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vii

viii Detailed Table of Contents

5. Time Series Analysis 43

5.1 Autoregressive and Moving Average Models 43

5.1.1 Autoregressive Model 43

5.1.2 Moving Average Models 45

5.1.3 Autocorrelation and Forecasting 47

5.2 Trends and Seasonality 49

5.3 Conditional Heteroskedasticity 51

5.4 Multivariate Time Series 54

5.5 References for Further Reading and Econometric

Software 57

5.6 Exercises 57

6. Fractals 59

6.1 Basic Definitions 59

6.2 Multifractals 63

6.3 References for Further Reading 67

6.4 Exercises 67

7. Nonlinear Dynamical Systems 69

7.1 Motivation 69

7.2 Discrete Systems: Logistic Map 71

7.3 Continuous Systems 75

7.4 Lorenz Model 79

7.5 Pathways to Chaos 82

7.6 Measuring Chaos 83

7.7 References for Further Reading 86

7.8 Exercises 86

8. Scaling in Financial Time Series 87

8.1 Introduction 87

8.2 Power Laws in Financial Data 88

8.3 New Developments 90

8.4 References for Further Reading 92

8.5 Exercises 92

9. Option Pricing 93

9.1 Financial Derivatives 93

9.2 General Properties of Options 94

9.3 Binomial Trees 98

9.4 Black-Scholes Theory 101

9.5 References for Further reading 105

ix

Detailed Table of Contents

9.6 Appendix. The Invariant of the Arbitrage-Free

Portfolio 105

9.7 Exercises 109

10. Portfolio Management 111

10.1 Portfolio Selection 111

10.2 Capital Asset Pricing Model (CAPM) 114

10.3 Arbitrage Pricing Theory (APT) 116

10.4 Arbitrage Trading Strategies 118

10.5 References for Further Reading 120

10.6 Exercises 120

11. Market Risk Measurement 121

11.1 Risk Measures 121

11.2 Calculating Risk 125

11.3 References for Further Reading 127

11.4 Exercises 127

12. Agent-Based Modeling of Financial Markets 129

12.1 Introduction 129

12.2 Adaptive Equilibrium Models 130

12.3 Non-Equilibrium Price Models 134

12.4 Modeling of Observable Variables 136

12.4.1 The Framework 136

12.4.2 Price-Demand Relations 138

12.4.3 Why Technical Trading May Be Successful 139

12.4.4 The Birth of a Liquid Market 141

12.5 References for Further Reading 143

12.6 Exercises 143

Comments 145

References 149

Answers to Exercises 159

Index 161

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Chapter 1

Introduction

This book is written for those physicists who want to work on Wall

Street but have not bothered to read anything about Finance. This is

a crash course that the author, a physicist himself, needed when he

landed a financial data analyst job and became fascinated with the

huge data sets at his disposal. More broadly, this book addresses the

reader with some background in science or engineering (college-level

math helps) who is willing to learn the basic concepts and quantitative

methods used in modern finance.

The book loosely consists of two parts: the â€˜â€˜appliedâ€™â€™ part and the

â€˜â€˜academicâ€™â€™ one. Two major fields, Econometrics and Mathematical

Finance, constitute the applied part of the book. Econometrics can be

broadly defined as the methods of model-based statistical inference in

financial economics [1]. This book follows the traditional definition

of Econometrics that focuses primarily on the statistical analysis of

economic and financial time series [2]. The other field is Mathematical

Finance [3, 4]. This term implies that finance has given a rise to

several new mathematical theories. The leading directions in

Mathematical Finance include portfolio theory, option-pricing

theory, and risk measurement.

The â€˜â€˜academicâ€™â€™ part of this book demonstrates that financial data

can be an area of exciting theoretical research, which might be of

interest to physicists regardless of their career motivation. This part

includes the Econophysics topics and the agent-based modeling of

1

2 Introduction

financial markets.1 Physicists use the term Econophysics to emphasize

the concepts of theoretical physics (e.g., scaling, fractals, and chaos)

that are applied to the analysis of economic and financial data. This

field was formed in the early 1990s, and it has been growing rapidly

ever since. Several books on Econophysics have been published to date

[5â€“11] as well as numerous articles in the scientific periodical journals

such as Physica A and Quantitative Finance.2 The agent-based model-

ing of financial markets was introduced by mathematically inclined

economists (see [12] for a review). Not surprisingly, physicists, being

accustomed to the modeling of â€˜â€˜anything,â€™â€™ have contributed into this

field, too [7, 10].

Although physicists are the primary audience for this book, two

other reader groups may also benefit from it. The first group includes

computer science and mathematics majors who are willing to work (or

have recently started a career) in the finance industry. In addition, this

book may be of interest to majors in economics and finance who are

curious about Econophysics and agent-based modeling of financial

markets. This book can be used for self-education or in an elective

course on Quantitative Finance for science and engineering majors.

The book is organized as follows. Chapter 2 describes the basics of

financial markets. Its topics include market price formation, returns

and dividends, and market efficiency. The next five chapters outline

the theoretical framework of Quantitative Finance: elements of math-

ematical statistics (Chapter 3), stochastic processes (Chapter 4), time

series analysis (Chapter 5), fractals (Chapter 6), and nonlinear dy-

namical systems (Chapter 7). Although all of these subjects have been

exhaustively covered in many excellent sources, we offer this material

for self-contained presentation.

In Chapter 3, the basic notions of mathematical statistics are

introduced and several popular probability distributions are listed.

In particular, the stable distributions that are used in analysis of

financial time series are discussed.

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