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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.

2.1 MARKET PRICE FORMATION

Millions of different financial assets (stocks, bonds, currencies,

options, and others) are traded around the world. Some financial

markets are organized in exchanges or bourses (e.g., New York

Stock Exchange (NYSE)). In other, so-called over-the-counter

(OTC) markets, participants operate directly via telecommunication

systems. Market data are collected and distributed by markets them-

selves and by financial data services such as Bloomberg and Reuters.

Modern electronic networks facilitate access to huge volumes of

market data in real time.

Market prices are formed with the trader orders (quotes) submitted

on the bid (buy) and ask (sell) sides of the market. Usually, there is a

5

6 Financial Markets

spread between the best (highest) bid and the best (lowest) ask prices,

which provides profits for the market makers. The prices seen on the

tickers of TV networks and on the Internet are usually the transaction

prices that correspond to the best prices. The very presence of trans-

actions implies that some traders submit market orders; they buy at

current best ask prices and sell at current best bid prices. The trans-

action prices represent the mere tip of an iceberg beneath which prices

of the limit orders reside. Indeed, traders may submit the sell orders at

prices higher than the best bid and the buy orders at prices lower than

the best ask. The limit orders reflect the trader expectations of future

price movement. There are also stop orders designated to limit pos-

sible losses. For an asset holder, the stop order implies selling assets if

the price falls to a predetermined value.

Holding assets, particularly holding derivatives (see Section 9.1), is

called long position. The opposite of long buying is short selling, which

means selling assets that the trader does not own after borrowing

them from the broker. Short selling makes sense if the price is

expected to fall. When the price does drop, the short seller buys the

same number of assets that were borrowed and returns them to the

broker. Short sellers may also use stop orders to limit their losses in

case the price grows rather than falls. Namely, they may submit the

stop order for triggering a buy when the price reaches a predeter-

mined value.

Limit orders and stop orders form the market microstructure: the

volume-price distributions on the bid and ask sides of the market. The

concept market liquidity is used to describe price sensitivity to market

orders. For instance, low liquidity means that the number of securities

available at the best price is smaller than a typical market order. In this

case, a new market order is executed within a range of available prices

rather than at a single best price. As a result, the best price changes its

value. Securities with very low liquidity may have no transactions and

few (if any) quotes for some time (in particular, the small-cap stocks off

regular trading hours). Market microstructure information usually is

not publicly available. However, the market microstructure may be

partly revealed in the price reaction to big block trades.

Any event that affects the market microstructure (such as submis-

sion, execution, or withdrawal of an order) is called a tick. Ticks are

recorded along with the time they are submitted (so-called tick-by-tick

7

Financial Markets

data). Generally, tick-by-tick data are not regularly spaced in time,

which leads to additional challenges for high-frequency data analysis

[1, 2]. Current research of financial data is overwhelmingly conducted

on the homogeneous grids that are defined with filtering and aver-

aging tick-by-tick data.

Another problem that complicates analysis of long financial time

series is seasonal patterns. Business hours, holidays, and even daylight

saving time shifts affect market activity. Introducing the dummy

variables into time series models is a general method to account for

seasonal effects (see Section 5.2). In another approach, â€˜â€˜operational

timeâ€™â€™ is employed to describe the non-homogeneity of business activ-

ity [2]. Non-trading hours, including weekends and holidays, may be

cut off from operational time grids.

2.2 RETURNS AND DIVIDENDS

2.2.1 SIMPLE COMPOUNDED RETURNS

AND

While price P is the major financial variable, its logarithm,

p Â¼ log (P) is often used in quantitative analysis. The primary reason

for using log prices is that simulation of a random price innovation

can move price into the negative region, which does not make sense.

In the mean time, negative logarithm of price is perfectly acceptable.

Another important financial variable is the single-period return (or

simple return) R(t) that defines the return between two subsequent

moments t and tÃ€1. If no dividends are paid,

R(t) Â¼ P(t)=P(t Ã€ 1) Ã€ 1 (2:2:1)

Return is used as a measure of investment efficiency.1 Its advantage is

that some statistical properties, such as stationarity, may be more

applicable to returns rather than to prices [3]. The simple return of a

portfolio, Rp (t), equals the weighed sum of returns of the portfolio

assets

X X

N N

Rp (t) Â¼ wip Â¼ 1,

wip Rip (t), (2:2:2)

iÂ¼1 iÂ¼1

where Rip and wip are return and weight of the i-th portfolio asset,

respectively; i Â¼ 1, . . . , N.

8 Financial Markets

The multi-period returns, or the compounded returns, define the

returns between the moments t and t Ã€ k Ã¾ 1. The compounded

return equals

R(t, k) Â¼ [R(t) Ã¾ 1] [R(t Ã€ 1) Ã¾ 1] . . . [R(t Ã€ k Ã¾ 1) Ã¾ 1] Ã¾ 1

Â¼ P(t)=P(t Ã€ k) Ã¾ 1 (2:2:3)

The return averaged over k periods equals

" #1=k

Y

kÃ€1

Ë‡

R(t, k) Â¼ (R(t Ã€ i) Ã¾ 1) Ã€1 (2:2:4)

iÂ¼0

If the simple returns are small, the right-hand side of (2.2.4) can be

reduced to the first term of its Taylor expansion:

1XkÃ€1

Ë‡

R(t, k) % R(t, i) (2:2:5)

k iÂ¼1

The continuously compounded return (or log return) is defined as:

r(t) Â¼ log [R(t) Ã¾ 1] Â¼ p(t) Ã€ p(t Ã€ 1) (2:2:6)

Calculation of the compounded log returns is reduced to simple

summation:

r(t, k) Â¼ r(t) Ã¾ r(t Ã€ 1) Ã¾ . . . Ã¾ r(t Ã€ k Ã¾ 1) (2:2:7)

However, the weighing rule (2.2.2) is not applicable to the log returns

since log of sum is not equal to sum of logs.

