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QUANTITATIVE FINANCE

FOR PHYSICISTS:

AN INTRODUCTION

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QUANTITATIVE FINANCE

FOR PHYSICISTS:

AN INTRODUCTION

ANATOLY B. SCHMIDT

AMSTERDAM â€¢ BOSTON â€¢ HEIDELBERG â€¢ LONDON

NEW YORK â€¢ OXFORD â€¢ PARIS â€¢ SAN DIEGO

SAN FRANCISCO â€¢ SINGAPORE â€¢ SYDNEY â€¢ TOKYO

Elsevier Academic Press

30 Corporate Drive, Suite 400, Burlington, MA 01803, USA

525 B Street, Suite 1900, San Diego, California 92101-4495, USA

84 Theobaldâ€™s Road, London WC1X 8RR, UK

This book is printed on acid-free paper.

Copyright # 2005, Elsevier Inc. All rights reserved.

No part of this publication may be reproduced or transmitted in any form or by any

means, electronic or mechanical, including photocopy, recording, or any information

storage and retrieval system, without permission in writing from the publisher.

Permissions may be sought directly from Elsevierâ€™s Science & Technology Rights

Department in Oxford, UK: phone: (Ã¾44) 1865 843830, fax: (Ã¾44) 1865 853333,

e-mail: permissions@elsevier.com.uk. You may also complete your request on-line

via the Elsevier homepage (http://elsevier.com), by selecting â€˜â€˜Customer Supportâ€™â€™

and then â€˜â€˜Obtaining Permissions.â€™â€™

Library of Congress Cataloging-in-Publication Data

Application submitted.

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN: 0-12-088464-X

For all information on all Elsevier Academic Press publications visit our Web site at

www.books.elsevier.com

Printed in the United States of America

04 05 06 07 08 09 9 8765 4321

Table of Contents

Chapter 1

Introduction 1

Chapter 2

Financial Markets 5

Chapter 3

Probability Distributions 17

Chapter 4

Stochastic Processes 29

Chapter 5

Time Series Analysis 43

Chapter 6

Fractals 59

Chapter 7

Nonlinear Dynamical Systems 69

Chapter 8

Scaling in Financial Time Series 87

v

vi Contents

Chapter 9

Option Pricing 93

Chapter 10

Portfolio Management 111

Chapter 11

Market Risk Measurement 121

Chapter 12

Agent-Based Modeling of Financial Markets 129

Comments 145

References 149

Answers to Exercises 159

Index 161

Detailed Table of Contents

1. Introduction 1

2. Financial Markets 5

2.1 Market Price Formation 5

2.2 Returns and Dividends 7

2.2.1 Simple and Compounded Returns 7

2.2.2 Dividend Effects 8

2.3 Market Efficiency 11

2.3.1 Arbitrage 11

2.3.2 Efficient Market Hypothesis 12

2.4 Pathways for Further Reading 14

2.5 Exercises 15

3. Probability Distributions 17

3.1 Basic Definitions 17

3.2 Important Distributions 20

3.3 Stable Distributions and Scale Invariance 25

3.4 References for Further Reading 27

3.5 Exercises 27

4. Stochastic Processes 29

4.1 Markov Processes 29

4.2 Brownian Motion 32

4.3 Stochastic Differential Equation 35

4.4 Stochastic Integral 36

4.5 Martingales 39

4.6 References for Further Reading 41

4.7 Exercises 41

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.

Chapter 4 begins with an introduction to the Wiener process, which

is the basis for description of the stochastic financial processes. Three

methodological approaches are outlined: one is rooted in the generic

Markov process, the second one is based on the Langevin equation,

and the last one stems from the discrete random walk. Then the basics

of stochastic calculus are described. They include the Itoâ€™s lemma and

3

Introduction

the stochastic integral in both the Ito and the Stratonovich forms.

Finally, the notion of martingale is introduced.

Chapter 5 begins with the univariate autoregressive and moving

average models, the classical tools of the time series analysis. Then the

approaches to accounting for trends and seasonality effects are dis-

cussed. Furthermore, processes with non-stationary variance (condi-

tional heteroskedasticity) are described. Finally, the specifics of the

multivariate time series are outlined.

