysis 3-4
Summary
Conclusion 4
Appendix 5-7
References 7
Bibliography 7

3. pg. 2
Stock Market
The stock market is, or stock exchange, is where the stocks/shares of a publicly are traded
through direct exchanges or bought directly. There are two sections to the exchange, the
first being the Primary section where IPO’s (Initial Public Offerings) are held. This is where a
company decides to sells share to the general public and where companies first trade to get
onto the market. The secondary market is where all other trades are made, by either
companies or individual investors. If an investor buys a share for £X amount and the share
raises to £X+£Y amount, where Y>£0, and then decides to sell; the profit he makes is called
capital gain. If the share prices decreases, for example, the share price goes from £X to £X-
£Y then this is called capital loss.[1]
Geometric Brownian Model (GBM)
The random walk is used to model the random movement of any given object; share or a
particle. A key point to take note of is that the object must move away from where it started,
therefore it is always moving in one direction or another. The type of random walk that
concerns us is a geometric random walk. This is where the future price of an object is,
𝑋𝑡 +1 = ( 𝑄) 𝑋𝑡 + 𝑋𝑡 ∶ 𝑄 = 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑖𝑛 𝑑𝑒𝑐𝑖𝑚𝑎𝑙 𝐹𝑜𝑟𝑚, {1}
𝑋𝑡 = 𝑂𝑏𝑗𝑒𝑐𝑡 𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑃𝑟𝑖𝑐𝑒, 𝑋𝑡+1 = 𝑂𝑏𝑗𝑒𝑐𝑡 𝐹𝑢𝑡𝑢𝑟𝑒 𝑃𝑟𝑖𝑐𝑒
A Brownian motion is an example of a geometric random walk, and an example of Brownian
motion if a system called the ‘Markov Chain.’ This system states that the object moving has
no memory of where it has moved before, therefore future values are all independent of past
values. A geometric Brownian motion is a Brownian motion which in continous in time.
Standard Charted PLC
Standard charted PLC is a multination bank operating in over 70 countries with over
80,000 employees. The services SC provide are: consumer banking, corporate banking,
investment banking, mortgages, private banking and wealth management.
Question One - Time-series Graph
Thisis a time-seriesgraphof the share pricesof SC from01/07/2015 to 01/01/2015. We can clearly
see fromthisgraph that the trendfor share pricesforthiscompanyis theyare generallydecreasing.
A sample of the data I usedto create thisgraph can be foundinAppendix A,andraw data from[2].
800
900
1000
1100
1200
1300
1400
1500
1600
1700
6/25/2013 10/3/2013 1/11/2014 4/21/2014 7/30/2014 11/7/2014
Time-seriesgraphfor share prices of StandardCharteredPLC