2.2.2 DIVIDEND EFFECTS

If dividends D(t Ã¾ 1) are paid within the period [t, t Ã¾ 1], the simple

return (see 2.2.1) is modified to

R(t Ã¾ 1) Â¼ [P(t Ã¾ 1) Ã¾ D(t Ã¾ 1) ]=P(t) Ã€ 1 (2:2:8)

The compounded returns and the log returns are calculated in the

same way as in the case with no dividends.

Dividends play a critical role in the discounted-cash-flow (or pre-

sent-value) pricing model. Before describing this model, let us intro-

duce the notion of present value. Consider the amount of cash K

invested in a risk-free asset with the interest rate r. If interest is paid

9

Financial Markets

every time interval (say every month), the future value of this cash

after n periods is equal to

FV Â¼ K(1 Ã¾ r)n (2:2:9)

Suppose we are interested in finding out what amount of money will

yield given future value after n intervals. This amount (present value)

equals

PV Â¼ FV=(1 Ã¾ r)n (2:2:10)

Calculating the present value via the future value is called discounting.

The notions of the present value and the future value determine the

payoff of so-called zero-coupon bonds. These bonds sold at their

present value promise a single payment of their future value at ma-

turity date.

The discounted-cash-flow model determines the stock price via its

future cash flow. For the simple model with the constant return

E[R(t) ] Â¼ R, one can rewrite (2.2.8) as

P(t) Â¼ E[{P(t Ã¾ 1) Ã¾ D(t Ã¾ 1)}=(1 Ã¾ R)] (2:2:11)

If this recursion is repeated K times, one obtains

" #

X

K

D(t Ã¾ i)=(1 Ã¾ R)i Ã¾ E[P(t Ã¾ K)=(1 Ã¾ R)K ]

P(t) Â¼ E (2:2:12)

iÂ¼1

In the limit K ! 1, the second term in the right-hand side of (2.2.12)

can be neglected if

lim E[P(t Ã¾ K)=(1 Ã¾ R)K ] Â¼ 0 (2:2:13)

K!1

Then the discounted-cash-flow model yields

" #

X1

D(t Ã¾ i)=(1 Ã¾ R)i

PD (t) Â¼ E (2:2:14)

iÂ¼1

Further simplification of the discounted-cash-flow model is based on

the assumption that the dividends grow linearly with rate G

E[D(t Ã¾ i) ] Â¼ (1 Ã¾ G)i D(t) (2:2:15)

Then (2.2.14) reduces to

10 Financial Markets

1Ã¾G

PD (t) Â¼ D(t) (2:2:16)

RÃ€G

Obviously, equation (2.2.16) makes sense only for R > G. The value

of R that may attract investors is called the required rate of return.

This value can be treated as the sum of the risk-free rate and the asset

risk premium. While the assumption of linear dividend growth is

unrealistic, equation (2.2.16) shows the high sensitivity of price to

change in the discount rate R when R is close to G (see Exercise 2). A

detailed analysis of the discounted-cash-flow model is given in [3].

If the condition (2.2.13) does not hold, the solution to (2.2.12) can

be presented in the form

P(t) Â¼ PD (t) Ã¾ B(t), B(t) Â¼ E[B(t Ã¾ 1)=(1 Ã¾ R) ] (2:2:17)

The term PD (t) has the sense of the fundamental value while the

function B(t) is often called the rational bubble. This term implies

that B(t) may lead to unbounded growthâ€”the â€˜â€˜bubble.â€™â€™ Yet, this

bubble is â€˜â€˜rationalâ€™â€™ since it is based on rational expectations of future

returns. In the popular Blanchard-Watson model

(

1 Ã¾ R B(t) Ã¾ e(t Ã¾ 1) with probability p, 0 < p < 1

p

B(t Ã¾ 1) Â¼ (2:2:18)

e(t Ã¾ 1) with probability 1 Ã€ p

where e(t) is an independent and identically distributed process (IID)2

with E[e(t) ] Â¼ 0. The specific of this model is that it describes period-

ically collapsing bubbles (see [4] for the recent research).

So far, the discrete presentation of financial data was discussed.

Clearly, market events have a discrete nature and price variations

cannot be smaller than certain values. Yet, the continuum presenta-

tion of financial processes is often employed [5]. This means that the

time interval between two consecutive market events compared to the

time range of interest is so small that it can be considered an infini-

tesimal difference. Often, the price discreteness can also be neglected

since the markets allow for quoting prices with very small differen-

tials. The future value and the present value within the continuous

presentation equal, respectively

FV Â¼ K exp (rt), PV Â¼ FV exp (Ã€rt) (2:2:19)

In the following chapters, both the discrete and the continuous pre-

sentations will be used.

11

Financial Markets

2.3 MARKET EFFICIENCY

2.3.1 ARBITRAGE

Asset prices generally obey the Law of One Price, which says that

prices of equivalent assets in competitive markets must be the same

[6]. This implies that if a security replicates a package of other

securities, the price of this security and the price of the package it

replicates must be equal. It is expected also that the asset price must

be the same worldwide, provided that it is expressed in the same

currency and that the transportation and transaction costs can be

neglected. Violation of the Law of One Price leads to arbitrage, which

means buying an asset and immediate selling it (usually in another

market) with profit and without risk. One widely publicized example

of arbitrage is the notable differences in prices of prescription drugs in

the USA, Europe, and Canada. Another typical example is the so-

called triangle foreign exchange arbitrage. Consider a situation in

which a trader can exchange one American dollar (USD) for one

Euro (EUR) or for 120 Yen (JPY). In addition, a trader can exchange

one EUR for 119 JPY. Hence, in terms of the exchange rates, 1 USD/

JPY > 1 EUR/JPY * 1 USD/EUR.3 Obviously, the trader who

operates, say 100000 USD, can make a profit by buying 12000000

JPY, then selling them for 12000000/119 % 100840 EUR, and then

buying back 100840 USD. If the transaction costs are neglected, this

operation will bring profit of about 840 USD.

The arbitrage with prescription drugs persists due to unresolved

legal problems. However, generally the arbitrage opportunities do not

exist for long. The triangle arbitrage may appear from time to time.

