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Elsevier Academic Press
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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

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

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
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


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

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

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.

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

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
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, ˜˜operational
time™™ 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.


While price P is the major financial variable, its logarithm,
p ¼ 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 tÀ1. If no dividends are paid,
R(t) ¼ P(t)=P(t À 1) À 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
Rp (t) ¼ wip ¼ 1,
wip Rip (t), (2:2:2)
i¼1 i¼1

where Rip and wip are return and weight of the i-th portfolio asset,
respectively; i ¼ 1, . . . , N.
8 Financial Markets

The multi-period returns, or the compounded returns, define the
returns between the moments t and t À k þ 1. The compounded
return equals
R(t, k) ¼ [R(t) þ 1] [R(t À 1) þ 1] . . . [R(t À k þ 1) þ 1] þ 1
¼ P(t)=P(t À k) þ 1 (2:2:3)
The return averaged over k periods equals
" #1=k
R(t, k) ¼ (R(t À i) þ 1) À1 (2:2:4)

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:
R(t, k) % R(t, i) (2:2:5)
k i¼1

The continuously compounded return (or log return) is defined as:
r(t) ¼ log [R(t) þ 1] ¼ p(t) À p(t À 1) (2:2:6)
Calculation of the compounded log returns is reduced to simple
r(t, k) ¼ r(t) þ r(t À 1) þ . . . þ r(t À k þ 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.

If dividends D(t þ 1) are paid within the period [t, t þ 1], the simple
return (see 2.2.1) is modified to
R(t þ 1) ¼ [P(t þ 1) þ D(t þ 1) ]=P(t) À 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
Financial Markets

every time interval (say every month), the future value of this cash
after n periods is equal to
FV ¼ K(1 þ 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)
PV ¼ FV=(1 þ 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) ] ¼ R, one can rewrite (2.2.8) as
P(t) ¼ E[{P(t þ 1) þ D(t þ 1)}=(1 þ R)] (2:2:11)
If this recursion is repeated K times, one obtains
" #
D(t þ i)=(1 þ R)i þ E[P(t þ K)=(1 þ R)K ]
P(t) ¼ E (2:2:12)

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 þ K)=(1 þ R)K ] ¼ 0 (2:2:13)

Then the discounted-cash-flow model yields
" #
D(t þ i)=(1 þ R)i
PD (t) ¼ E (2:2:14)

Further simplification of the discounted-cash-flow model is based on
the assumption that the dividends grow linearly with rate G
E[D(t þ i) ] ¼ (1 þ G)i D(t) (2:2:15)
Then (2.2.14) reduces to
10 Financial Markets

PD (t) ¼ D(t) (2:2:16)
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) ¼ PD (t) þ B(t), B(t) ¼ E[B(t þ 1)=(1 þ 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 growth”the ˜˜bubble.™™ Yet, this
bubble is ˜˜rational™™ since it is based on rational expectations of future
returns. In the popular Blanchard-Watson model
1 þ R B(t) þ e(t þ 1) with probability p, 0 < p < 1
B(t þ 1) ¼ (2:2:18)
e(t þ 1) with probability 1 À p
where e(t) is an independent and identically distributed process (IID)2
with E[e(t) ] ¼ 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 ¼ K exp (rt), PV ¼ FV exp (Àrt) (2:2:19)
In the following chapters, both the discrete and the continuous pre-
sentations will be used.
Financial Markets

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 ˜˜best™™ price. In our example, it is a finite
number of Yens available at the exchange rate USD/JPY ¼ 120. As
soon as they all are taken, the exchange rate USD/JPY falls to the
equilibrium value 1 USD/JPY ¼ 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

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 company™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
Three forms of EMH are discerned in modern economic literature.
In the ˜˜weak™™ form of EMH, current prices reflect all information on
past prices. Then the technical analysis seems to be helpless. In the
˜˜strong™™ 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 ˜˜semi-
strong™™ 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 Bachelier™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 [9“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
Financial Markets

