UNIVERSITY OF NOTTINGHAM

Geometric Brownian Motion

An analysis of model applicability to stock market indices

Adam Goodwin, Jamie Baptiste, Alex Hirst, Vikesh Nathan and Chiu Tsan Leung

Recent financial crises have been strong motives for extended research into the financial modeling of stock pricing. Geometric Brownian Motion (GBM) is now a widely used process for stock price modeling. Our investigation into GBM tests the assumptions of this process to see whether it can be accepted as a good model for underlying assets in option pricing. Theoretically, GBM seems like a good model due to its Markov property; however our results produced an interesting outcome. GBM was found to be a good model for shorter time periods but for longer time periods, the assumptions of the model fail. We also tested GBM on different market indices and found the FTSE 100 held for an extended period of time whereas the Nikkei 225 and the Hang Seng did not. Given the testing used for our investigation, we were able to implement these tests through a real time updating software package in R. This was produced with the intention that given a set of data from the given indices, the user is shown all graphical and numerical results from our testing. The user is then able to see whether the process can be accepted as a good model for their chosen stock or index and time period.

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Contents 1 Introduction ......................................................................................................... 2

1.1 Background ..................................................................................................... 2

1.2 Objectives ....................................................................................................... 2

1.3 Report overview ................................................................................................ 3

2 Geometric Brownian Motion model ............................................................................... 4

2.1 Brownian Motion ............................................................................................... 4

2.2 The assumptions ............................................................................................... 4

2.3 Using the natural log of the returns .......................................................................... 5

3 Methodology ......................................................................................................... 6

3.1 Testing the assumptions of the Geometric Brownian Motion model ..................................... 6

3.1.1 The normality assumption ............................................................................... 6

3.1.2 The constant variance assumption ...................................................................... 7

3.1.3 The constant mean assumption .......................................................................... 8

3.1.4 The assumption of independence ....................................................................... 9

3.2 The data ......................................................................................................... 9

3.3 Sampling methods ........................................................................................... 10

4 Results and discussion ............................................................................................ 12

4.1 Summary Table of Analysis ................................................................................. 12

4.2 Key findings ................................................................................................... 12

4.2.1 All model assumptions held on a quarterly basis .................................................... 12

4.2.2 All assumptions held for FTSE 100 annual periods ................................................. 17

4.2.3 Constant variance and normality did not hold for all periods ...................................... 19

4.2.4 Constant mean and independence held for all periods ............................................. 20

4.2.5 10 year weekly log return data was consistent with daily log return data ....................... 21

4.3 Further discussion ........................................................................................... 23

5 Further development ............................................................................................. 25

6 Software ............................................................................................................ 26

7 Conclusions ........................................................................................................ 27

8 Appendix ........................................................................................................... 28

8.1 Software user guide .......................................................................................... 28

8.2 Derivation of Geometric Brownian Motion ................................................................ 29

8.3 P value tables ................................................................................................. 31

8.4 Additional figures ............................................................................................ 38

9 References ......................................................................................................... 41

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1 Introduction

1.1 Background

Real option valuation plays a key role in capital investment decisions. In simple terms an option is the right, but not the obligation, to undertake business initiatives such as expansion, contraction, deferring or abandoning a capital investment project. These projects may increase the wealth of the business’ shareholders and being able to value these potential money making projects is key. This firm believes that certain stock market indices are useful underlying assets to price real options and this highlights the importance of ensuring the selected valuation model is accurate in forecasting stock prices. Our study focuses on Geometric Brownian Motion (GBM) as the underlying model for predicting these price movements. Since the introduction of Brownian Motion by Osborne (1959) in the appraisal of common stocks, Geometric Brownian Motion has been considered one of the most important models in valuing the growth of a stock over time. Samuelson (1965) was the first to advocate the application of the GBM model to predict price behaviour. Within its vast array of applications it is perhaps most well-known for its use in modelling stock prices in the Black-Scholes options pricing formulae; arguably the most important concept in modern financial theory. First introduced by Black and Scholes in the 1970’s within their conceptual framework, they assume that stock prices follow a Geometric Brownian Motion with constant drift and volatility. Here, drift can be defined by the overall trend the stock price has. The magnitude of the rise or fall in stock price increases as the magnitude of drift increases. The accuracy of GBM has come under some scrutiny with this investigation being a consequence. Many scholars have questioned the validity of the GBM model’s assumptions, which include:

1. Normality of returns 2. Independence – the Markov property 3. Constant drift 4. Constant volatility

Arguments for the use of GBM include:

- The Markov property can be defined as a process in which predictions of future activity can be made solely on the present knowledge where historical activity is irrelevant. This is in line with the weak form of the Efficient Market Hypothesis.

- The process exhibits the same variation in its path as actual stock prices. - The process prohibits negative values, which is appropriate as stock prices can

never be negative. - The model is relatively simple to use in calculations.

1.2 Objectives

The main objective of this study is to confirm or refute the previously mentioned assumptions of the GBM process in predicting stock market movements. The study

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focuses on three indices: the FTSE 100, the Nikkei 225 and the Hang Seng. These indices are the most relevant to the shipping company concerned, who need a real option to be valued. A working piece of software will also be produced that enables the user to implement the statistical techniques used in the investigation, over user defined time horizons on the stated indices.

1.3 Report overview

In the subsequent pages a brief summary is given of the reviewed literature which led to

the testing of the previously mentioned assumptions. The methodology follows, starting

with the theoretical background of the GBM model, proceeding to discuss the

assumptions and the approach taken to test them. The methodology section also

elaborates on the choice of tests and sampling techniques of the data in question and is

followed by the analysis itself. The analysis is divided into five sections, with each section

corresponding to a key finding. For each key finding, an in-depth analysis and discussion

of the results is provided. The analysis section is completed with further discussion of the

findings. In the next section, a description of the software developed is given, including a

simple user guide in the appendix. The final sections contain an overall conclusion of the

investigation as well as suggestions for further developments that have been identified.

The developments include potential improvements of the GBM model and further testing.

An appendix and references are included which offer more in-depth information behind

the report.

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2 Geometric Brownian Motion model

Section two expands on standard Brownian Motion and GBM. The assumptions are also

explained and the rationale behind the use of the natural logarithmic returns is clarified.

2.1 Brownian Motion

Brownian motion, 𝐵𝑡 at time 𝑡, also known as a Wiener process, is a stochastic process

with the following properties:

1. 𝐵0 = 0

2. Non-overlapping increments are independent

3. 𝐵𝑡 − 𝐵𝑠~𝑁(0, 𝑡 − 𝑠) for 0 < 𝑠 < 𝑡 are independent normally distributed

random variables with zero mean and variance 𝑡 − 𝑠.

4. The function 𝑡 ⟼ 𝐵𝑡 is a continuous function.

A stochastic process 𝑆𝑡 is said to follow a Geometric Brownian Motion if it satisfies the

following stochastic differential equation (SDE)

𝑑𝑆𝑡 = 𝜇𝑆𝑡𝑑𝑡 + 𝜎𝑆𝑡𝑑𝐵𝑡

where 𝐵𝑡 is standard Brownian motion and µ and 𝜎 are constants known as drift and

volatility respectively. Throughout this report 𝜇 and 𝜎 will always refer to the drift and

volatility parameters, unless stated otherwise.

By solving the above SDE (see appendix for full derivation), the solution is given by

following formula

𝑆𝑡 = 𝑆0 exp ((𝜇 −𝜎2

2) 𝑡 + 𝜎𝐵𝑡) (*)

Equation (*) is more commonly described as Geometric Brownian motion.

2.2 The assumptions

In order to test whether Geometric Brownian Motion is a good model for stock prices, statistical testing must be carried out on the four key assumptions of a GBM process highlighted earlier:

1. The natural log returns of stock prices follow a normal distribution

2. Stock prices are independent random variables

3. The natural log returns have constant mean

4. The natural log returns have constant variance

A group of stock prices within a certain timeframe will follow a Geometric Brownian Motion with drift parameter 𝜇 and volatility parameter 𝜎 if the natural log of the ratio of the price at time 𝑡 in the future to the present will follow a normal distribution with parameters 𝜇𝑡 and 𝜎2𝑡. From initial findings and expectations:

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- Expected returns of a stock price are independent of past stock prices, which is a

sensible assumption to make in reality.

- Geometric Brownian Motion prohibits negative values, which is appropriate as stock prices can never be negative.

- However for GBM, volatility is assumed to be constant whereas for real stocks,

volatility is changing over time. Therefore there are initial issues with this

assumption.