Foreign exchange traders make a living, in part, by finding such

opportunities. They rush to exchange USD for JPY. It is important

to remember that, as it was noted in Section 2.1, there is only a finite

number of assets at the â€˜â€˜bestâ€™â€™ price. In our example, it is a finite

number of Yens available at the exchange rate USD/JPY Â¼ 120. As

soon as they all are taken, the exchange rate USD/JPY falls to the

equilibrium value 1 USD/JPY Â¼ 1 EUR/JPY * 1 USD/EUR, and the

arbitrage vanishes. In general, when arbitrageurs take profits, they act

in a way that eliminates arbitrage opportunities.

12 Financial Markets

2.3.2 EFFICIENT MARKET HYPOTHESIS (EMH)

Efficient market is closely related to (the absence of) arbitrage. It

might be defined as simply an ideal market without arbitrage, but there

is much more to it than that. Let us first ask what actually causes price

to change. The share price of a company may change due to its new

earnings report, due to new prognosis of the company performance, or

due to a new outlook for the industry trend. Macroeconomic and

political events, or simply gossip about a companyâ€™s management,

can also affect the stock price. All these events imply that new infor-

mation becomes available to markets. The Efficient Market Theory

states that financial markets are efficient because they instantly reflect

all new relevant information in asset prices. Efficient Market Hypoth-

esis (EMH) proposes the way to evaluate market efficiency. For

example, an investor in an efficient market should not expect earnings

above the market return while using technical analysis or fundamental

analysis.4

Three forms of EMH are discerned in modern economic literature.

In the â€˜â€˜weakâ€™â€™ form of EMH, current prices reflect all information on

past prices. Then the technical analysis seems to be helpless. In the

â€˜â€˜strongâ€™â€™ form, prices instantly reflect not only public but also private

(insider) information. This implies that the fundamental analysis

(which is what the investment analysts do) is not useful either. The

compromise between the strong and weak forms yields the â€˜â€˜semi-

strongâ€™â€™ form of EMH according to which prices reflect all publicly

available information and the investment analysts play important role

in defining fair prices.

Two notions are important for EMH. The first notion is the

random walk, which will be formally defined in Section 5.1. In short,

market prices follow the random walk if their variations are random

and independent. Another notion is rational investors who immedi-

ately incorporate new information into fair prices. The evolution of

the EMH paradigm, starting with Bachelierâ€™s pioneering work on

random price behavior back in 1900 to the formal definition of

EMH by Fama in 1965 to the rigorous statistical analysis by Lo

and MacKinlay in the late 1980s, is well publicized [9â€“13]. If prices

follow the random walk, this is the sufficient condition for EMH.

However, as we shall discuss further, the pragmatic notion of market

13

Financial Markets

efficiency does not necessarily require prices to follow the random

walk.

Criticism of EMH has been conducted along two avenues. First, the

thorough theoretical analysis has resulted in rejection of the random

walk hypothesis for the weekly U.S. market returns during 1962â€“1986

[12]. Interestingly, similar analysis for the period of 1986â€“1996 shows

that the returns conform more closely to the random walk. As the

authors of this research, Lo and MacKinlay, suggest, one possible

reason for this trend is that several investment firms had implemented

statistical arbitrage trading strategies5 based on the market inefficien-

cies that were revealed in early research. Execution of these strategies

could possibly eliminate some of the arbitrage opportunities.

Another reason for questioning EMH is that the notions of â€˜â€˜fair

priceâ€™â€™ and â€˜â€˜rational investorsâ€™â€™ do not stand criticism in the light of

the financial market booms and crashes. The â€˜â€˜irrational exuberanceâ€™â€™

in 1999â€“2000 can hardly be attributed to rational behavior [10]. In

fact, empirical research in the new field â€˜â€˜behavioral financeâ€™â€™ demon-

strates that investor behavior often differs from rationality [14, 15].

Overconfidence, indecisiveness, overreaction, and a willingness to

gamble are among the psychological traits that do not fit rational

behavior. A widely popularized example of irrational human behav-

ior was described by Kahneman and Tversky [16]. While conducting

experiments with volunteers, they asked participants to make choices

in two different situations. First, participants with $1000 were given a

choice between: (a) gambling with a 50% chance of gaining $1000 and

a 50% chance of gaining nothing, or (b) a sure gain of $500. In the

second situation, participants with $2000 were given a choice be-

tween: (a) a 50% chance of losing $1000 and a 50% of losing nothing,

and (b) a sure loss of $500. Thus, the option (b) in both situations

guaranteed a gain of $1500. Yet, the majority of participants chose

option (b) in the first situation and option (a) in the second one.

Hence, participants preferred sure yet smaller gains but were willing

to gamble in order to avoid sure loss.

Perhaps Keynesâ€™ explanation that â€˜â€˜animal spiritsâ€™â€™ govern investor

behavior is an exaggeration. Yet investors cannot be reduced to

completely rational machines either. Moreover, actions of different

investors, while seemingly rational, may significantly vary. In part,

this may be caused by different perceptions of market events and

14 Financial Markets

trends (heterogeneous beliefs). In addition, investors may have differ-

ent resources for acquiring and processing new information. As a

result, the notion of so-called bounded rationality has become popular

in modern economic literature (see also Section 12.2).

Still the advocates of EMH do not give up. Malkiel offers the

following argument in the section â€˜â€˜What do we mean by saying markets

are efficientâ€™â€™ of his book â€˜â€˜A Random Walk down Wall Streetâ€™â€™ [9]:

â€˜â€˜No one person or institution has yet to provide a long-term,

consistent record of finding risk-adjusted individual stock

trading opportunities, particularly if they pay taxes and

incur transactions costs.â€™â€™

Thus, polemics on EMH changes the discussion from whether

prices follow the random walk to the practical ability to consistently

â€˜â€˜beat the market.â€™â€™

Whatever experts say, the search of ideas yielding excess returns

never ends. In terms of the quantification level, three main directions

in the investment strategies may be discerned. First, there are qualita-

tive receipts such as â€˜â€˜Dogs of the Dowâ€™â€™ (buying 10 stocks of the Dow

Jones Industrial Average with highest dividend yield), â€˜â€˜January

Effectâ€™â€™ (stock returns are particularly high during the first two Janu-

ary weeks), and others. These ideas are arguably not a reliable profit

source [9].