efficiency does not necessarily require prices to follow the random
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 1962“1986
[12]. Interestingly, similar analysis for the period of 1986“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 ˜˜fair
price™™ and ˜˜rational investors™™ do not stand criticism in the light of
the financial market booms and crashes. The ˜˜irrational exuberance™™
in 1999“2000 can hardly be attributed to rational behavior [10]. In
fact, empirical research in the new field ˜˜behavioral finance™™ 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 Keynes™ explanation that ˜˜animal spirits™™ 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 ˜˜What do we mean by saying markets
are efficient™™ of his book ˜˜A Random Walk down Wall Street™™ [9]:
˜˜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.™™
Thus, polemics on EMH changes the discussion from whether
prices follow the random walk to the practical ability to consistently
˜˜beat the market.™™
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 ˜˜Dogs of the Dow™™ (buying 10 stocks of the Dow
Jones Industrial Average with highest dividend yield), ˜˜January
Effect™™ (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
˜˜channel,™™ ˜˜head and shoulders,™™ 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., [17“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.

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
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 [9“11].6 On the
MBA level, much of the material pertinent to this chapter is given
in [3].

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 1996“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) ¼ PU (t) À 0:5 where PU is standard
uniform distribution (explain why the relation e(t) ¼ PU (t)
cannot be used). Use p ¼ 0:75 and R ¼ 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 ¼ 1.7705, EUR/USD ¼ 1.1914,
EUR/GBP ¼ 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.,
[1“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.

Consider the random variable (or variate) X. The probability dens-
ity function P(x) defines the probability to find X between a and b

b) ¼ P(x)dx
Pr(a X (3:1:1)

The probability density must be a non-negative function and must
satisfy the normalization condition


P(x)dx ¼ 1 (3:1:2)

where the interval [Xmin , Xmax ] is the range of all possible values of X.
In fact, the infinite limits [À1, 1] can always be used since P(x) may

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

b) ¼
Pr(X P(x)dx (3:1:3)

Obviously, probability satisfies the condition
Pr(X > b) ¼ 1 À 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)
m  E[X] ¼ xP(x)dx (3:1:5)

Median of X is the value, M, for which
Pr(X > M) ¼ Pr(X < M) ¼ 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 ˜˜lucky™™ ticket winning one million dollars and 999
˜˜losers.™™ 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
E[Xt jIt ] ¼ 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
Var[X]  s2 ¼ (x À m)2 P(x)dx (3:1:8)
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
mn  E[Xn ] ¼ 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 ¼ m2 À m2 .
Two other important parameters, skewness S and kurtosis K, are
related to the third and fourth moments, respectively,

S ¼ E[(x À m)3 ]=s3 , K ¼ E[(x À 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 ¼ K À 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)
b c

c) ¼
Pr(X b, Y h(x, y)dxdy (3:1:11)
À1 À1

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

h(x, y)dxdy ¼ 1 (3:1:12)
À1 À1
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) ¼
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 ¼ E[(x À mX )(y À mY )] ¼ E[xy] À mX mY (3:1:13)
Obviously, covariance reduces to variance if X ¼ Y: sXX ¼ 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) ¼ Cov(x:y)=(sX sY ) (3:1:14)
The values of the correlation coefficient are within the range [ À 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 ¼ E[(xi À mi )(xj À mj )] (3:1:15)

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 ¼ (3:2:1)
1=(b À a), a x b
The uniform distribution has the following mean and higher-order
mU ¼ 0, s2 U ¼ (b À a)2 =12, SU ¼ 0, KeU ¼ À6=5 (3:2:2)
The case with a ¼ 0 and b ¼ 1 is called the standard uniform distribu-
tion. Many computer languages and software packages have a library
function for calculating the standard uniform distribution.
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 ¼ 1 À p (so-called Ber-
noulli trials)
PB (n; N, p) ¼ CNn pn qNÀn ¼ CNn pn (1 À p)NÀn , CNn ¼ (3:2:3)
n!(N À n)!
The factor CNn is called the binomial coefficient. Mean and higher-
order moments for the binomial distribution are equal, respectively,
mB ¼ Np, s2 B ¼ Np(1 À p), SB ¼ (q À p)=sB , KeB ¼ (1 À 6pq)=sB 2
In the case of large N and large (N À n), the binomial distribution
approaches the form
PB (n) ¼ p¬¬¬¬¬¬ exp [À(x À mB )2 =2s2 B ], N ! 1, (N À n) ! 1 (3:2:5)
that coincides with the normal (or Gaussian) distribution (see 3.2.9). In
the case with p ( 1, the binomial distribution approaches the Poisson
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 ¼ pN
 n n  n NÀn
PP (n, N) ¼ 1À (3:2:6)
n!(N À n)! N N
As the number of trials N becomes very large (N ! 1), equation
(3.2.6) approaches the limit
PP (n) ¼ nn eÀn =n! (3:2:7)
Mean, variance, skewness, and excess kurtosis of the Poisson distri-
bution are equal, respectively,
mP ¼ s2 ¼ n, SP ¼ nÀ1=2 , KeP ¼ nÀ1 (3:2:8)