- From initial research undertaken, it is believed real stock price returns are not

normally distributed. As explained later, the distribution of data has fatter tails

and extreme price changes are more likely.

2.3 Using the natural log of the returns

Define the natural log of the returns as

𝑟𝑡 = ln (𝑆𝑡

𝑆𝑡−1)

where 𝑆𝑡 is the price of the stock at time 𝑡. In order to assess the assumptions of Geometric Brownian Motion, it is more convenient to work with 𝑟𝑡. By using equation (*) from section 2.1, the formula can be rearranged in terms of 𝑟𝑡 from time 0 to time 𝑡

ln (𝑆𝑡

𝑆0) = (𝜇 −

𝜎2

2) 𝑡 + 𝜎𝐵𝑡

where 𝐵𝑡~𝑁(0, 𝑡).

Using the Markov property the consecutive log returns can be written as

ln (𝑆𝑡

𝑆𝑡−1) = (𝜇 −

𝜎2

2) 𝑡 + 𝜎𝐵1

where 𝐵1~𝑁(0,1) . It can be shown that ln (𝑆𝑡

𝑆𝑡−1) ~𝑁(𝜇, 𝜎2) , so that 𝑟𝑡 is normally

distributed and the normality assumption can be tested using 𝑟𝑡.

Similarly, when testing for constant mean and variance, 𝑟𝑡 can be used. Assuming 𝑟𝑡 is

normally distributed, maximum likelihood estimation shows that 𝜇 and 𝜎2 can be

estimated by the sample mean and variance. Therefore, tests of constant mean and

variance will determine whether 𝜇 and 𝜎2 are constant over time.

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3 Methodology

In order to assess the validity of the Geometric Brownian Motion model assumptions, an analysis will be conducted on each of them, using a variety of plots and statistically rigorous hypothesis tests. The current section will outline each of these methods. When performing these tests for the investigation, a 5% significance level will be used as a benchmark.

3.1 Testing the assumptions of the Geometric Brownian Motion model

3.1.1 The normality assumption

Histograms are a very useful graphical method to visualize the data. In particular, the shape of the graph will provide an initial assessment as to whether or not normality is present. For a slightly more informative plot to assess normality, quantile-quantile plots (QQ plots) will also be plotted. In combination with the below hypothesis testing, these will provide insight into the underlying distribution of the data. Shapiro-Wilk The Shapiro-Wilk test for normality is a very popular test to determine whether a data sample is normally distributed. As such, the test will be used in this analysis. The hypotheses for the Shapiro-Wilk test are as follows

𝐻0 :The data are normally distributed 𝐻1 :The data are not normally distributed

When using the test, for a sufficiently small p-value, the assumption of normality cannot reasonably hold. To carry out the test, the following test statistic should be calculated

𝑊 = (∑ 𝑎𝑖𝑟(𝑖)

𝑛𝑖=1 )2

∑ (𝑦𝑖 − �̅�)2𝑛𝑖=1

where 𝒓 is the sample of interest, 𝑟(𝑖) is the 𝑖𝑡ℎ ordered value of the sample, �̅� is the

sample mean and 𝑎𝑖 is the 𝑖𝑡ℎelement of the vector defined by

𝒂 = 𝒎𝑇𝑉−1

(𝒎𝑇𝑉−𝟏𝑉−1𝒎)1/2

where 𝒎 is the vector of expected values of the standard normal order statistics and 𝑉 is the covariance matrix of these order statistics. The coefficients 𝑎𝑖 are determined by the sample size and in practice are already calculated in tables of values. A full derivation of the statistic is given in the original paper on the test (Shapiro and Wilk, 1965). The reasoning behind the statistic is that for normally distributed data, the numerator and denominator of 𝑊 will be equal. The smaller the value of 𝑊, the less likely the data are normal. For the following analysis, computer software will perform the test.

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A key limitation of the Shapiro-Wilk test is its sensitivity to deviation from normality in larger samples. The test will therefore more readily reject the null hypothesis even if the data are normally distributed. QQ plots will provide a basis to assess whether the test has correctly rejected the assumption of normality. Jarque-Bera The Jarque-Bera test for normality assesses whether the skewness and excess kurtosis of a sample are zero. In normally distributed data, the skewness and excess kurtosis values are zero. The Jarque-Bera test defines skewness as

𝑆 = �̂�3

�̂�3=

1𝑛

∑ (𝑟𝑖 − �̅�)3𝑛𝑖=1

(1𝑛

∑ (𝑟𝑖 − �̅�)2)𝑛𝑖=1

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and the excess kurtosis as

𝐾 = �̂�4

�̂�4− 3 =

1𝑛

∑ (𝑟𝑖 − �̅�)4𝑛𝑖=1

(1𝑛

∑ (𝑟𝑖 − �̅�)2)𝑛𝑖=1

2 − 3

where �̅� is the sample mean. Informally, kurtosis can be defined as ‘the degree of peakedness of a distribution’. Using the above definitions the Jarque-Bera statistic is then defined as

𝑛

6(𝑆2 +

1

4𝐾2)

The hypotheses are as follows

𝐻0 :The skewness and excess kurtosis are equal to zero 𝐻1: The skewness and excess kurtosis are not equal to zero

Therefore, if the null hypothesis is rejected, it is unlikely the data are normally distributed. To perform the test, the sample skewness and excess kurtosis should be calculated and then substituted into the test statistic formula. Again, software will be used in the analysis to perform these calculations.

3.1.2 The constant variance assumption

As an exploratory analysis, the log return data will be plotted. The plot will visually show any obvious cases of non-constant variance. To assess constant variance, two tests will be used: the Levene test and the Bartlett test. Both tests split the data into groups and compute their corresponding statistics. If the variances of each group are the same, this implies that variance is constant across the whole sample. The hypotheses for each test are also the same and are

𝐻0: 𝜎12 = 𝜎2

2 = ⋯ = 𝜎𝑘2 (The variances are equal)

𝐻1: 𝜎𝑖2 ≠ 𝜎𝑗

2 for at least one pair (𝑖, 𝑗) (At least two variances are not equal)

where 𝑘 is the number of groups the sample is divided into and 𝑖, 𝑗 ∈ {1,2, … , 𝑘}, 𝑖 ≠ 𝑗.

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With both of these tests, the number of groups to split the sample into is arbitrary and therefore the result may change depending upon the number of groups. Caution should be used when deciding the group number 𝑘. The Bartlett test The Bartlett test has been shown to be sensitive to departures from normality. For this reason, the analysis will include the use of the Levene test, which is less sensitive to normality departures. The Bartlett test statistic is defined as

𝑇 = (𝑁 − 𝑘) ln 𝑠𝑝

2 − ∑ (𝑁𝑖 − 1) ln 𝑠𝑖2𝑘

𝑖=1

1 +1

3(𝑘 − 1)[(∑

1𝑁𝑖 − 1

𝑘𝑖=1 ) −

1𝑁 − 𝑘

]

Where 𝑁 is the total sample size, 𝑘 is the number of groups, 𝑁𝑖 is the sample size of group 𝑖, 𝑠𝑖

2 is the sample variance of group 𝑖 and 𝑠𝑝2 is the pooled variance defined here as

𝑠𝑝2 =

1

𝑁 − 𝑘∑(𝑁𝑖 − 1)𝑠𝑖

2

𝑘

𝑖=1

The Bartlett test statistic approximately has a 𝜒2 distribution. The Levene test For the Levene test, there is a choice of using the mean, median and trimmed mean in calculating the statistic. Due to the more powerful properties of using the median (Brown and Forsythe, 1974), this analysis will use the median as the central measure for the statistic. Suppose the sample is given by 𝑋, then the statistic is

𝑊 = (𝑁 − 𝑘

𝑘 − 1)

∑ 𝑁𝑖(�̅�𝑖. − �̅�..)2𝑘

𝑖=1

∑ ∑ (𝑍𝑖𝑗 − �̅�𝑖.)2𝑁𝑖

𝑗=1𝑘𝑖=1

𝑍𝑖𝑗 = |𝑋𝑖𝑗 − �̃�𝑖.|

where 𝑁 is the total sample size, 𝑘 is the number of groups, 𝑁𝑖 is the sample size of group 𝑖, �̃�𝑖. is the median of group 𝑖, �̅�𝑖. is the mean of the 𝑍𝑖𝑗 in group 𝑖 and �̅�.. is the mean of all

the 𝑍𝑖𝑗 . The statistic for the test approximately has an F distribution.

