Statistical Analysis of the Log Returns of Financial Assets

Statistical Analysis of the Log Returns of Financial Assets

Leo Quigley Student ID Number: 0442372 BSc in Financial Mathematics Supervisor: Dr. Volkert Paulsen Second Reader: Dr. David Ramsey

April 9, 2008

Abstract

In many models of financial mathematics, such as the mean-variance model for portfolio selection and asset pricing models, the independence and identical normal distribution of the asset returns is the cornerstone assumption on which these are built. Empirical studies have shown that the returns of an asset don't actually follow a normal distribution but in fact they have fatter tails than the normal can capture. There is evidence that the asset returns not only display this so-called heavy tailed behaviour but are also possibly skewed in their distributions. Empirical research has also found that returns display alternating periods of high and low volatility contradicting the idea of independent and identical distribution.

Acknowledgments I would like to thank my family for all of their support throughout my time in college.

Thanks to everyone who I have become friends with in college who have made this the best four years of my life.

I would also like to thank my supervisor, Dr. Volkert Paulsen, for both suggesting this topic and assisting me throughout.

Contents

1 Introduction

3

1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Outline of Paper . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Overview of Returns of Financial Assets

6

2.1 Properties of Stock Prices . . . . . . . . . . . . . . . . . . . . 6

2.2 Defining a Financial Asset Return . . . . . . . . . . . . . . . . 9

2.3 Statistical Properties of Returns . . . . . . . . . . . . . . . . . 13

3 Random Walk Approach and Normality of Returns

16

3.1 Random Walk Hypothesis . . . . . . . . . . . . . . . . . . . . 16

3.1.1 Market Efficiency . . . . . . . . . . . . . . . . . . . . . 17

3.1.2 Definition of a Random Walk . . . . . . . . . . . . . . 19

3.1.3 Applying the Hypothesis to Financial Series Data . . . 19

3.2 Testing for Normality . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1 Overview of Normality in Returns . . . . . . . . . . . . 21

3.2.2 Exploratory Data Analysis . . . . . . . . . . . . . . . . 24

1

CONTENTS

2

3.2.3 Statistical Tests of Normality . . . . . . . . . . . . . . 47

4 Extreme Value Theory Approach

66

4.1 Extreme Value Theory . . . . . . . . . . . . . . . . . . . . . . 66

4.1.1 Fisher-Tippett Theorem . . . . . . . . . . . . . . . . . 68

4.1.2 Generalisalized Extreme Value Distribution . . . . . . 70

4.1.3 General Pareto Distribution . . . . . . . . . . . . . . . 70

4.2 Peak Over Threshold Method . . . . . . . . . . . . . . . . . . 71

4.2.1 Introduction to Peak Over Threshold . . . . . . . . . . 71

4.2.2 Pickands-Balkema-De Hann Theorem . . . . . . . . . . 72

4.2.3 POT Using GPD Approach . . . . . . . . . . . . . . . 73

4.2.4 Application of POT to the Tails . . . . . . . . . . . . . 75

5 Time Series Approach

89

5.1 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2 Correlation and Autocorrelation. . . . . . . . . . . . . . . . . 91

6 Conclusions

97

6.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download