The normal (Gaussian) distribution has the form
PN (x) ¼ p¬¬¬¬¬¬ exp [À(x À m)2 =2s2 ] (3:2:9)
22 Probability Distributions

It is often denoted N(m, s). Skewness and excess kurtosis of the
normal distribution equals zero. The transform z ¼ (x À m)=s con-
verts the normal distribution into the standard normal distribution
PSN (z) ¼ p¬¬¬¬¬¬ exp [Àz2 =2] (3:2:10)
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)
p¬¬¬¬¬¬ exp (Àx2 =2)dx ¼ 0:5 erf(z= 2) (3:2:11)

Then the cumulative distribution function for the standard normal
distribution equals
PrSN (z) ¼ 0:5[1 þ 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 ¼ [À2 ln x1 )]1=2 cos (2px2 ), y2 ¼ [À2 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
mi , s2
mN ¼ ¼ (3:2:14)
i¼1 i, j¼1

The lognormal distribution is a distribution in which the logarithm of a
variate has the normal form
Probability Distributions

p¬¬¬¬¬¬ exp [À( ln x À m)2 =2s2 ]
PLN (x) ¼ (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 ¼ exp (m þ 0:5s2 ),
s2 ¼ [ exp (s2 ) À 1] exp (2m þ s2 ),

SLN ¼ [ exp (s2 ) À 1]1=2 [ exp (s2 ) þ 2],
KeLN ¼ exp (4s2 ) þ 2 exp (3s2 ) þ 3 exp (2s2 ) À 6 (3:2:16)
The Cauchy distribution (Lorentzian) is an example of the stable distri-
bution (see the next section). It has the form
PC (x) ¼ (3:2:17)
p[b2 þ (x À m)2 ]
The specific of the Cauchy distribution is that all its moments are
infinite. The case with b ¼ 1 and m ¼ 0 is named the standard Cauchy
PC (x) ¼ (3:2:18)
p[1 þ 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 1996“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) ¼ Pr(X x) for a random variable X exists,
then the cumulative distribution of the maximum values of
X, Hj (x) ¼ Pr(Xmax x) has the following asymptotic form
exp [À(1 þ j(x À mmax )=smax )À1=j ], j 6¼ 0,
Hj (x) ¼ (3:2:19)
exp [À exp (À(x À mmax )=smax )], j¼0

where 1 þ j(x À mmax )=smax > 0 in the case with j 6¼ 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





’6 ’4 ’2 0 2 4
Figure 3.1 The standardized distribution of the weekly returns of the S&P
500 SPDR (SPY) for 1996“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 ¼ 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.
Probability Distributions

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)
F(q) ¼ 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 À gjqja [1 À ibd tan (pa=2)], if a 6¼ 1
ln FL (q) ¼ (3:3:2)
imq À gjqj[1 þ 2ibd ln (jqj)=p)], if a ¼ 1

In (3.3.2), d ¼ q=jqj and the distribution parameters must satisfy the
following conditions
2, À 1
0<a 1, g > 0 (3:3:3)
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 ¼ 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 ¼ 2. For the
normal distribution
ln FN (q) ¼ imq À 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 ¼ s2 =2 (where s2 is variance). The Cauchy distribution is defined
26 Probability Distributions

with the parameters a ¼ 1 and b ¼ 0. Its characteristic function
ln FC (q) ¼ imq À 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) $ jxjÀ(1þa) (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-
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) ¼ Lf(ax) (3:3:7)
The solution to equation (3.3.7) is simply the power law
f(x) ¼ xn (3:3:8)
where n ¼ Àln (L)= ln (a). The power-law function f(x) (3.3.8) is scale-
free since the ratio f(ax)=f(x) ¼ 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
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
Probability Distributions

distribution yet yields finite variance is named truncated Levy flight.
This distribution is defined as [2]

jxj > ˜
fTL (x) ¼ (3:3:9)
CfL (x), À˜ x ˜
In (3.3.9), fL (x) is the Levy distribution ˜ is the cutoff length, and C is
the normalization constant. Sometimes the exponential cut-off is used
at large distances [3]
fTL (x) $ exp ( À ljxj), l > 0, jxj > ˜ (3:3:10)
Since fTL (x) has finite variance, it converges to the normal distribu-
tion according to the central limit theorem.