3.1.3 The constant mean assumption

To test for constant mean, the analysis will include a fitted linear regression model of the form 𝑦 = 𝛼 + 𝛽𝑥 and a hypothesis test will be conducted on the coefficient of 𝑥. Using the above form of the linear model, the hypothesis test is

𝐻0: 𝛽 = 0 𝐻1: 𝛽 ≠ 0

Therefore, if the 𝑥 coefficient of the model is zero, the model becomes 𝑦 = 𝛼 which is a constant. Using least squares regression, 𝛼 is approximated by the mean of the sample and implies the mean of the sample is constant.

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For this test to be valid, there is an assumption that the residuals (휀𝑖 ) of the model are 휀𝑖~𝑁(0, 𝜎2). For each hypothesis test conducted, the corresponding QQ plot of the

residuals will be plotted to check if the assumption of the residuals holds. Stationarity testing Within time series analysis, many models are based on the concept of stationarity and in practical terms second order stationarity is used. Second order stationarity holds within a data set or model if:

1. The mean is constant for the whole series

2. The variance is constant for the whole series

3. The autocovariance function only depends on the distance between time

points.

To supplement the above tests on assessing constant mean and variance, testing for stationarity will also be used. The analysis conducted will use the Priestley-Subba Rao test of stationarity. The test uses spectral densities and an analysis of variance table. The sum of squares are then compared to a 𝜒2distribution. Further information can be found in Priestley and Subba Rao’s paper (1969). For the Priestley-Subba Rao test, the hypotheses are

𝐻0: The data are stationary 𝐻1: The data are not stationary

If the test rejects the null hypothesis, the data cannot reasonably be considered stationary. Therefore the mean and variance are not constant for the data.

3.1.4 The assumption of independence To assess independence, the turning point test will be used. The turning point test determines whether or not a time series is random. If the data is random, each point will be independent. The procedure counts the number of turning points: this is the number of times the sequence ‘changes direction’. More formally, the hypotheses are

𝐻0:The data are independent, identically distributed random variables 𝐻1:The data are not independent, identically distributed random variables

Therefore if the test rejects the null hypothesis, the data cannot reasonably be independent. An important limitation of the statistic is that it is only valid for large samples. For large samples, it is approximately normally distributed with a mean (2𝑛 − 4)/3 and a variance (16𝑛 − 29)/90 , where 𝑛 is the sample size. However, within the analysis performed, the minimum sample size considered is 63 and the approximation should reasonably hold.

3.2 The data

The data used in this analysis consists of time series of the FTSE 100, the Nikkei and the Hang Seng indices at their adjusted daily and weekly closing prices. The data has been

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obtained directly from Yahoo Finance covering respective periods from 3rd Jan 1984 – 1st Feb 2015 (FTSE), 4th Jan 1984 – 1st Feb 2015 (Nikkei) and 31st Dec 1986 – 1st Feb 2015 (Hang Seng). Adjusted closing prices are used because they give a more accurate valuation of the index, since dividends paid to shareholders and stock splits are also accounted for. As discussed in section two, if the Geometric Brownian Motion model holds, the log of the returns, either daily or weekly, should be normally distributed. The log returns should also have constant mean and variance and the data should be independent. As such, the log returns were calculated for all of the data, resulting in both daily and weekly log returns data. The log return data was then sampled and analysed using the methods discussed in section 3.1.

3.3 Sampling methods

The focus of the investigation is to see whether the Geometric Brownian Motion model assumptions hold. To do this, different time intervals are considered: quarterly periods, annual periods, five year periods and ten year periods. To standardize the time periods, an annual period has length of 252 days: the average number of trading days in a calendar year. Therefore a quarterly period has 63 data points, a five year period has 1,260 data points and a ten year period has 2520 data points. The majority of the analysis was conducted on daily log return data however weekly log return data for ten year periods was also analyzed. The sampling methods for the data are detailed in the following paragraphs. Quarterly and annual sampling To sample quarterly and annual periods, a predetermined number of ten samples for each index was taken. The number represents the time limitations on the project. Each of the available indices were split into ten equal intervals and a random sample from each interval was taken. By systematically sampling, overlapping samples are avoided but are still random within the interval. The sample function in R was used to randomly sample each interval. 5 Year For five year periods, a randomized approach was less feasible due to the limited data available and a systematic sample was taken. Every interval of length 1,260 was taken, from the start of the data to the final available full period of 1,260 days. Using this method, six samples from the FTSE 100, five samples from the Hang Seng index and six samples from the Nikkei 225 index were obtained. 10 Year For ten year periods, the same approach as five year periods was applied, taking sequential periods of 2,520 days. There were three samples from the FTSE 100, two samples from the Hang Seng index and three samples from the Nikkei 225 index. As mentioned, weekly returns were also considered. The standardized time periods are no longer applicable for these samples but the same approach for ten year daily log returns were used. However, the starting and ending dates were made to match up for

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each sample. The reason for matching the physical dates was to ensure the same time period for daily ten year log returns and weekly ten year log returns matched, allowing for fair comparison. There were three samples from the FTSE 100, two samples from the Hang Seng index and three samples from the Nikkei 225 index. The number of data points varied slightly in each sample in order to match the dates.

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4 Results and discussion

Within this section, the results of the analysis will be highlighted. A summary table detailing when assumptions of the model hold is outlined, followed by a detailed analysis and discussion of the key findings. P value tables and additional graphs can be found in the appendix.

4.1 Summary Table of Analysis

The following table summarizes our analysis into a simple ✓(Yes) or ✗(No) and it identifies over which indices and time horizons the validity of Geometric Brownian Motion is most applicable, without having to extensively review our analysis.

4.2 Key findings

The test results identified a number of significant trends throughout the analysis and are detailed here.

4.2.1 All model assumptions held on a quarterly basis

For every test implemented, nearly every sample accepted the assumption under scrutiny, across every index. There is strong evidence to suggest that for short time periods the model assumptions held. Each index is now further analyzed. Hang Seng For the assumption of normality, seven out of ten samples accepted the null hypotheses and only samples two, four and seven were rejected for both the Shapiro-Wilk and the Jarque-Bera tests. Observation of the QQ plots also suggested that samples two and especially seven had heavier tails than a normal distribution and sample four had a slight negative skew. Outlying points could explain the inconsistencies in samples two and four.

Index Sample Normality Independence Constant Mean

Constant Variance

FTSE 100 Quarterly ✓ ✓ ✓ ✓

Annually ✓ ✓ ✓ ✓

5 Year ✗ ✓ ✓ ✗

10 Year ✗ ✓ ✓ ✗ Nikkei Quarterly ✓ ✓ ✓ ✓

Annually ✗ ✓ ✓ ✗

5 Year ✗ ✓ ✓ ✗

10 Year ✗ ✓ ✓ ✗

Hang Seng Quarterly ✓ ✓ ✓ ✓

Annually ✗ ✓ ✓ ✗

5 Year ✗ ✓ ✓ ✗

10 Year ✗ ✓ ✓ ✗

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Histograms for samples two and four showed a slight positive and negative skewness respectively and therefore did not match a normal distribution. Sample seven was also slightly negatively skewed. Despite these inconsistencies, there was evidence to suggest that normality of these samples held.

Fig. 1

Fig. 2

Fig.2 highlights the wider tails shown in sample seven. Notice how the far left and far right points do not lie on the straight line, indicating departures from normality.

For the constant variance assumption, all samples accepted the null hypothesis for the Levene test. However for the Bartlett test, only seven out of ten samples accepted the null hypothesis. Samples two, four and six were rejected. Notice that the same samples rejected for normality, namely two and four. Outliers may also be able to account for these samples. All but one sample accepted the null hypothesis for constant mean. Sample four rejected the null hypothesis and had a p value of 0.03912.

Fig. 3

Fig. 4

Analysis of the QQ plot in fig.3 shows an outlying point, numbered 59. This challenged the validity of this test for sample 4. The remaining samples produced linear QQ plots like fig.4, which was further confirmation of the assumption of constant mean.

The stationarity test confirmed the mentioned tests for constant mean and variance. Eight out of ten samples accepted the null hypothesis. All log returns graphs for the eight samples appeared to have a constant mean and variance, supporting the test results. Samples two and four were rejected.