4. pg. 3
Question 2 - Theory of Methods
A lot of the information here has been gainedfrom [3]. The stochastic process of Share
prices from t=0 onwards, {St}t≤0, is a GBM, from t=0-T : T=future time andwith parameter μ
and σ, if the Log(St) is a Brownian motion with an initial value of Log(S0) with parameter µ-
0.5σ2 and σ. We can derive from this that;
𝑅𝑡 = 𝐿𝑛 (
𝑠 𝑡+𝛿𝑡
𝑆 𝑡
) = 𝐿𝑛( 𝑆 𝑡+𝛿𝑡
) − 𝐿𝑛( 𝑆 𝑡
) ~ 𝑁 ([ 𝜇 −
1
2
𝜎2] 𝛿𝑡, 𝜎2
𝛿𝑡) {2}
𝑊ℎ𝑒𝑟𝑒 𝛿𝑡 = 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡𝑖𝑚𝑒 = 1
365⁄ 𝑌𝑒𝑎𝑟𝑠
The Return calculated is the relative return from one share price to the next. So 1.00 stands
for 100%, I.e the share price hasn’t changed, or 0.9 is 90%, it is then transformed by taking
the natural log as shown in equation two. You can see a sample of the data in appendix A
which shows the data I used to calculate the returns and the returns itself.
The returns (Rt) as we can see are calculatedover different time i ntervals, one day to the
next and another day to the next, therefore these returns should be independent of each
other. Hence, if the returns are not independent or not normally distributed, the Ln(St) does
not satisfy the conditions to be a Brownian motion, which then implies the stochastic
process of share prices (St) is not a Geometric Brownian motion. I am standardising the
returns before doing these test as it makes the results easier to interpret. It does not affect
the results since it only shifts the data to a mean of 0
Using this information, I will be testing the returns for normality and indepe ndence. To test
normality I will use: Shapiro-Wilk, Kolmogorov-Smirnov tests, Q-Q plots, and a histogram
to test normality using Rt. To test for independence, I will use Chi-squared test for
association after I code the returns in to categories. I will also use regression coefficient
test using the Rt values.
Question 3 an 4 combined (Tests and conclusions with a final conclusion
together)- Analysis of Tests
Normality of returns
All the outputs from SPSS which I derived my re sults from with be in Appendix B-D
Tests for normality; Shapiro-Wilk, Kolmogorov-Smirnov
𝐻0: 𝑇ℎ𝑒 𝑟𝑒𝑡𝑢𝑟𝑛𝑠 (𝑅𝑡) 𝑎𝑟𝑒 𝑛𝑜𝑟𝑚𝑎𝑙𝑙𝑦 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑒𝑑
𝐻1: 𝑇ℎ𝑒 𝑟𝑒𝑡𝑢𝑟𝑛𝑠 (𝑅𝑡) 𝑎𝑟𝑒 𝑛𝑜𝑡 𝑛𝑜𝑟𝑚𝑎𝑙𝑙𝑦 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑒𝑑
P=0.000, Reject H0.
From this we can derive that there is very strong evidence (P=0.000,***) to reject the fact
that the returns are normally distributed.
The Q-Q plot is in appendix C, from this I would suggest that the returns are not normally
distributed because of the slight curve/pattern in the data.
The histogram along with the data used to create the graph is in appendix D. To create this
histogram, I first normalized my Rt values, and then allocated these new values (using
rounding techniques) to the BIN range and then plotted this on a histogram . I conclude
from this test that the data is slightly positively skewedand therefore not normally
distributed.
From all of these tests, I would conclude there the returns are not normally distributed.

5. pg. 4
Independence of returns
The data for this test is found in appendix E
To test for independence, I have tested the Rt and Rt-1 values to see if there is a correlation
between the present and future value.
𝐻0: 𝐵1 = 0, 𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑛𝑜 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛𝑠ℎ𝑖𝑝, 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑡ℎ𝑒 𝑟𝑒𝑡𝑢𝑟𝑛𝑠 𝑎𝑟𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡
𝐻1: 𝐵1 ≠ 0, 𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛𝑠ℎ𝑖𝑝 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑟𝑒𝑡𝑢𝑟𝑛𝑠, 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑡ℎ𝑒𝑦 𝑎𝑟𝑒 𝑛𝑜𝑡 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡
P=0.271, Accept H0.
I can derive from this that there is no evidence to reject H0 therefore we accept H0 which
suggests that the data is independent.
Pearson’s Chi-squared test for association, data in appendix F,
𝐻0: 𝑁𝑜 𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑖𝑜𝑛 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑅𝑡+1 𝑎𝑛𝑑 𝑅𝑡, 𝐹𝑢𝑡𝑢𝑟𝑒 𝑎𝑛𝑑 𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑅𝑒𝑡𝑢𝑟𝑛𝑠
𝐻1: 𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎𝑛 𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑖𝑜𝑛 𝑅𝑡+1 𝑎𝑛𝑑 𝑅𝑡, 𝐹𝑢𝑡𝑢𝑟𝑒 𝑎𝑛𝑑 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑅𝑒𝑡𝑢𝑟𝑛𝑠
P=0.725, Accept H0.
From this, I conclude that there is no evidence to reject H0 therefore I shall aceept H0 and
state there is no association between future and present returns.
Conclusion
I conclude from all the results and analysis I have performed above, that the returns (Rt) are
independent from each other, I.e the present value has no effect on the future value. Since
Rt is not normally distributed, this implies that Ln(St) is not a Brownian motion. The theory,
on page 3, also states that if Ln(St) is not a Brownian motion, then St, our share price, does
not follow a Geometric Brownian Motion.