Then there are relatively simple patterns of technical analysis, such as

â€˜â€˜channel,â€™â€™ â€˜â€˜head and shoulders,â€™â€™ and so on (see, e.g., [7]). There has

been ongoing academic discussion on whether technical analysis is able

to yield persistent excess returns (see, e.g., [17â€“19] and references

therein). Finally, there are trading strategies based on sophisticated

statistical arbitrage. While several trading firms that employ these strat-

egies have proven to be profitable in some periods, little is known about

persistent efficiency of their proprietary strategies. Recent trends indi-

cate that some statistical arbitrage opportunities may be fading [20].

Nevertheless, one may expect that modern, extremely volatile markets

will always provide new occasions for aggressive arbitrageurs.

2.4 PATHWAYS FOR FURTHER READING

In this chapter, a few abstract statistical notions such as IID and

random walk were mentioned. In the next five chapters, we take a short

15

Financial Markets

tour of the mathematical concepts that are needed for acquaintance

with quantitative finance. Those readers who feel confident in their

mathematical background may jump ahead to Chapter 8.

Regarding further reading for this chapter, general introduction to

finance can be found in [6]. The history of development and valid-

ation of EMH is described in several popular books [9â€“11].6 On the

MBA level, much of the material pertinent to this chapter is given

in [3].

EXERCISES

1. Familiarize yourself with the financial market data available on

the Internet (e.g., http://www.finance.yahoo.com). Download the

weekly closing prices of the exchange-traded fund SPDR that

replicates the S&P 500 index (ticker SPY) for 1996â€“2003. Cal-

culate simple weekly returns for this data sample (we shall use

these data for other exercises).

2. Calculate the present value of SPY for 2004 if the asset risk

premium is equal to (a) 3% and (b) 4%. The SPY dividends in

2003 were $1.63. Assume the dividend growth rate of 5% (see

Exercise 5.3 for a more accurate estimate). Assume the risk-free

rate of 3%. What risk premium was priced in SPY in the end of

2004 according to the discounted-cash-flow theory?

3. Simulate the rational bubble using the Blanchard-Watson

model (2.2.18). Define e(t) Â¼ PU (t) Ã€ 0:5 where PU is standard

uniform distribution (explain why the relation e(t) Â¼ PU (t)

cannot be used). Use p Â¼ 0:75 and R Â¼ 0:1 as the initial values

for studying the model sensitivity to the input parameters.

4. Is there an arbitrage opportunity for the following set of the

exchange rates: GBP/USD Â¼ 1.7705, EUR/USD Â¼ 1.1914,

EUR/GBP Â¼ 0.6694?

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Chapter 3

Probability Distributions

This chapter begins with the basic notions of mathematical statistics

that form the framework for analysis of financial data (see, e.g.,

[1â€“3]). In Section 3.2, a number of distributions widely used in statis-

tical data analysis are listed. The stable distributions that have become

popular in Econophysics research are discussed in Section 3.3.

3.1 BASIC DEFINITIONS

Consider the random variable (or variate) X. The probability dens-

ity function P(x) defines the probability to find X between a and b

Ã°

b

b) Â¼ P(x)dx

Pr(a X (3:1:1)

a

The probability density must be a non-negative function and must

satisfy the normalization condition

XÃ°

max

P(x)dx Â¼ 1 (3:1:2)

Xmin

where the interval [Xmin , Xmax ] is the range of all possible values of X.

In fact, the infinite limits [Ã€1, 1] can always be used since P(x) may

17

18 Probability Distributions

be set to zero outside the interval [Xmin , Xmax ]. As a rule, the infinite

integration limits are further omitted.

Another way of describing random variable is to use the cumulative

distribution function

Ã°

b

b) Â¼

Pr(X P(x)dx (3:1:3)

Ã€1

Obviously, probability satisfies the condition

Pr(X > b) Â¼ 1 Ã€ Pr(X b) (3:1:4)

Two characteristics are used to describe probable values of random

variable X: mean (or expectation) and median. Mean of X is the

average of all possible values of X that are weighed with the prob-

ability density P(x)

Ã°

m E[X] Â¼ xP(x)dx (3:1:5)

Median of X is the value, M, for which

Pr(X > M) Â¼ Pr(X < M) Â¼ 0:5 (3:1:6)

Median is the preferable characteristic of the most probable value for

strongly skewed data samples. Consider a sample of lottery tickets

that has one â€˜â€˜luckyâ€™â€™ ticket winning one million dollars and 999

â€˜â€˜losers.â€™â€™ The mean win in this sample is $1000, which does not

realistically describe the lottery outcome. The median zero value is a

much more relevant characteristic in this case.

The expectation of a random variable calculated using some avail-

able information It (that may change with time t) is called conditional

expectation. The conditional probability density is denoted by P(xjIt ).

Conditional expectation equals

Ã°

E[Xt jIt ] Â¼ xP(xjIt )dx (3:1:7)

Variance, Var, and the standard deviation, s, are the conventional

estimates of the deviations from the mean values of X

Ã°

Var[X] s2 Â¼ (x Ã€ m)2 P(x)dx (3:1:8)

19

Probability Distributions

In financial literature, the standard deviation of price is used to

characterize the price volatility.

The higher-order moments of the probability distributions are

defined as

Ã°

mn E[Xn ] Â¼ xn P(x)dx (3:1:9)

According to this definition, mean is the first moment (m m1 ), and

variance can be expressed via the first two moments, s2 Â¼ m2 Ã€ m2 .

Two other important parameters, skewness S and kurtosis K, are

related to the third and fourth moments, respectively,

S Â¼ E[(x Ã€ m)3 ]=s3 , K Â¼ E[(x Ã€ m)4 ]=s4 (3:1:10)

Both parameters, S and K, are dimensionless. Zero skewness implies

that the distribution is symmetrical around its mean value. The posi-

tive and negative values of skewness indicate long positive tails and

long negative tails, respectively. Kurtosis characterizes the distribu-

tion peakedness. Kurtosis of the normal distribution equals three.