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Sample five was stationary under the test and it is clear from fig.5 the data are centred about one value. The variance also looked constant overall, although looked slightly smaller between days one and fifteen compared to the rest of the sample. At first viewing,

fig.6 does seem to be stationary, since the data is centred about zero, indicating a constant mean. However, there is a noticeable spike at day four, which may be an outlier. Overall, after accounting for extreme data points, all quarterly samples were stationary. For independence the turning point test accepted the null hypothesis in nine out of ten samples, giving strong evidence to support the assumption. Only sample seven was rejected. Nikkei 225 When testing normality, only samples one, two and three were rejected for both the

Shapiro-Wilk test and the Jarque-Bera test, indicating consistency between them. For

instance, sample one is positively skewed and did not appear to come from a normal

distribution, as shown in fig.7. Fig.8 shows there are possible outlying points that could

explain the skewness of the data. There is evidence that normality held for quarterly time

periods.

Fig. 5

Fig. 6

Fig. 7

Fig. 8

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Constant variance was found in nine out of ten samples for the Levene test, except for

sample one. The Bartlett test was consistent but samples two and three were also

rejected. There is strong evidence to suggest that constant volatility held.

The stationarity test accepted the null hypothesis in eight out of ten samples. Samples

two and three were rejected with p values of 0.0000 and 0.0103 (to 4 d.p.) respectively.

Fig. 10

Fig. 11

On closer inspection of the log returns plots, fig.10 shows that there are extreme returns

at day twenty six and twenty seven for sample two that may have caused the rejection of

stationarity. However, even with these points removed, between days one to twenty five

and twenty eight to sixty three the variances looked different and stationarity still may

not have held.

Fig.11 is clearly stationary and the other samples had very similar graphs. Therefore

stationarity reasonably held.

There is strong evidence to support the assumption of independence, with nine out of ten samples accepting the null hypothesis. FTSE 100 When considering normality, sample nine was rejected for both hypothesis tests with a p value of 0.0001 and 0.0000 for the Shapiro-Wilk test and the Jarque-Bera test respectively. By looking at fig.12, the results could be explained by outlying points located in the bottom left and top right of the plot. Fig.13 also shows a slight negative skew, with the outlying point clearly shown at 0.06. With the exception of sample nine, normality reasonably held.

The test for constant mean accepted the null hypothesis confirming a constant mean in all ten of the samples. The corresponding QQ plots presented linearity throughout, as shown in fig.9, which further validated the assumption of constant mean.

Fig. 9

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Fig. 12

Fig. 13

Constant volatility is less certain for FTSE, since the Levene test rejected three out of ten samples and the Bartlett test rejected four out of ten samples. Samples five, seven and nine were rejected for both tests and eight was also rejected for the Bartlett test.

Fig. 14

Fig. 15

From fig.14, is it unclear why sample five was rejected, since the mean and variance appeared constant. Fig.15 easily explains why sample nine was rejected, since the variance increased significantly after day forty. Constant variance did hold; however further testing should be considered. The test for constant mean accepted the null hypothesis, confirming a constant mean in nine out of ten samples. Sample nine was the only sample that rejected constant drift, with a p value of 0.0219.

Fig. 16

Fig. 17

In fig.16, the outlying point numbered 53, along with the slight variation along the line indicated departure from normality in the residuals. The deviation questioned the validity of this test for sample nine. The remaining nine samples produced linear QQ plots shown in fig.17, which further validated the constant mean assumption.

Stationarity was accepted in seven out of ten samples. Samples two, eight and nine were rejected. Sample two is considered here.

17

Fig. 18

For sample two, it was unclear from the log returns plot in fig.18 why stationarity was rejected. There seemed to be constant variance although the mean may not be constant since there was a slight upward trend between day one and thirty. However, the test had a p value of 0.0317 and stationarity may still have held. Overall, stationarity reasonably held for quarterly periods on the FTSE 100.

Independence was accepted for all samples and there was strong evidence the assumption held.

4.2.2 All assumptions held for FTSE 100 annual periods Normality The test for normality rejected the null hypothesis for all ten samples in the Shapiro-Wilk test. This was inconsistent with the results of the Jarque-Bera test, in which six of the ten samples were accepted: samples three, five, six and eight were rejected. According to the QQ plots, samples one and eight had heavier tails indicating departures from normality. However, both plots contain a number of outliers, which might explain this. Additional analysis of the histograms identified further differences. Samples two and six had a slight negative skew while the other samples followed a normal distribution. Overall there was evidence to accept the assumption of normality; however further testing would be required for a more definitive result.

Fig. 19

Fig. 20

Fig.19 shows a QQ plot of sample eight which showed departure from normality and outliers in the tails. Fig.20 is a histogram plot of sample six which suggested the data had a strong negative skew.

Rather interestingly the Levene and Bartlett tests for constant variance accepted the null hypothesis for all ten of their respective samples. All the p values were close to or equal to one, which implied that all samples strongly accepted the assumption of constant variance.

18

The test for constant mean accepted the null hypothesis in all of the ten samples. The respective QQ plots of the residuals were linear throughout which indicated that all the data samples were normally distributed, validating the assumption of constant mean as shown in fig.21.

Fig. 21

Stationarity was accepted in eight out of ten samples: only samples six and eight were rejected, exhibiting unique behaviour not seen in any other samples.

For instance, the log returns plot for sample six, indicated a seasonal behavioural pattern with four distinct troughs during the trading year, which can be seen in fig.22. The p value for sample six was 0.0450, so the test was only just rejected. Stationarity might have held throughout all the samples if the apparent seasonal trend in samples six and eight was removed.

The turning point test results for independence found for six of the ten samples the null hypothesis was accepted. Again there was some evidence to accept the assumption of independence, however further testing would be required for a more definitive conclusion. Perhaps removing sample six, which added a degree of uncertainty throughout the analysis of the FTSE 100 annual data, would increase the proportion of accepted samples for normality, independence and stationarity. Discussion A key finding between indices lies within the model assumptions holding for the FTSE 100 annual periods and not with the Nikkei 225 or the Hang Seng. The first natural difference is geographical location. Each market has its own different factors affecting the risk but risks associated with certain locations in the world will only affect certain markets. The FTSE 100 may not have quite as much risk as the Asian markets and could explain why the investigation has concluded constant variance holds for annual periods. There have been more noticeable stock market declines with the Hang Seng and the Nikkei 225 than the FTSE 100. More declines and instability will lead to a higher volatility and a greater possibility for extreme returns. Over longer periods of time, there is more room for market conditions to change and for risk to increase or decrease. The FTSE has been more stable for longer periods and could explain why annual periods have constant variance.

Fig. 22

19

Similarly, if conditions are stable, extreme returns are less likely. The normality assumption was rejected for the other indices due to wider tails in the data. The FTSE 100, being less volatile, does not have wide tails in the histogram plots.

4.2.3 Constant variance and normality did not hold for all periods With the exception of the annual samples for the FTSE 100 index, after quarterly intervals, the constant variance and normality assumptions did not hold. Annual periods Both the Nikkei 225 and the Hang Seng had very few samples accepting the null hypothesis of constant variance. Stationarity was rejected in nearly every sample. The log returns plots showed there was non-constant variance throughout all of the samples. Similarly, both normality tests rejected nearly every sample. The QQ plots confirmed the distributions had wider tails and could not be considered normal. Five year and ten year periods All but one sample rejected the constant variance hypothesis for both tests over the five year period. Only one FTSE 100 sample did not. All ten year samples were rejected for constant variance. All samples for both five and ten years were rejected for stationarity too and the log returns plots confirmed non-constant variance. Every sample was rejected for normality using both tests and the QQ plots confirmed the data had heavy tails. The histograms should be viewed with caution since much of the data is lost in the plot. Discussion The assumption of constant variance on the surface seems very unlikely in practice. Market risk is constantly changing as a result of numerous factors, some of which will only be relevant to certain markets. For example, European markets are heavily influenced by European Union policies. The geographical proximity of certain indices to emerging markets, which are by nature more volatile, is also a factor. One example of non-constant volatility for all markets is during times of economic and financial instability. Stock market crash periods tend to have higher volatility than during boom periods. Take the ten year sample three for the FTSE 100. Between days 1250-1500, there is a huge variation in the log returns. This period reflects the global financial crisis in 2007 and 2008, when there was huge uncertainty in the market.

20

Fig. 23

Similar crashes have occurred in the Nikkei 225 and the Hang Seng during 1998 and 2008. If Geometric Brownian Motion is used it may only be reasonably applied in a stable market or periods of stability. The normality assumption not holding for longer periods is unusual, since with larger samples normality would be expected by the central limit theorem; provided the data come from an identical distribution. Many of the histograms presented show wider tails, which suggests extreme returns are more frequent than if normally distributed. With changing market risk, extreme returns would be more common. With normality not holding, various other distributions have been suggested. Fama (1964) included the use of t distributions to better account for the wider tails of the data. There has also been evidence that log return data is log normally distributed, which could explain why the Jarque-Bera test rejected many samples, since the data was very skewed.

