Chapter 1: Introduction
Modeling of the stock market returns and inter-relationship between the prices of different stocks has become a hot topic of many researchers (Mantegna and Stanley, 1995; Mantegna, 1991; Levy and Solomon, 1996; Bak et al, 1997). These scholars used mathematical instruments associated with the graph theory to estimate the relationships between the stock returns.
Samuelson (1965) has shown several decades ago that the stock market returns should be modeled as a stochastic or random process. The randomness of returns was also supported by the efficient market hypothesis, which was a mainstream theory of financial markets up until the 2000's when behavioural school of finance started gaining strength.
On the one hand, Lo (1991) has demonstrated that stock returns have very low cross correlations and this fact emphasises the randomness of returns. However, other researchers such as Ross (1976) argue that cross correlations exist and are explained by economic theory.
1.1. Aims and Objectives
The aim of the dissertation is to find out which stocks that comprise FTSE 100 share price index have a high degree of cross correlation. This aim will be reached by exploring all constituents of the FTSE 100 share price index, modeling the minimum spanning tree and estimating cross correlation between the stocks.
Chapter 2: Literature Review
This chapter of the research presents the arguments of the key researchers who investigated the cross correlations in the stock market. This chapter will help to set the context of the research and compare the outcomes of the analysis conducted further with the results obtained by the previous attempts to explore the stock market behaviour and inter-relationship of the stock returns.
2.1. Stock Returns Network and Graph Theory
Graph theory has originated in mathematics and was later widely adopted in the computer science and modeling of financial time series. The graphs represent a model of inter-relationships between data points. These relationships are shown with lines or arrows that connect the points of data. These points may be represented by any objects such as stock prices. The graphs may be divided into directed and undirected. The latter imply that the data objects do not show any dependencies and are rather similar in their nature. In this case, undirected graphs are recommended to draw with lines or arcs. Directed graphs are indicated with arrows, which show the direction of dependency (Bondy and Murty, 2008; Harary and Palmer, 1973; Chartrand, 1985).
The graph theory is widely implemented in logistics. For example, it may be used to model the least costly airline routes, delivery routes, road construction and other field. The stock market can also be presented as a network in which individual stocks comprise portfolios in which investors allocate their funds.
Even in the early theory of optimal portfolio selection, Markowitz (1952) suggests comprising a portfolio of stocks based on the correlations and co-variances among them. In this respect the stock prices and stock returns represent the vertices in the graphs. The graph for six random stocks would have the following form:
This is an undirected graph that shows static links between different objects, which could be represented by stock prices or stock returns (Zhuang and Ye, 2008). In order to comprise a portfolio of minimum variance, graph theory may also be implemented.
One of the elements in the graph theory that help investors to comprise such a portfolio is minimum spanning trees. It is a subgraph that provides the optimal path to all vertices, i.e. there is a minimum total distance between all the vertices (Karger et al, 1995; Pettie and Ramachandran, 2002).
The minimum spanning tree is found after summarizing the weights of all vertices or data objects and minimizing this number. For example, the provided nine vertices may be connected in different ways. However, the minimum spanning tree will be represented by the green lines that show the total minimum path from each point to the others.
Minimum spanning trees also find implication not only in the finance but also in other industries in order to optimize logistics or find out the least costly way (Graham and Hell, 1985).
2.2. Cross-Correlation of Stock Returns
The study of cross correlation in the stock market has become very popular recently. The researchers implemented random matrix theory (RMT) to study cross correlations (Laloux et al, 1999; Laloux et al, 2000; Plerou et al, 2000; Sharifi et al, 2004).
The cross correlation matrices have been analysed in the past by comparing the eigen values. The structure of the correlation matrix was determined to be significant after reviewing the deviations of the eigen values from those predicted by RMT.
Drozdz et al (2001) previously attempted to analyse cross correlation between the Dow Jones Industrial Average constituents and DAX constituents. They also implemented the eigen value and random matrix theory as their methodology. The researchers have concluded that there were statistically significant cross correlations between the two stock markets.
Chapter 3: Methodology
This chapter discusses the methods by which the stocks comprising FTSE 100 share price index will be analysed. The relationships between the one hundred stocks will be established by means of the cross correlation matrix estimated in statistical software MatLab. Cross correlation matrix can be viewed as a matrix filled with correlation coefficients.
Correlation coefficients demonstrate the degree of linear association between the given random variables. This correlation may be positive or negative. Positive correlation indicates that the two variables tend to move in the same direction or changes in one of them are associated with the changes in another variable in the same direction. Negative correlation implies the movement of variables in the opposite directions.
Cross correlation are modeled in MatLab using the following formula:
Where xn and yn are the processes that are stationary; E implies the expected value (Orfanidis, 1996).
Chapter 4: Results and Analysis
This chapter of the research project provides the outcomes of the calculations and modeling that were explained in the previous section. The methods are applied to the constituents of the FTSE 100 share price index. The results are presented by the output of the statistical software MatLab that has been used for the calculation of the cross correlations and finding the minimum spanning tree of the stocks that comprise FTSE 100 share price index.
The results of the dissertation will be achieved in accordance with the following time table.
Table 1 Time Table of Research
Target Period
Task to be Achieved
Week 1
Complete Literature Review and Study Available Models
Week 2
Discuss and Choose Methodology of the Research
Week 3
Applying the Methods to FTSE 100 index constituents
Week 4
Discuss the limitations of the research
Week 5
Conduct a discussion of the findings and results
Week 6
Make conclusions
Week 7
Make recommendations
Week 8
Complete the introduction and abstract
Week 9
Proofreading and submission
Chapter 5: Discussion and Conclusions
The last chapter of the dissertation has a goal to present a discussion of the results. The literature review has shown that there have been different findings achieved by previous researchers. Some of these findings will be consistent with the results found in this dissertation while others will contradict to the results. The differences may be explained by the different stock markets used in investigation, different selection of stocks or share price indexes and different methods of finding association between the stock returns. The chapter also concludes on the aims and objectives set in the introduction chapter. It provides a summary of the outcomes and what implication they have on financial world. This conclusion is followed by recommendations for future studies.