The excess kurtosis, Ke Â¼ K Ã€ 3, is often used as a measure of devi-

ation from the normal distribution. In particular, positive excess

kurtosis (or leptokurtosis) indicates more frequent medium and large

deviations from the mean value than is typical for the normal distri-

bution. Leptokurtosis leads to a flatter central part and to so-called

fat tails in the distribution. Negative excess kurtosis indicates frequent

small deviations from the mean value. In this case, the distribution

sharpens around its mean value while the distribution tails decay

faster than the tails of the normal distribution.

The joint distribution of two random variables X and Y is the

generalization of the cumulative distribution (see 3.1.3)

Ã°Ã°

b c

c) Â¼

Pr(X b, Y h(x, y)dxdy (3:1:11)

Ã€1 Ã€1

In (3.1.11), h(x, y) is the joint density that satisfies the normalization

condition

Ã°Ã°

11

h(x, y)dxdy Â¼ 1 (3:1:12)

Ã€1 Ã€1

20 Probability Distributions

Two random variables are independent if their joint density function

is simply the product of the univariate density functions: h(x, y) Â¼

f (x)g(y). Covariance between two variates provides a measure of their

simultaneous change. Consider two variates, X and Y, that have the

means mX and mY , respectively. Their covariance equals

Cov(x, y) sXY Â¼ E[(x Ã€ mX )(y Ã€ mY )] Â¼ E[xy] Ã€ mX mY (3:1:13)

Obviously, covariance reduces to variance if X Â¼ Y: sXX Â¼ sX 2 .

Positive covariance between two variates implies that these variates

tend to change simultaneously in the same direction rather than in

opposite directions. Conversely, negative covariance between two

variates implies that when one variate grows, the second one tends

to fall and vice versa. Another popular measure of simultaneous

change is the correlation coefficient

Corr(x, y) Â¼ Cov(x:y)=(sX sY ) (3:1:14)

The values of the correlation coefficient are within the range [ Ã€ 1, 1].

In the general case with N variates X1 , . . . , XN (where N > 2),

correlations among variates are described with the covariance matrix.

Its elements equal

Cov(xi , xj ) sij Â¼ E[(xi Ã€ mi )(xj Ã€ mj )] (3:1:15)

3.2 IMPORTANT DISTRIBUTIONS

There are several important probability distributions used in quan-

titative finance. The uniform distribution has a constant value within

the given interval [a, b] and equals zero outside this interval

0, x < a and x > b

PU Â¼ (3:2:1)

1=(b Ã€ a), a x b

The uniform distribution has the following mean and higher-order

moments

mU Â¼ 0, s2 U Â¼ (b Ã€ a)2 =12, SU Â¼ 0, KeU Â¼ Ã€6=5 (3:2:2)

The case with a Â¼ 0 and b Â¼ 1 is called the standard uniform distribu-

tion. Many computer languages and software packages have a library

function for calculating the standard uniform distribution.

21

Probability Distributions

The binomial distribution is a discrete distribution of obtaining n

successes out of N trials where the result of each trial is true with

probability p and is false with probability q Â¼ 1 Ã€ p (so-called Ber-

noulli trials)

N!

PB (n; N, p) Â¼ CNn pn qNÃ€n Â¼ CNn pn (1 Ã€ p)NÃ€n , CNn Â¼ (3:2:3)

n!(N Ã€ n)!

The factor CNn is called the binomial coefficient. Mean and higher-

order moments for the binomial distribution are equal, respectively,

mB Â¼ Np, s2 B Â¼ Np(1 Ã€ p), SB Â¼ (q Ã€ p)=sB , KeB Â¼ (1 Ã€ 6pq)=sB 2

(3:2:4)

In the case of large N and large (N Ã€ n), the binomial distribution

approaches the form

1

PB (n) Â¼ pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ exp [Ã€(x Ã€ mB )2 =2s2 B ], N ! 1, (N Ã€ n) ! 1 (3:2:5)

2psB

that coincides with the normal (or Gaussian) distribution (see 3.2.9). In

the case with p ( 1, the binomial distribution approaches the Poisson

distribution.

The Poisson distribution describes the probability of n successes in

N trials assuming that the fraction of successes n is proportional to

the number of trials: n Â¼ pN

n n n NÃ€n

N!

PP (n, N) Â¼ 1Ã€ (3:2:6)

n!(N Ã€ n)! N N

As the number of trials N becomes very large (N ! 1), equation

(3.2.6) approaches the limit

PP (n) Â¼ nn eÃ€n =n! (3:2:7)

Mean, variance, skewness, and excess kurtosis of the Poisson distri-

bution are equal, respectively,

mP Â¼ s2 Â¼ n, SP Â¼ nÃ€1=2 , KeP Â¼ nÃ€1 (3:2:8)

P

The normal (Gaussian) distribution has the form

1

PN (x) Â¼ pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ exp [Ã€(x Ã€ m)2 =2s2 ] (3:2:9)

2ps

22 Probability Distributions

It is often denoted N(m, s). Skewness and excess kurtosis of the

normal distribution equals zero. The transform z Â¼ (x Ã€ m)=s con-

verts the normal distribution into the standard normal distribution

1

PSN (z) Â¼ pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ exp [Ã€z2 =2] (3:2:10)

2p

Note that the probability for the standard normal variate to assume

the value in the interval [0, z] can be used as the definition of the error

function erf(x)

Ã°

z

pï¬ƒï¬ƒï¬ƒ

1

pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ exp (Ã€x2 =2)dx Â¼ 0:5 erf(z= 2) (3:2:11)

2p

0

Then the cumulative distribution function for the standard normal

distribution equals

pï¬ƒï¬ƒï¬ƒ

PrSN (z) Â¼ 0:5[1 Ã¾ erf(z= 2)] (3:2:12)

According to the central limit theorem, the probability density distri-

bution for a sum of N independent random variables with finite

variances and finite means approaches the normal distribution as N

grows to infinity. Due to exponential decay of the normal distribu-

tion, large deviations from its mean rarely appear. The normal distri-

bution plays an extremely important role in all kinds of applications.