4.2.4 Constant mean and independence held for all periods

For every period in each index, the assumption of constant mean and independence of returns held. Further detail for each period is now detailed. Annual None of the annual samples for both the Nikkei 225 and the Hang Seng rejected the null hypothesis of constant mean. The corresponding QQ plots of the residuals for the daily returns all appear to be linear indicating normality in the residuals, validating the null hypothesis that constant mean held. Similarly results of the turning point test accepted nearly all of the samples for both the Nikkei and the Hang Seng on the assumption of independence. Having previously discussed the FTSE 100 in section 4.2.2, overall there was strong evidence to assume constant mean and independence held annually for all three indices. Five year and ten year periods All of the five year and ten year samples accepted the constant mean hypothesis for all three indices, with the exception of sample one of the Nikkei 225 index. Analysis of the corresponding QQ plot of the residuals showed heavy tails, signifying departure from normality which reduced the validity of the test. A lesser degree of certainty was observed when using the turning point test for independence where nearly all samples accepted the null hypothesis. Overall the data was independent for five and ten year periods.

21

Discussion It is a commonly held belief that future stock prices and returns are independent of past returns and prices. The results of the investigation are consistent with this belief. In an efficient market, individuals cannot predict prices based upon past returns due to information being available to everyone. This idea is formed from the Efficient Market Hypothesis. If the Efficient Market Hypothesis (EMH) theory holds, investors would not be able to make riskless profits from market inefficiencies and overall their expected return would be zero. The assumption of constant mean supports this idea, along with the results from the experiment. Given that GBM is an extension of Brownian Motion, which is the limit of a random walk process, EMH leads naturally to the random walk hypothesis (RWH). This states that stock market prices evolve according to a random walk and therefore cannot be predicted. If expected returns are constant, the RWH and constant expected returns version of the EMH are identical.

4.2.5 10 year weekly log return data was consistent with daily log return data

As summarized in the above table, the use of weekly log return data over a ten year period reaffirmed and in some cases strengthened the results of our initial investigation into the daily log return data. In compliance with the daily log return data the results refuted the assumption of normality and constant variance. Both the Shapiro-Wilk and the Jarque-Bera normality tests produced extremely small p-values, the associated QQ plots were heavy tailed which further verified the rejection of normality. The results of the Levene and Bartlett tests for constant variance were again consistent in strongly rejecting the assumption of constant variance. The test for constant mean accepted the null hypothesis throughout the weekly log return data with the exception of a single sample. The QQ plots of the residuals for the weekly returns all appear to be roughly linear with slightly heavy tails. This suggested that the residuals showed departure from normality and questioned the validity of test conducted.

Index 10 year sample

Normality Independence Constant Mean

Constant Variance

FTSE 100 Daily ✗ ✓ ✓ ✗

Weekly ✗ ✓ ✓ ✗

Nikkei Daily ✗ ✓ ✓ ✗

Weekly ✗ ✓ ✓ ✗

Hang Seng Daily ✗ ✓ ✓ ✗

Weekly ✗ ✓ ✓ ✗

22

All the weekly returns samples rejected the hypothesis of stationarity. Each test produced extremely small p values and the log returns plots clearly showed there is non-constant variance throughout. However, constant mean is apparent. Finally, the turning point test for independence from past data accepted nearly all of the samples on the assumption, across all three indices. For two of the three indices an additional was accepted compared to the daily returns data, strengthening this assumption.

23

4.3 Further discussion

A further financial interpretation of the above results is now considered. To further deepen the interpretation, one can also examine the interdependence of global stock markets, to observe whether or not spillovers are asymmetric. Asymmetric here refer to the way that bad news in one market spreads to another. Take for instance the daily log

returns for the three indices from June 1st 1987. Towards the end of 1987 there transpired a momentous event in financial history, known as Black Monday. In the wake of this crash, global markets plummeted, shedding enormous value in a very short period. This explains the clear correlation between the log returns plot of the three indices shown in fig.24, which all exhibit a large, predominantly negative variance from a centred mean around the same date.

Stochastic volatility models Our analysis identified flaws within the GBM model, namely constant variance and normality not holding. From an economic perspective, these flaws were expected. Without uncertainty (volatility) and risk, there is no driving factor for investor behaviour. To assume volatility to be constant in time was naïve, and stochastic volatility models may resolve the limitations of the GBM model. Unlike local volatility models that assume variance to be a function of both current asset price and time, stochastic volatility models are those where the variance of the stochastic process is also a random process in itself. A selection of these models will now be discussed. Heston Model The Heston stochastic volatility model would be a logical progression because it allows for non-normal log returns. The volatility is also mean reverting and takes into account that returns and implied volatility are negatively correlated. The latter is known as the leverage effect. The model is also more effective over longer time periods where the GBM model failed. This is due to its failure to create a short term skew as strong as the one given by the market. Another type of modeling, the GARCH models, are similar however

Fig. 24

24

they assume that the randomness of the variance process adjusts with the variance. On the other hand, the Heston model adjusts with the square root of the variance.

SABR model The SABR model aims to capture the volatility smile, which is the implied volatility pattern arising when pricing financial assets. A simplified example can be seen in fig. 25, where implied volatility decreases initially as the strike price increases but then begins to increase again after a certain point. Summary of other models In summary of this section, it is important to remember that these models both have their own advantages and disadvantages, like the GBM model. Stochastic volatility models are powerful tools only when the underlying assumptions are thoroughly understood and carefully applied. Although it has been suggested that these are a reasonable next step in modelling stock prices, a similar level of analysis in this investigation should be undertaken in order to confirm this.

Fig. 25

25

5 Further development Due to the time constrained nature of the project, there were several areas of

improvement within the investigation.

The experiment only considered one method of testing for independence between the log

returns data. Further testing should be carried out to fortify the assumption that returns

are independent. Graphical methods such as autocorrelation and partial autocorrelation

graphs would also be beneficial for future analysis.

The investigation used daily log returns data and weekly log returns data for ten year

periods. The Geometric Brownian Motion model does not make any assumptions about

the time intervals of the process and therefore can, in principle, be applied to any log

returns period. Further investigation of weekly log returns data should be carried out,

along with other log returns periods: quarterly, bi-annually and annually.

Three stock market indices were considered in the analysis. For both of the Asian indices,

the assumptions of the model only held for quarterly periods. The FTSE 100 however held

both quarterly and annually. Further investigation into different indices, for example

European and American indices may provide an explanation as to which markets are

most suitable to model stock prices using Geometric Brownian Motion.

26

6 Software In order to replicate the findings of the analysis already outlined, software was produced

using the computer program R and its numerous packages. The software, Stock Market

Analyzer (SMA), allows the user to select any stock or index mentioned on Yahoo Finance

and any time period for the stock data. SMA then implements a selection of the tests

outlined within this report and displays numerous graphs for interpretation.

The software is laid out in accordance with the assumptions tested in this report, with a

tab for each assumption. It also provides help boxes to aid the user in interpreting the

results of tests and graphs. The application shows six different test results along with six

plots of which the user can interpret to decide whether Geometric Brownian Motion is a

good model for their desired stock or index.

Key features

Fully automated software – SMA automatically gathers the user’s choice of stock prices from Yahoo Finance, provided that the device using the software is connected to the internet. This negates the need for the user to have a large database of stock prices on their local drive and also means the user does not need to manually input the stock prices they wish to analyze.

Data is always up to date – As the stock prices are automatically collected from

Yahoo Finance, the stock prices are updated daily and therefore are in real time and do not require any manual adjustment.

Mathematical/financial knowledge – The help button was introduced so that

someone without a high level of statistical knowledge can use the app and interpret some of the results the application gives. In addition to this, if the user does not know any stock tickers to examine, the application provides a link so that the user can search for a specific stock ticker.

Download report button – The download button allows the user to download the

results to their local drive. They can use this to create a large database of statistical results, or interpret the results another time when an internet connection is not present.

Simple user interface – One of the main objectives when creating the application

was to make a simple user interface so that a wide range of people can use the application and understand how it works.

Error detecting – The application calculates when a user incorrectly inputs a stock

symbol or when they select an infeasible date range, and it notifies them with message.

Multi-device functionality – The online version of the app works on desktop and mobile devices.

27

For more information, see the user guide outlined in the appendix and the software files.