In Chapter 6, the basic definitions of the fractal theory are dis-

cussed. The concept of multifractals, which has been receiving a lot of

attention in recent financial time series research, is also introduced.

Chapter 7 describes the elements of nonlinear dynamics that are

important for agent-based modeling of financial markets. To illustrate

the major concepts in this field, two classical models are discussed: the

discrete logistic map and the continuous Lorenz model. The main

pathways to chaos and the chaos measures are also outlined.

Those readers who do not need to refresh their knowledge of the

mathematical concepts may skip Chapters 3 through 7.3

The other five chapters are devoted to financial applications. In

Chapter 8, the scaling properties of the financial time series are

discussed. The main subject here is the power laws manifesting in

the distributions of returns. Alternative approaches in describing the

scaling properties of the financial time series including the multifrac-

tal models are also outlined.

The next three chapters, Chapters 9 through 11, relate specifically

to Mathematical Finance. Chapter 9 is devoted to the option pricing.

It starts with the general properties of stock options, and then the

option pricing theory is discussed using two approaches: the method

of the binomial trees and the classical Black-Scholes theory.

Chapter 10 is devoted to the portfolio theory. Its basics include the

capital asset pricing model and the arbitrage pricing theory. Finally,

several arbitrage trading strategies are listed. Risk measurement is the

subject of Chapter 11. It starts with the concept of value at risk, which

is widely used in risk management. Then the notion of coherent risk

measure is introduced and one such popular measure, the expected

tail losses, is described.

Finally, Chapter 12 is devoted to agent-based modeling of financial

markets. Two elaborate models that illustrate two different

4 Introduction

approaches to defining the price dynamics are described. The first one

is based on the supply-demand equilibrium, and the other approach

employs an empirical relation between price change and excess

demand. Discussion of the model derived in terms of observable

variables concludes this chapter.

The bibliography provides the reader with references for further

reading rather than with a comprehensive chronological review. The

reference list is generally confined with recent monographs and

reviews. However, some original work that either has particularly

influenced the author or seems to expand the field in promising

ways is also included.

In every chapter, exercises with varying complexity are provided.

Some of these exercises simply help the readers to get their hands on

the financial market data available on the Internet and to manipulate

the data using Microsoft Excel software.4 Other exercises provide a

means of testing the understanding of the bookâ€™s theoretical material.

More challenging exercises, which may require consulting with ad-

vanced textbooks or implementation of complicated algorithms, are

denoted with an asterisk. The exercises denoted with two asterisks

offer discussions of recent research reports. The latter exercises may

be used for seminar presentations or for course work.

A few words about notations. Scalar values are denoted with the

regular font (e.g., X) while vectors and matrices are denoted with

boldface letters (e.g., X). The matrix transposes are denoted with

primes (e.g., X0 ) and the matrix determinants are denoted with vertical

bars (e.g., jXj). The following notations are used interchangeably:

X(tk ) X(t) and X(tkÃ€1 ) X(t Ã€ 1). E[X] is used to denote the ex-

pectation of the variable X.

The views expressed in this book may not reflect the views of my

former and current employers. While conducting the Econophysics

research and writing this book, I enjoyed support from Blake LeBaron,

Thomas Lux, Sorin Solomon, and Eugene Stanley. I am also indebted

to anonymous reviewers for attentive analysis of the bookâ€™s drafts.

Needless to say, I am solely responsible for all possible errors present in

this edition. I will greatly appreciate all comments about this book;

please send them to a_b_schmidt@hotmail.com.

Alec Schmidt

Cedar Knolls, NJ, June 2004

Chapter 2

Financial Markets

This chapter begins with a description of market price formation. The

notion of return that is widely used for analysis of the investment

efficiency is introduced in Section 2.2. Then the dividend effects on

return and the present-value pricing model are described. The next big

topic is market efficiency (Section 2.3). First, the notion of arbitrage is

defined. Then the Efficient Market Hypothesis, both the theory and

its critique, are discussed. The pathways for further reading in Section

2.4 conclude the chapter.

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