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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

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