7 Conclusions Our results produced five key findings:

1. All model assumptions held on a quarterly basis 2. All assumptions held for FTSE 100 annual periods 3. Constant variance and normality did not hold for all periods 4. Constant mean and independence held for all periods 5. Weekly log return data was consistent with daily log return data

A significant difference was found between the findings of the different market indices. The FTSE 100 results were not consistent with the Nikkei 225 and the Hang Seng indices. All model assumptions held for the FTSE 100 annual periods whereas they did not for the Nikkei 225 and the Hang Seng. This could be a result of different and unique business and financial risks in certain geographical locations. The Hang Seng and the Nikkei 225 are both Asian markets which could be more volatile and unstable than the FTSE 100. Normality was shown to be rejected by the hypothesis tests and showed heavier tails on QQ plots for longer time periods. Therefore, stock pricing did not follow a normal distribution for these periods, and the normality assumption did not hold for longer time periods. Constant variance is an unrealistic assumption in practical terms. The risks in the market are constantly changing, especially during periods of economic and financial instability. This can explain samples which show huge variations in log returns. The results of the testing were somewhat similar to the theoretical perspective. Independence of stock pricing is a common belief and matches the Efficient Market Hypothesis so with no surprise, this assumption held. As a further insight into log returns, weekly log data was used for ten year samples to see whether the results would be consistent with daily log returns, as was expected. All tests found that the same assumptions were heavily rejected using weekly log returns and was therefore consistent with the daily log returns. Overall, all assumptions held for quarterly time periods so GBM can be accepted as a good model for underlying assets in option pricing for shorter time periods. However, for five year and ten year time periods, the normality and constant variance assumptions do not hold, so GBM would not be a good model. Further testing using different indices is needed to confirm whether GBM is a good model for annual periods.

28

8 Appendix

8.1 Software user guide

Please note, method two only works for standalone PCs. For example, it will not work on a networked university computer. This is usually due to security settings of the

network.

29

8.2 Derivation of Geometric Brownian Motion Consider a stochastic process 𝑆𝑡 that satisfies the following equation

𝑑𝑆𝑡 = 𝜇𝑆𝑡𝑑𝑡 + 𝜎𝑆𝑡𝑑𝐵𝑡

where 𝐵𝑡is a Brownian motion and 𝜇 and 𝜎 are constants.

To solve, we require a solution to be function in terms of t and 𝐵𝑡, so that 𝑆𝑡 = 𝑔(𝑡, 𝐵𝑡).

Consider the following theorem (a proof can be found in Kulkarni, Modeling and

analysis of stochastic systems)

Let {𝑋𝑡, 𝑡 ≥ 0} be a diffusion process. Let 𝑔(𝑡, 𝑥) be a function that is continuously

differentiable in t and twice continuously differentiable in x. The stochastic process {𝑔(𝑡, 𝑋𝑡), 𝑡 ≥ 0} satisfies the following stochastic differential equation

𝑑𝑔(𝑡, 𝑋𝑡) = (𝛿𝑔(𝑡, 𝑋𝑡)

𝛿𝑡+ 𝜇(𝑡, 𝑋𝑡)

𝛿𝑔(𝑡, 𝑋𝑡)

𝛿𝑥+

1

2𝜎2(𝑡, 𝑋𝑡)

𝛿2𝑔(𝑡, 𝑋𝑡)

𝛿𝑥2) 𝑑𝑡

+ 𝜎(𝑡, 𝑥)𝛿𝑔(𝑥, 𝑋𝑡)

𝛿𝑥𝑑𝐵𝑡

This equation is known as Ito’s formula. Now, when 𝐵𝑡 = 𝑋𝑡 the above formula reduces

to

𝑑𝑔(𝑡, 𝐵𝑡) = (𝛿𝑔(𝑡, 𝐵𝑡)

𝛿𝑡+

1

2

𝛿2𝑔(𝑡, 𝐵𝑡)

𝛿𝑥2) 𝑑𝑡 +

𝛿𝑔(𝑡, 𝐵𝑡)

𝛿𝑥𝑑𝐵𝑡

From the above adjusted Ito’s formula, we have that 𝑔(𝑡, 𝐵𝑡) must satisfy

𝛿𝑔(𝑡, 𝑥)

𝛿𝑡+

1

2

𝛿2𝑔(𝑡, 𝑥)

𝛿𝑥2= 𝜇𝑔(𝑡, 𝑥)

(1)

𝛿𝑔(𝑡, 𝑥)

𝛿𝑥= 𝜎𝑔(𝑡, 𝑥)

(2)

Equation (2) implies that

𝑔(𝑡, 𝑥) = 𝑓(𝑡)𝑒𝜎𝑥 (3)

where 𝑓(𝑡) is to be found. Substitution into (1) gives

𝑑𝑓(𝑡)

𝑑𝑡𝑒𝜎𝑥 +

1

2𝑓(𝑡)𝜎2𝑒𝜎𝑥 = 𝜇𝑓(𝑡)𝑒𝜎𝑥

⇔ 𝑑𝑓(𝑡)

𝑑𝑡= (𝜇 −

1

2𝜎2) 𝑓(𝑡)

So that

30

𝑓(𝑡) = 𝑐𝑒(𝜇−12

𝜎2)𝑡

where 𝑐 is a constant. Substituting 𝑓(𝑡) into (3) gives

𝑔(𝑡, 𝑥) = 𝑐𝑒(𝜇−12

𝜎2)𝑡+𝜎𝑥

Therefore, the general solution is

𝑆𝑡 = 𝑐𝑒(𝜇−12

𝜎2)𝑡+𝜎𝐵𝑡

To find 𝑐, let 𝑡 = 0 and, from the properties of a standard Brownian motion, 𝐵0 = 0 we

have 𝑆0 = 𝑐. Hence the solution is

𝑆𝑡 = 𝑆0𝑒(𝜇−12

𝜎2)𝑡+𝜎𝐵𝑡

31

8.3 P value tables Please note all p values and statistics are to 4d.p and rejected samples at a 5% level are highlighted.

Quarterly Annual

Sam

ple

FT

SE

Han

g Se

ng

Nik

kei

FT

SE

Han

g Se

ng

Nik

kei

P

value

W

Stat

P

value

W

Stat

P

value

W

Stat

P

value

W

Stat

P

value

W

Stat

P

value

W

Stat

1 0.6586 0.9854 0.2117 0.9744 0.0000 0.8792 0.0025 0.9817 0.0000 0.4706 0.0000 0.9334

2 0.3210 0.9781 0.0022 0.9339 0.0000 0.7329 0.0321 0.9879 0.0000 0.8777 0.0000 0.9078

3 0.0534 0.9627 0.1520 0.9715 0.0195 0.9540 0.0001 0.9736 0.0000 0.9287 0.0072 0.9843

4 0.8625 0.9893 0.0000 0.7983 0.7731 0.9875 0.0017 0.9807 0.0000 0.9129 0.0000 0.9511

5 0.5856 0.9840 0.9361 0.9913 0.6286 0.9848 0.0000 0.9653 0.0000 0.9493 0.0001 0.9736

6 0.8849 0.9898 0.2531 0.9759 0.9139 0.9906 0.0000 0.9459 0.0000 0.9681 0.0000 0.9646

7 0.7402 0.9869 0.0079 0.946 0.5594 0.9835 0.0008 0.9789 0.0001 0.9740 0.0001 0.9732

8 0.1747 0.9727 0.0834 0.9665 0.3186 0.9780 0.0000 0.9339 0.0003 0.9759 0.0000 0.9182

9 0.0001 0.9057 0.6591 0.9854 0.2665 0.9764 0.0152 0.9861 0.5665 0.9949 0.5907 0.9950

10 0.1273 0.9700 0.7435 0.9869 0.8201 0.9884 0.0080 0.9846 0.0119 0.9855 0.0000 0.9665

Table 1 Shapiro-Wilk test results for quarterly and annual periods

5 years 10 years

Sam

ple

FT

SE

Han

g Se

ng

Nik

kei

FT

SE

Han

g Se

ng

Nik

kei

P

value

W

Stat

P

value

W

Stat

P

value

W

Stat

P

value

W

Stat

P

value

W

Stat

P

value

W

Stat

1 0.0000 0.8399 0.0000 0.5631 0.0000 0.7833 0.0000 0.8921 0.0000 0.7067 0.0000 0.8864

2 0.0000 0.9767 0.0000 0.9546 0.0000 0.9466 0.0000 0.9684 0.0000 0.9011 0.0000 0.9833

