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QUANTITATIVE FINANCE
FOR PHYSICISTS:
AN INTRODUCTION
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QUANTITATIVE FINANCE
FOR PHYSICISTS:
AN INTRODUCTION



ANATOLY B. SCHMIDT




AMSTERDAM • BOSTON • HEIDELBERG • LONDON
NEW YORK • OXFORD • PARIS • SAN DIEGO
SAN FRANCISCO • SINGAPORE • SYDNEY • 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 Theobald™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 Elsevier™s Science & Technology Rights
Department in Oxford, UK: phone: (þ44) 1865 843830, fax: (þ44) 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 ˜˜Customer Support™™
and then ˜˜Obtaining Permissions.™™

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 ˜˜applied™™ part and the
˜˜academic™™ 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 ˜˜academic™™ 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
[5“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 ˜˜anything,™™ 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 Ito™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 book™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(tkÀ1 )  X(t À 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 book™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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