The Box-Miller method is often used for modeling the normal distri-

bution with given uniform distribution [4]. Namely, if two numbers

x1 and x2 are drawn from the standard uniform distribution, then

y1 and y2 are the standard normal variates

y1 Â¼ [Ã€2 ln x1 )]1=2 cos (2px2 ), y2 Â¼ [Ã€2 ln x1 )]1=2 sin (2px2 ) (3:2:13)

Mean and variance of the multivariate normal distribution with N

variates can be easily calculated via the univariate means mi and

covariances sij

X X

N N

mi , s2

mN Â¼ Â¼ (3:2:14)

sij

N

iÂ¼1 i, jÂ¼1

The lognormal distribution is a distribution in which the logarithm of a

variate has the normal form

23

Probability Distributions

1

pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ exp [Ã€( ln x Ã€ m)2 =2s2 ]

PLN (x) Â¼ (3:2:15)

xs 2p

Mean, variance, skewness, and excess kurtosis of the lognormal dis-

tribution can be expressed in terms of the parameters s and m

mLN Â¼ exp (m Ã¾ 0:5s2 ),

s2 Â¼ [ exp (s2 ) Ã€ 1] exp (2m Ã¾ s2 ),

LN

SLN Â¼ [ exp (s2 ) Ã€ 1]1=2 [ exp (s2 ) Ã¾ 2],

KeLN Â¼ exp (4s2 ) Ã¾ 2 exp (3s2 ) Ã¾ 3 exp (2s2 ) Ã€ 6 (3:2:16)

The Cauchy distribution (Lorentzian) is an example of the stable distri-

bution (see the next section). It has the form

b

PC (x) Â¼ (3:2:17)

p[b2 Ã¾ (x Ã€ m)2 ]

The specific of the Cauchy distribution is that all its moments are

infinite. The case with b Â¼ 1 and m Â¼ 0 is named the standard Cauchy

distribution

1

PC (x) Â¼ (3:2:18)

p[1 Ã¾ x2 ]

Figure 3.1 depicts the distribution of the weekly returns of the ex-

change-traded fund SPDR that replicates the S&P 500 index (ticker

SPY) for 1996â€“2003 in comparison with standard normal distribution

and the standard Cauchy distributions (see Exercise 3).

The extreme value distributions can be introduced with the Fisher-

Tippett theorem. According to this theorem, if the cumulative distri-

bution function F(x) Â¼ Pr(X x) for a random variable X exists,

then the cumulative distribution of the maximum values of

X, Hj (x) Â¼ Pr(Xmax x) has the following asymptotic form

(

exp [Ã€(1 Ã¾ j(x Ã€ mmax )=smax )Ã€1=j ], j 6Â¼ 0,

Hj (x) Â¼ (3:2:19)

exp [Ã€ exp (Ã€(x Ã€ mmax )=smax )], jÂ¼0

where 1 Ã¾ j(x Ã€ mmax )=smax > 0 in the case with j 6Â¼ 0: In (3.2.19),

mmax and smax are the location and scale parameters, respectively;

j is the shape parameter and 1=j is named the tail index. The

24 Probability Distributions

0.5

0.4

0.3

0.2

SPY

Normal

Cauchy

0.1

0

âˆ’6 âˆ’4 âˆ’2 0 2 4

Figure 3.1 The standardized distribution of the weekly returns of the S&P

500 SPDR (SPY) for 1996â€“2003 in comparison with the standard normal

and the standard Cauchy distributions.

Fisher-Tippett theorem does not define the values of the parameters

mmax and smax . However, special methods have been developed for

their estimation [5].

It is said that the cumulative distribution function F(x) is in the

domain of attraction of Hj (x). The tail behavior of the distribution

F(x) defines the shape parameter. The Gumbel distribution corres-

ponds to the case with j Â¼ 0. Distributions with thin tails, such as

normal, lognormal, and exponential distributions, have the Gumbel

domain of attraction. The case with j > 0 is named the Frechet

distribution. Domain of the Frechet attraction corresponds to distri-

butions with fat tails, such as the Cauchy distribution and the Pareto

distribution (see the next Section). Finally, the case with j < 0 defines

the Weibull distribution. This type of distributions (e.g., the uniform

distribution) has a finite tail.

25

Probability Distributions

3.3 STABLE DISTRIBUTIONS AND SCALE

INVARIANCE

The principal property of stable distribution is that the sum of

variates has the same distribution shape as that of addends (see,

e.g., [6] for details). Both the Cauchy distribution and the normal

distribution are stable. This means, in particular, that the sum of

two normal distributions with the same mean and variance is also the

normal distribution (see Exercise 2). The general definition for

the stable distributions was given by Levy. Therefore, the stable

distributions are also called the Levy distributions.

Consider the Fourier transform F(q) of the probability distribution

function f(x)

Ã°

F(q) Â¼ f(x)eiqx dx (3:3:1)

The function F(q) is also called the characteristic function of the

stochastic process. It can be shown that the logarithm of the charac-

teristic function for the Levy distribution has the following form

(

imq Ã€ gjqja [1 Ã€ ibd tan (pa=2)], if a 6Â¼ 1

ln FL (q) Â¼ (3:3:2)

imq Ã€ gjqj[1 Ã¾ 2ibd ln (jqj)=p)], if a Â¼ 1

In (3.3.2), d Â¼ q=jqj and the distribution parameters must satisfy the

following conditions

2, Ã€ 1

0<a 1, g > 0 (3:3:3)

b

The parameter m corresponds to the mean of the stable distribution

and can be any real number. The parameter a characterizes the

distribution peakedness. If a Â¼ 2, the distribution is normal. The

parameter b characterizes skewness of the distribution. Note that

skewness of the normal distribution equals zero and the parameter

b does not affect the characteristic function with a Â¼ 2. For the

normal distribution

ln FN (q) Â¼ imq Ã€ gq2 (3:3:4)

The non-negative parameter g is the scale factor that characterizes the

spread of the distribution. In the case of the normal distribution,

g Â¼ s2 =2 (where s2 is variance). The Cauchy distribution is defined

26 Probability Distributions

with the parameters a Â¼ 1 and b Â¼ 0. Its characteristic function

equals

ln FC (q) Â¼ imq Ã€ gjqj (3:3:5)

The important feature of the stable distributions with a < 2 is that

they exhibit the power-law decay at large absolute values of the

argument x

fL (jxj) $ jxjÃ€(1Ã¾a) (3:3:6)

The distributions with the power-law asymptotes are also named the

Pareto distributions. Many processes exhibit power-law asymptotic

behavior. Hence, there has been persistent interest to the stable distri-

butions.