6. pg. 5
Appendix
Appendix A
Date Open High Low Close Volume
Adj
Close
01/07/2013 1444.5 1471 1422.5 1468 4140700 1391.64
02/07/2013 1471.5 1471.5 1448 1463.5 2165000 1387.37
03/07/2013 1444 1446.5 1409.5 1427.5 3448800 1353.25
Return-
Ln(present/past)
Standardised Return-
Ln(present/past)
Sample Mean
Of Returns
Standard
Dev of
returns
-0.003070103 -0.145623331 -0.001075517 0.013696883
-0.024906165 -1.739859176
0.047204386 3.52488254
Appendix B
Tests of Normality
Kolmogorov-Smirnova Shapiro-Wilk
Statistic df Sig. Statistic df Sig.
LogReturn .074 393 .000 .935 393 .000
a. Lilliefors Significance Correction
Appendix C

7. pg. 6
2
0000
1
0
1
0
2
0
1
5
77
17
22
36
42
45
61
49
22
14
21
16
77
3
2
0
1
00
1
000000
0
10
20
30
40
50
60
70
-4.75
-4
-3.25
-2.5
-1.75
-1
-0.25
0.5
1.25
2
2.75
3.5
4.25
5
FREQUENCY
BIN RANGE
Histogram tesing for
normal distribution
Frequency
Appendix D
Bin
range
-5 Frequency Cumulative %
-4.75 2 0.51%
-4.5 0 0.51%
-4.25 0 0.51%
-4 0 0.51%
-3.75 0 0.51%
-3.5 1 0.77%
-3.25 0 0.77%
-3 1 1.02%
-2.75 0 1.02%
-2.5 2 1.53%
-2.25 0 1.53%
-2 1 1.79%
-1.75 5 3.06%
-1.5 7 4.85%
-1.25 7 6.63%
-1 17 10.97%
-0.75 22 16.58%
-0.5 36 25.77%
-0.25 42 36.48%
0 45 47.96%
0.25 61 63.52%
0.5 49 76.02%
0.75 22 81.63%
1 14 85.20%
1.25 21 90.56%
1.5 16 94.64%
1.75 7 96.43%
2 7 98.21%
2.25 3 98.98%
2.5 2 99.49%
2.75 0 99.49%
3 1 99.74%
3.25 0 99.74%
3.5 0 99.74%
3.75 1 100.00%
4 0 100.00%
4.25 0 100.00%
4.5 0 100.00%
4.75 0 100.00%
5 0 100.00%
More 0 100.00%

8. pg. 7
Appendix E
Coefficientsa
Model
Unstandardized Coefficients
Standardized
Coefficients
t Sig.B Std. Error Beta
1 (Constant) .001 .050 .015 .988
StandardReturnOnestepBac
k
.056 .050 .056 1.105 .270
a. DependentVariable:StandardReturn_2
Model Summary
Model R
R
Square
Adjusted R
Square
Std. Error of the
Estimate
Change Statistics
R Square
Change
F
Change df1 df2
Sig. F
Change
1 .056a
.003 .001 .99969980167 .003 1.220 1 391 .270
a. Predictors:(Constant),StandardReturnOnestepBack
Appendix F
Chi-Square Tests
Value df
Asymp. Sig. (2-
sided)
Pearson Chi-Square 1.318a
3 .725
Likelihood Ratio 1.309 3 .727
Linear-by-Linear Association .225 1 .636
N of Valid Cases 138
a. 1 cells (12.5%) have expected countless than 5. The minimum
expected count is 4.71.
References
Yahoo! Finance, Historical Share Prices, Available:
http://finance.yahoo.com/q/hp?s=STAN.L+Historical+Prices . Last accessedon
30/01/2015
Bibliography
Investopedia. (not known). Stock Market definiton. Available:
http://www.investopedia.com/terms/s/stockmarket.asp. Last accessed30/01/2015.
Universitat wien, Brownian motion and Geometric Brownian Motion, Available:
http://homepage.univie.ac.at/kujtim.avdiu/dateien/BrownianMotion.pdf, Last accessedon
30/01/2015