The Feller™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].

1. Calculate the correlation coefficients between the prices of
Microsoft (MSFT), Intel (INTC), and Wal-Mart (WMT). Use
monthly closing prices for the period 1994“2003. What do you
think of the opposite signs for some of these coefficients?
2. Familiarize yourself with Microsoft Excel™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) ¼ 0:95 using the NORMSINV function; (c)
calculate x such that Pr(Z ! x) ¼ 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 ¼ 0 and s ¼ 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 ¼ 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 þ bX2 $ CX þ 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 Langevin™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 Ito™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.

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

30 Stochastic Processes

f(x1 , t1 ; x2 , t2 ; . . . xk , tk jxkþ1 , tkþ1 ; xkþ2 , tkþ2 ; . . . ) ¼
f(x1 , t1 ; x2 , t2 ; . . . xkþ1 , tkþ1 ; . . . )=f(xkþ1 , tkþ1 ; xkþ2 , tkþ2 ; . . . ) (4:1:1)
In (4.1.1) and further in this section, t1 > t2 > . . . tk > tkþ1 > . . .
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
f(x1 , t1 ; x2 , t2 ; . . . ) ¼ f(x1 , t1 )f(x2 , t2 ) . . .  f(xi , ti ) (4:1:2)

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 jxkþ1 , tkþ1 ; xkþ2 , tkþ2 ; . . . ) ¼
f(x1 , t1 ; x2 , t2 ; . . . xk , tk jxkþ1 , tkþ1 ) (4:1:3)
This means that evolution of the system is determined with the initial
condition (i.e., with the value xkþ1 at time tkþ1 ). It follows for the
Markov process that
f(x1 , t1 ; x2 , t2 ; x3 , t3 ) ¼ 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
f(x1 , t1 jx3 , t3 ) ¼ f(x1 , t1 ; x2 , t2 jx3 , t3 )dx2
¼ 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 ) ¼ 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
f(x1 , t1 jx3 , t3 ) ¼ 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
Stochastic Processes

1 @2
@ @
f(x, tjx0 , t0 ) ¼À [A(x, t)f(x, tjx0 , t0 )] þ [D(x, t)f(x, tjx0 , t0 )]þ
2 @x2
@t @x
[R(xjz, t)f(z, tjx0 , t0 ) ÀR(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
A(x, t) ¼ lim (z À x)f(z, t þ Dtjx, t)dz (4:1:9)
Dt!0 Dt
(z À x)2 f(z, t þ Dtjx, t)dz
D(x, t) ¼ 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
R(xjz, t) ¼ lim f(x, t þ 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 ) ¼ À [A(x, t)f(x, tjx0 , t0 )]
@t @x
1 @2
þ [D(x, t)f(x, tjx0 , t0 )]
2 @x2
This equation with A(x, t) ¼ 0 and D ¼ const is reduced to the
diffusion equation that describes the Brownian motion
D @2
f(x, tjx0 , t0 ) ¼ f(x, tjx0 , t0 ) (4:1:13)
2 @x2
Equation (4.1.13) has the analytic solution in the Gaussian form
f(x, tjx0 , t0 ) ¼ [2pD(t À t0 )]À1=2 exp [À(x À x0 )2 =2D(t À t0 )] (4:1:14)
Mean and variance for the distribution (4.1.14) equal
E[x(t)] ¼ x0 , Var[x(t)] ¼ E[(x(t) À x0 )2 ] ¼ s2 ¼ D(t À t0 ) (4:1:15)
The diffusion equation (4.1.13) with D ¼ 1 describes the standard
Wiener process for which
E[(x(t) À x0 )2 ] ¼ t À 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.

In mathematical statistics, the notion of the Brownian motion is
used for describing the generic stochastic process. Yet, this term
referred originally to Brown™s observation of random motion of
pollen in water. Random particle motion in fluid can be described
using different theoretical approaches. Einstein™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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