The power-law distributions describe the scale-free processes. Scale

invariance of a distribution means that it has a similar shape on

different scales of independent variables. Namely, function f(x) is

scale-invariant to transformation x ! ax if there is such parameter

L that

f(x) Â¼ Lf(ax) (3:3:7)

The solution to equation (3.3.7) is simply the power law

f(x) Â¼ xn (3:3:8)

where n Â¼ Ã€ln (L)= ln (a). The power-law function f(x) (3.3.8) is scale-

free since the ratio f(ax)=f(x) Â¼ L does not depend on x. Note that the

parameter a is closely related to the fractal dimension of the function

f(x). The fractal theory will be discussed in Chapter 6.

Unfortunately, the moments of stable processes E[xn ] with power-

law asymptotes (i.e., when a < 2) diverge for n ! a. As a result, the

mean of a stable process is infinite when a 1. In addition, variance

of a stable process is infinite when a < 2. Therefore, the normal

distribution is the only stable distribution with finite mean and finite

variance.

The stable distributions have very helpful features for data analysis

such as flexible description of peakedness and skewness. However, as it

was mentioned previously, the usage of the stable distributions in

financial applications is often restricted because of their infinite vari-

ance at a < 2. The compromise that retains flexibility of the Levy

27

Probability Distributions

distribution yet yields finite variance is named truncated Levy flight.

This distribution is defined as [2]

jxj > â€˜

0,

fTL (x) Â¼ (3:3:9)

CfL (x), Ã€â€˜ x â€˜

In (3.3.9), fL (x) is the Levy distribution â€˜ is the cutoff length, and C is

the normalization constant. Sometimes the exponential cut-off is used

at large distances [3]

fTL (x) $ exp ( Ã€ ljxj), l > 0, jxj > â€˜ (3:3:10)

Since fTL (x) has finite variance, it converges to the normal distribu-

tion according to the central limit theorem.

3.4 REFERENCES FOR FURTHER READING

The Fellerâ€™s textbook is the classical reference to the probability

theory [1]. The concept of scaling in financial data has been advocated

by Mandelbrot since the 1960s (see the collection of his work in [7]).

This problem is widely discussed in the current Econophysics litera-

ture [2, 3, 8].

3.5 EXERCISES

1. Calculate the correlation coefficients between the prices of

Microsoft (MSFT), Intel (INTC), and Wal-Mart (WMT). Use

monthly closing prices for the period 1994â€“2003. What do you

think of the opposite signs for some of these coefficients?

2. Familiarize yourself with Microsoft Excelâ€™s statistical tools. As-

suming that Z is the standard normal distribution: (a) calculate

Pr(1 Z 3) using the NORMSDIST function; (b) calculate x

such that Pr(Z x) Â¼ 0:95 using the NORMSINV function; (c)

calculate x such that Pr(Z ! x) Â¼ 0:15; (d) generate 100 random

numbers from the standard normal distribution using Tools/

Data Analysis/Random Number Generation. Calculate the

sample mean and standard variance. How do they differ from

the theoretical values of m Â¼ 0 and s Â¼ 1, respectively? (e) Do

the same for the standard uniform distribution as in (d).

28 Probability Distributions

(f) Generate 100 normally distributed random numbers x using

the function x Â¼ NORMSINV(z) where z is taken from a sample

of the standard uniform distribution. Explain why it is possible.

Calculate the sample mean and the standard deviation. How do

they differ from the theoretical values of m and s, respectively?

3. Calculate mean, standard deviation, excess kurtosis, and skew

for the SPY data sample from Exercise 2.1. Draw the distribu-

tion function of this data set in comparison with the standard

normal distribution and the standard Cauchy distribution.

Compare results with Figure 3.1.

Hint: (1) Normalize returns by subtracting their mean and divid-

ing the results by the standard deviation. (2) Calculate the histo-

gram using the Histogram tool of the Data Analysis menu. (3)

Divide the histogram frequencies with the product of their sum and

the bin size (explain why it is necessary).

4. Let X1 and X2 be two independent copies of the normal distri-

bution X $ N(m, s2 ). Since X is stable, aX1 Ã¾ bX2 $ CX Ã¾ D.

Calculate C and D via given m, s, a, and b.

Chapter 4

Stochastic Processes

Financial variables, such as prices and returns, are random time-

dependent variables. The notion of stochastic process is used to de-

scribe their behavior. Specifically, the Wiener process (or the Brownian

motion) plays the central role in mathematical finance. Section 4.1

begins with the generic path: Markov process ! Chapmen-Kolmo-

gorov equation ! Fokker-Planck equation ! Wiener process. This

methodology is supplemented with two other approaches in Section

4.2. Namely, the Brownian motion is derived using the Langevinâ€™s

equation and the discrete random walk. Then the basics of stochastic

calculus are described. In particular, the stochastic differential equa-

tion is defined using the Itoâ€™s lemma (Section 4.3), and the stochastic

integral is given in both the Ito and the Stratonovich forms

(Section 4.4). Finally, the notion of martingale, which is widely popu-

lar in mathematical finance, is introduced in Section 4.5.