3 0.0000 0.9923 0.0000 0.9231 0.0000 0.9710 0.0000 0.9040 - - 0.0000 0.9219

4 0.0000 0.9857 0.0000 0.9551 0.0000 0.9907 - - - - - -

5 0.0000 0.9532 0.0000 0.9221 0.0000 0.8964 - - - - - -

6 0.0000 0.9189 - - 0.0000 0.9631 - - - - - -

Table 2 Shapiro-Wilk test results for 5 and 10 year periods

32

Quarterly Annual

Sam

ple

FT

SE

Han

g Se

ng

Nik

kei

FT

SE

Han

g Se

ng

Nik

kei

P

value

JB

Stat

P

value

JB

Stat

P

value

JB

Stat

P

value

JB

Stat

P

value

JB

Stat

P

value

JB

Stat

1 0.9968 0.0064 0.2065 3.1551 0.0000 57.040 0.2978 2.4224 0.0000 12561

4 0.0000 191.31

2 0.2511 2.7636 0.0000 34.856 0.0000 556.72 0.4162 1.7530 0.0000 922.97 0.0000 355.09

3 0.0955 4.6965 0.0735 5.2200 0.0000 19.980 0.0022 12.246 0.0000 311.95 0.0000 23.143

4 0.9265 0.1527 0.0000 384.21 0.5434 1.2198 0.4828 1.4564 0.0000 886.39 0.0000 160.30

5 0.9157 0.1762 0.8746 0.2681 0.6225 0.9480 0.0000 33.388 0.0000 164.38 0.0000 49.564

6 0.6845 0.7583 0.3850 1.9088 0.6824 0.7644 0.0000 79.054 0.0000 80.150 0.0000 130.47

7 0.9657 0.0699 0.0001 17.718 0.5767 1.1010 0.1593 3.6734 0.0000 22.278 0.0000 29.040

8 0.0540 5.835 0.2765 2.5710 0.8865 0.2409 0.0000 79.516 0.0000 27.557 0.0000 275.56

9 0.0000 33.872 0.5928 1.0458 0.6252 0.9395 0.2721 2.6033 0.4450 1.6194 0.7856 0.4827

10 0.2392 2.8613 0.8668 0.2858 0.7918 0.4669 0.5713 1.1195 0.0164 8.2173 0.0000 67.079

Table 3 Jarque-Bera test results for quarterly and annual periods

5 years 10 years

Sam

ple

FT

SE

Han

g Se

ng

Nik

kei

FT

SE

Han

g Se

ng

Nik

kei

P

value

JB

Stat

P

value

JB

Stat

P

value

JB

Stat

P

value

JB

Stat

P

value

JB

Stat

P

value

JB

Stat

1 0.0000 45579 0.0000 15643

31 0.0000

14749

5.5 0.0000 59896 0.0000

22900

84 0.0000 26562

2 0.0000 433.38 0.0000 701.54 0.0000 1554.2 0.0000 739.55 0.0000 14461 0.0000 369.64

3 0.0000 48.254 0.0000 3689.4 0.0000 351.49 0.0000 8137.6 - - 0.0000 7866.3

4 0.0000 100.68 0.0000 1424.8 0.0000 91.096 - - - - - -

5 0.0000 960.93 0.0000 2593.7 0.0000 4635.8 - - - - - -

6 0.0000 2306.2 - - 0.0000 1394.7 - - - - - -

Table 4 Jarque-Bera test results for 5 and 10 year periods

33

Quarterly Annual Sa

mp

le

FT

SE

Han

g Se

ng

Nik

kei

FT

SE

Han

g Se

ng

Nik

kei

P

value

W

Stat

P

value

W

Stat

P

value

W

Stat

P

value

W

Stat

P

value

W

Stat

P

value

W

Stat

1 0.6239 0.4754 0.3437 1.0866 0.2219 1.5426 0.9991 0.0073 0.0003 6.4532 0.0985 2.1176

2 0.2759 1.3148 0.4152 0.8915 0.0004 8.9474 1 0.0000 0.0000 5.6573 0.0000 19.797

3 0.3294 1.1307 0.8339 0.1822 0.0788 2.6479 0.9990 0.0079 0.0021 5.0426 0.0010 5.6041

4 0.7170 0.3345 0.0871 2.5399 0.2370 1.4736 1.00 0.0050 0.0726 2.3538 0.0000 12.927

5 0.0161 4.4184 0.5614 0.5827 0.3913 0.9526 1 0 0.0091 3.9300 0.0007 5.9088

6 0.9054 0.0995 0.2916 1.2571 0.3107 1.1914 0.9996 0.0047 0.0038 4.5866 0.1205 1.9600

7 0.0409 3.3667 0.5857 0.5396 0.7440 0.2971 0.9991 0.0078 0.5245 0.7478 0.0479 2.6733

8 0.0620 2.9098 0.7464 0.2939 0.7690 0.2638 1 0.0000 0.0341 2.9323 0.0000 15.523

9 0.0009 7.9349 0.6590 0.4198 0.3376 1.1051 0.9999 0.0013 0.0019 5.1258 0.4857 0.8166

10 0.5741 0.5599 0.5654 0.5754 0.3396 1.0989 0.9995 0.0049 0.0344 2.9256 0.0007 5.8170

Table 5 Levene test results for quarterly and annual periods

5 years 10 years

Sam

ple

FT

SE

Han

g Se

ng

Nik

kei

FT

SE

Han

g Se

ng

Nik

kei

P

value

W

Stat

P

value

W

Stat

P

value

W

Stat

P

value

W

Stat

P

value

W

Stat

P

value

W

Stat

1 0.0000 7.0836 0.0000 6.2095 0.0000 21.592 0.0000 7.8634 0.0002 6.5479 0.0000 52.68

2 0.0532 2.3408 0.0000 22.726 0.0000 31.988 0.0000 87.813 0.0000 72.684 0.0000 23.129

3 0.0000 18.389 0.0000 26.412 0.0000 14.057 0.0000 76.426 - - 0.0000 41.381

4 0.0000 19.694 0.0000 21.241 0.0002 5.538 - - - - - -

5 0.0000 30.945 0.0000 57.138 0.0000 56.336 - - - - - -

6 0.0000 29.014 0.0000 6.2095 0.0000 6.6598 - - - - - -

Table 6 Levene test results for 5 and 10 year periods

34

Quarterly Annual Sa

mp

le

FT

SE

Han

g Se

ng

Nik

kei

FT

SE

Han

g Se

ng

Nik

kei

P

value

T

Stat

P

value

T

Stat

P

value

T

Stat

P

value

T

Stat

P

value

T

Stat

P

value

T

Stat

1 0.4074 1.7959 0.0822 4.9965 0.0057 10.333 0.9999 0.0060 0.0000 293.19 0.0002 20.029

2 0.3020 2.398 0.0415 6.3649 0.0000 64.760 1 0.0024 0.0000 53.588 0.0000 121.82

3 0.2241 2.9915 0.2398 2.8556 0.0025 11.947 0.9998 0.0095 0.0000 28.801 0.0005 17.746

4 0.9742 0.0523 0.0000 22.071 0.0834 4.9678 1 0.0058 0.0000 31.023 0.0000 50.527

5 0.0084 9.57 0.4938 1.4111 0.1074 4.4623 1 0.0003 0.0117 15.940 0.0000 37.052

6 0.7977 0.452 0.0370 6.592 0.3596 2.0456 1 0.0060 0.0000 24.822 0.0030 13.955

7 0.0149 8.407 0.0929 4.7519 0.4561 1.5699 0.9998 0.0095 0.1254 5.7326 0.0053 12.721

8 0.0062 10.172 0.7023 0.7069 0.9354 0.1336 1 0.0029 0.0752 16.869 0.0000 104.05

9 0.0000 26.352 0.7244 0.6449 0.3661 2.0097 0.9999 0.0042 0.0012 15.827 0.4786 2.4816

10 0.3624 2.0298 0.3040 2.3811 0.6854 0.7555 1 0.0058 0.0024 9.4420 0.0000 29.423

Table 7 Bartlett test results for quarterly and annual periods

5 years 10 years

Sam

ple

FT

SE

Han

g Se

ng

Nik

kei

FT

SE

Han

g Se

ng

Nik

kei

P

value

T

Stat

P

value

T

Stat

P

value

T

Stat

P

value

T

Stat

P

value

T

Stat

P

value

T

Stat

1 0.0000 230.23 0.0000 493.42 0.0000 461.66 0.0000 181.28 0.0000 435.10 0.0000 395

2 0.0669 8.7793 0.0000 145.77 0.0000 254.15 0.0000 486.54 0.0000 671.31 0.0000 101.37

3 0.0000 100.54 0.0000 284.09 0.0000 87.235 0.0000 638.32 - - 0.0000 365.93

4 0.0000 109.02 0.0000 176.46 0.0002

22.461 - - - - - -

5 0.0000 241.49 0.0000 484.33 0.0000 532.08 - - - - - -

6 0.0000 294.58 0.0000 493.42 0.0000 46.917 - - - - - -

Table 8 Bartlett test results for 5 and 10 year periods

35

Quarterly Annual Sa

mp

le

FT

SE

Han

g Se

ng

Nik

kei

FT

SE

Han

g Se

ng

Nik

kei

P

value

T

Stat

P

value

T

Stat

P

value

T

Stat

P

value

T

Stat

P

value

T

Stat

P

value

T

Stat

1 0.0925 2.921 0.2205 1.5330 0.3026 1.0810 0.3788 0.7774 0.1664 1.9260 0.4466 0.5812

2 0.4116 0.6835 0.7462 0.1057 0.9729 0.0012 0.3765 0.7849 0.4963 0.4641 0.7573 0.0957

3 0.5443 0.3718 0.7702 0.0861 0.0990 2.8060 0.6857 0.1642 0.3772 0.7827 0.3192 0.9963

4 0.9464 0.0046 0.0391 4.4450 0.4372 0.6115 0.5284 0.3986 0.4996 0.4571 0.2081 1.5930

5 0.7684 0.0875 0.2745 1.2160 0.3216 0.9985 0.6363 0.2242 0.7348 0.1150 0.7145 0.1341

6 0.6263 0.2396 0.3835 0.7704 0.7911 0.0708 0.4868 0.4849 0.6705 0.1814 0.3329 0.9412

7 0.3416 0.9187 0.3485 0.8927 0.8409 0.0407 0.9304 0.0076 0.0692 3.3310 0.5026 0.4508

8 0.9088 0.0132 0.5082 0.4431 0.5599 0.3436 0.9095 0.0130 0.9600 0.0025 0.5632 0.3351

9 0.0219 5.537 0.9635 0.0021 0.8242 0.0498 0.9026 0.0150 0.1015 2.6450 0.5632 0.3351

10 0.8995 0.0161 0.9415 0.0054 0.0671 3.4760 0.8533 0.0343 0.5261 0.4031 0.3737 0.7942

Table 9 Constant mean test results for quarterly and annual periods

5 years 10 years

Sam

ple

FT

SE

Han

g Se

ng

Nik

kei

FT

SE

Han

g Se

ng

Nik

kei

P

value

T

Stat

P

value

T

Stat

P

value

T

Stat

P

value

T

Stat

P

value

T

Stat

P

value

T

Stat

1 0.3923 0.9525 0.5910 0.2890 0.7098 0.1385 0.5764 0.3122 0.6651 0.1875 0.0484 3.8990

2 0.9729 0.0012 0.3315 0.9437 0.7979 0.0656 0.1018 2.6800 0.6206 0.2451 0.9782 0.0007

3 0.1620 1.9580 0.7700 0.0856 0.9974 0.0000 0.6168 0.2505 - - 0.5502 0.3570

4 0.3761 0.7840 0.0623 3.4830 0.3893 0.7417 - - - - - -

5 0.3565 0.8509 0.9299 0.0078 0.3654 0.8197 - - - - - -

6 0.4339 0.6128 0.5910 0.2890 0.2190 1.5120 - - - - - -

Table 10 Constant mean test results for 5 and 10 year periods

36

Quarterly Annual Sa

mp

le

FT

SE

Han

g Se

ng

Nik

kei

FT

SE

Han

g Se

ng

Nik

kei

P

value

Z

Stat

P

value

Z

Stat

P

value

Z

Stat

P

value

Z

Stat

P

value

Z

Stat

P

value

Z

Stat

1 0.4793 0.7075 0.4070 -0.829 0.0432 -2.021 0.1938 -1.300 0.9597 0.0506 0.0802 -1.749

2 0.6133 -0.505 0.8398 -0.202 0.0432 -2.021 0.0188 -2.349 0.0004 -3.513 0.0802 -1.749

3 0.4188 -0.809 0.1264 -1.529 0.6133 -0.505 0.2715 1.0996 0.9201 0.1004 0.4841 -0.700

4 0.7598 0.3057 0.2214 -1.223 0.6860 0.404 0.0494 -1.965 0.3936 0.8531 0.5158 0.6498

5 0.6133 -0.505 0.6074 0.5138 0.1571 -1.415 0.7624 0.3023 0.8409 -0.201 0.9601 0.0500

6 0.686 0.4043 0.6810 -0.411 0.4793 0.7075 0.0373 2.0828 0.6129 0.5059 0.7264 0.3499

7 0.2214 1.2228 0.0086 -2.628 0.6133 -0.505 0.2715 1.0996 0.0119 -2.514 0.3955 -0.850

8 0.7598 0.3057 0.0858 -1.718 0.4188 -0.809 0.6502 0.4535 0.3955 -0.850 0.6893 -0.400

9 0.2214 1.2228 0.1889 1.3139 0.3122 1.0107 0.0005 3.4987 0.5158 0.650 0.9601 0.0500

10 0.7598 -0.306 0.6860 0.4043 0.8398 -0.202 0.4498 -0.756 0.9601 0.0500 0.1617 1.3995

Table 11 Turning point test results for quarterly and annual periods

5 years 10 years

Sam

ple

FT

SE

Han

g Se

ng

Nik

kei

FT

SE

Han

g Se

ng

Nik

kei

P

value

Z

Stat

P

value

Z

Stat

P

value

Z

Stat

P

value

Z

Stat

P

value

Z

Stat

P

value

Z

Stat

1 0.0374 -2.081 0.0105 -2.560 0.0023 -3.053 0.0279 -2.199 0.0323 -2.141 0.0026 -3.009

2 0.2828 -1.074 0.6061 -0.516 0.2120 -1.248 0.0169 -2.390 0.3575 -0.920 0.5814 -0.551

3 0.0107 -2.551 0.1853 -1.325 0.6720 0.4235 0.3133 1.0125 - - 0.0777 1.7645

4 0.4199 -0.807 0.9821 -0.022 0.2120 -1.248 - - - - - -

5 0.4469 0.7605 0.562 0.5799 0.1740 1.3596 - - - - - -

6 0.4875 0.6942 - - 0.2465 1.159 - - - - - -

Table 12 Turning point test results for 5 and 10 year periods

37

Quarterly Annual 5 years 10 years Sa

mp

le

FT

SE

Han

g Se

ng

Nik

kei

FT

SE

Han

g Se

ng

Nik

kei

FT

SE

Han

g Se

ng

Nik

kei

FT

SE

Han

g Se

ng

Nik

kei

P values

1 0.7663 0.1561 0.2686 0.9464 0.0000 0.0068 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

2 0.0317 0.0251 0.0000 0.8573 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

3 0.5752 0.2758 0.0103 0.1385 0.0002 0.0004 0.0000 0.0000 0.0000 0.0000 - 0.0000

4 0.3154 0.0012 0.0795 0.5376 0.0015 0.0000 0.0000 0.0000 0.0000 - - -

5 0.1519 0.1738 0.5914 0.1653 0.0000 0.0000 0.0000 0.0000 0.0000 - - -

6 0.8962 0.2470 0.4472 0.0450 0.0000 0.0000 0.0000 - 0.0000 - - -

7 0.2146 0.3322 0.3981 0.8283 0.1739 0.0038 - - - - - -

8 0.0490 0.4296 0.9933 0.0000 0.0089 0.0000 - - - - - -

9 0.0000 0.8522 0.6964 0.7916 0.0076 0.0619 - - - - - -

10 0.3390 0.3010 0.8880 0.8809 0.0406 0.0003 - - - - - -

Table 13 Stationarity test results for quarterly, annual, 5 year and 10 year periods

38

8.4 Additional figures

Samples mentioned within the text

Further FTSE 100 annual sample analysis

From above, sample one was rejected by the Shapiro-Wilk test but not by the Jarque-Bera test. This may be due to Shapiro-Wilk’s sensitivity to normality, since there are a few outlying points in the above graph. Sample eight gives an example of where both tests have correctly rejected the assumption of normality.

39

Further examples of QQ plots for larger samples

10 year weekly QQ plot examples

40

Log returns plot examples for larger samples

41

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