4.1 MARKOV PROCESSES

Consider a process X(t) for which the values x1 , x2 , . . . are meas-

ured at times t1 , t2 , . . . Here, one-dimensional variable x is used

for notational simplicity, though extension to multidimensional

systems is trivial. It is assumed that the joint probability density

f(x1 , t1 ; x2 , t2 ; . . . ) exists and defines the system completely. The con-

ditional probability density function is defined as

29

30 Stochastic Processes

f(x1 , t1 ; x2 , t2 ; . . . xk , tk jxkÃ¾1 , tkÃ¾1 ; xkÃ¾2 , tkÃ¾2 ; . . . ) Â¼

f(x1 , t1 ; x2 , t2 ; . . . xkÃ¾1 , tkÃ¾1 ; . . . )=f(xkÃ¾1 , tkÃ¾1 ; xkÃ¾2 , tkÃ¾2 ; . . . ) (4:1:1)

In (4.1.1) and further in this section, t1 > t2 > . . . tk > tkÃ¾1 > . . .

unless stated otherwise. In the simplest stochastic process, the present

has no dependence on the past. The probability density function for

such a process equals

Y

f(x1 , t1 ; x2 , t2 ; . . . ) Â¼ f(x1 , t1 )f(x2 , t2 ) . . . f(xi , ti ) (4:1:2)

i

The Markov process represents the next level of complexity, which

embraces an extremely wide class of phenomena. In this process, the

future depends on the present but not on the past. Hence, its condi-

tional probability density function equals

f(x1 , t1 ; x2 , t2 ; . . . xk , tk jxkÃ¾1 , tkÃ¾1 ; xkÃ¾2 , tkÃ¾2 ; . . . ) Â¼

f(x1 , t1 ; x2 , t2 ; . . . xk , tk jxkÃ¾1 , tkÃ¾1 ) (4:1:3)

This means that evolution of the system is determined with the initial

condition (i.e., with the value xkÃ¾1 at time tkÃ¾1 ). It follows for the

Markov process that

f(x1 , t1 ; x2 , t2 ; x3 , t3 ) Â¼ f(x1 , t1 jx2 , t2 )f(x2 , t2 jx3 , t3 ) (4:1:4)

Using the definition of the conditional probability density, one can

introduce the general equation

Ã°

f(x1 , t1 jx3 , t3 ) Â¼ f(x1 , t1 ; x2 , t2 jx3 , t3 )dx2

Ã°

Â¼ f(x1 , t1 jx2 , t2 ; x3 , t3 )f(x2 , t2 jx3 , t3 )dx2 (4:1:5)

For the Markov process,

f(x1 , t1 jx2 , t2 ; x3 , t3 ) Â¼ f(x1 , t1 jx2 , t2 ), (4:1:6)

Then the substitution of equation (4.1.6) into equation (4.1.5) leads to

the Chapmen-Kolmogorov equation

Ã°

f(x1 , t1 jx3 , t3 ) Â¼ f(x1 , t1 jx2 , t2 )f(x2 , t2 jx3 , t3 )dx2 (4:1:7)

This equation can be used as the starting point for deriving the

Fokker-Planck equation (see, e.g., [1] for details). First, equation

(4.1.7) is transformed into the differential equation

31

Stochastic Processes

1 @2

@ @

f(x, tjx0 , t0 ) Â¼Ã€ [A(x, t)f(x, tjx0 , t0 )] Ã¾ [D(x, t)f(x, tjx0 , t0 )]Ã¾

2 @x2

@t @x

Ã°

[R(xjz, t)f(z, tjx0 , t0 ) Ã€R(zjx, t)f(x, tjx0 , t0 )]dz (4:1:8)

In (4.1.8), the drift coefficient A(x, t) and the diffusion coefficient

D(x, t) are equal

Ã°

1

A(x, t) Â¼ lim (z Ã€ x)f(z, t Ã¾ Dtjx, t)dz (4:1:9)

Dt!0 Dt

Ã°

1

(z Ã€ x)2 f(z, t Ã¾ Dtjx, t)dz

D(x, t) Â¼ lim (4:1:10)

Dt!0 Dt

The integral in the right-hand side of the Chapmen-Kolmogorov

equation (4.1.8) is determined with the function

1

R(xjz, t) Â¼ lim f(x, t Ã¾ Dtjz, t) (4:1:11)

Dt!0 Dt

It describes possible discontinuous jumps of the random variable. Neg-

lecting this term in equation (4.1.8) yields the Fokker-Planck equation

@ @

f(x, tjx0 , t0 ) Â¼ Ã€ [A(x, t)f(x, tjx0 , t0 )]

@t @x

(4:1:12)

1 @2

Ã¾ [D(x, t)f(x, tjx0 , t0 )]

2 @x2

This equation with A(x, t) Â¼ 0 and D Â¼ const is reduced to the

diffusion equation that describes the Brownian motion

D @2

@

f(x, tjx0 , t0 ) Â¼ f(x, tjx0 , t0 ) (4:1:13)

2 @x2

@t

Equation (4.1.13) has the analytic solution in the Gaussian form

f(x, tjx0 , t0 ) Â¼ [2pD(t Ã€ t0 )]Ã€1=2 exp [Ã€(x Ã€ x0 )2 =2D(t Ã€ t0 )] (4:1:14)

Mean and variance for the distribution (4.1.14) equal

E[x(t)] Â¼ x0 , Var[x(t)] Â¼ E[(x(t) Ã€ x0 )2 ] Â¼ s2 Â¼ D(t Ã€ t0 ) (4:1:15)

The diffusion equation (4.1.13) with D Â¼ 1 describes the standard

Wiener process for which

E[(x(t) Ã€ x0 )2 ] Â¼ t Ã€ t0 (4:1:16)

32 Stochastic Processes

The notions of the generic Wiener process and the Brownian motion

are sometimes used interchangeably, though there are some fine

differences in their definitions [2, 3]. I shall denote the Wiener process

with W(t) and reserve this term for the standard version (4.1.16), as it

is often done in the literature.

The Brownian motion is the classical topic of statistical physics.

Different approaches for introducing this process are described in the

next section.

4.2 BROWNIAN MOTION

In mathematical statistics, the notion of the Brownian motion is

used for describing the generic stochastic process. Yet, this term

referred originally to Brownâ€™s observation of random motion of

pollen in water. Random particle motion in fluid can be described

using different theoretical approaches. Einsteinâ€™s original theory of

the Brownian motion implicitly employs both the Chapman-Kolmo-

gorov equation and the Fokker-Planck equation [1]. However, choos-

ing either one of these theories as the starting point can lead to the

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