Mathematical Methods for Finance E-Book

Sergio M. Focardi

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Frank J. Fabozzi Series
• Sprache: Englisch

Seitenzahl: 405

Veröffentlichungsjahr: 2013

Beschreibung

The mathematical and statistical tools needed in the rapidly growing quantitative finance field With the rapid growth in quantitative finance, practitioners must achieve a high level of proficiency in math and statistics. Mathematical Methods and Statistical Tools for Finance, part of the Frank J. Fabozzi Series, has been created with this in mind. Designed to provide the tools needed to apply finance theory to real world financial markets, this book offers a wealth of insights and guidance in practical applications. It contains applications that are broader in scope from what is covered in a typical book on mathematical techniques. Most books focus almost exclusively on derivatives pricing, the applications in this book cover not only derivatives and asset pricing but also risk management--including credit risk management--and portfolio management. * Includes an overview of the essential math and statistical skills required to succeed in quantitative finance * Offers the basic mathematical concepts that apply to the field of quantitative finance, from sets and distances to functions and variables * The book also includes information on calculus, matrix algebra, differential equations, stochastic integrals, and much more * Written by Sergio Focardi, one of the world's leading authors in high-level finance Drawing on the author's perspectives as a practitioner and academic, each chapter of this book offers a solid foundation in the mathematical tools and techniques need to succeed in today's dynamic world of finance.

Leseprobe

Contents

Cover

Series Page

Title Page

Copyright Page

Dedication

Preface

About the Authors

Chapter 1: Basic Concepts

INTRODUCTION

SETS AND SET OPERATIONS

DISTANCES AND QUANTITIES

FUNCTIONS

VARIABLES

KEY POINTS

Chapter 2: Differential Calculus

INTRODUCTION

LIMITS

CONTINUITY

TOTAL VARIATION

THE NOTION OF DIFFERENTIATION

COMMONLY USED RULES FOR COMPUTING DERIVATIVES

HIGHER-ORDER DERIVATIVES

TAYLOR SERIES EXPANSION

CALCULUS IN MORE THAN ONE VARIABLE

KEY POINTS

Chapter 3: Integral Calculus

INTRODUCTION

RIEMANN INTEGRALS

LEBESGUE-STIELTJES INTEGRALS

INDEFINITE AND IMPROPER INTEGRALS

THE FUNDAMENTAL THEOREM OF CALCULUS

INTEGRAL TRANSFORMS

CALCULUS IN MORE THAN ONE VARIABLE

KEY POINTS

Chapter 4: Matrix Algebra

INTRODUCTION

VECTORS AND MATRICES DEFINED

SQUARE MATRICES

DETERMINANTS

SYSTEMS OF LINEAR EQUATIONS

LINEAR INDEPENDENCE AND RANK

HANKEL MATRIX

VECTOR AND MATRIX OPERATIONS

FINANCE APPLICATION

EIGENVALUES AND EIGENVECTORS

DIAGONALIZATION AND SIMILARITY

SINGULAR VALUE DECOMPOSITION

KEY POINTS

Chapter 5: Probability

INTRODUCTION

REPRESENTING UNCERTAINTY WITH MATHEMATICS

PROBABILITY IN A NUTSHELL

OUTCOMES AND EVENTS

PROBABILITY

MEASURE

RANDOM VARIABLES

INTEGRALS

DISTRIBUTIONS AND DISTRIBUTION FUNCTIONS

RANDOM VECTORS

STOCHASTIC PROCESSES

PROBABILISTIC REPRESENTATION OF FINANCIAL MARKETS

INFORMATION STRUCTURES

FILTRATION

KEY POINTS

Chapter 6: Probability

INTRODUCTION

CONDITIONAL PROBABILITY AND CONDITIONAL EXPECTATION

MOMENTS AND CORRELATION

COPULA FUNCTIONS

SEQUENCES OF RANDOM VARIABLES

INDEPENDENT AND IDENTICALLY DISTRIBUTED SEQUENCES

SUM OF VARIABLES

GAUSSIAN VARIABLES

APPPROXIMATING THE TAILS OF A PROBABILITY DISTRIBUTION: CORNISH-FISHER EXPANSION AND HERMITE POLYNOMIALS

THE REGRESSION FUNCTION

FAT TAILS AND STABLE LAWS

KEY POINTS

Chapter 7: Optimization

INTRODUCTION

MAXIMA AND MINIMA

LAGRANGE MULTIPLIERS

NUMERICAL ALGORITHMS

CALCULUS OF VARIATIONS AND OPTIMAL CONTROL THEORY

STOCHASTIC PROGRAMMING

APPLICATION TO BOND PORTFOLIO: LIABILITY-FUNDING STRATEGIES

KEY POINTS

Chapter 8: Difference Equations

INTRODUCTION

THE LAG OPERATOR L

HOMOGENEOUS DIFFERENCE EQUATIONS

RECURSIVE CALCULATION OF VALUES OF DIFFERENCE EQUATIONS

NONHOMOGENEOUS DIFFERENCE EQUATIONS

SYSTEMS OF LINEAR DIFFERENCE EQUATIONS

SYSTEMS OF HOMOGENEOUS LINEAR DIFFERENCE EQUATIONS

KEY POINTS

Chapter 9: Differential Equations

INTRODUCTION

DIFFERENTIAL EQUATIONS DEFINED

ORDINARY DIFFERENTIAL EQUATIONS

SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

CLOSED-FORM SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS

NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS

NONLINEAR DYNAMICS AND CHAOS

PARTIAL DIFFERENTIAL EQUATIONS

KEY POINTS

Chapter 10: Stochastic Integrals

INTRODUCTION

THE INTUITION BEHIND STOCHASTIC INTEGRALS

BROWNIAN MOTION DEFINED

PROPERTIES OF BROWNIAN MOTION

STOCHASTIC INTEGRALS DEFINED

SOME PROPERTIES OF ITÔ STOCHASTIC INTEGRALS

MARTINGALE MEASURES AND THE GIRSANOV THEOREM

KEY POINTS

Chapter 11: Stochastic Differential Equations

INTRODUCTION

THE INTUITION BEHIND STOCHASTIC DIFFERENTIAL EQUATIONS

ITÔ PROCESSES

STOCHASTIC DIFFERENTIAL EQUATIONS

GENERALIZATION TO SEVERAL DIMENSIONS

SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATIONS

DERIVATION OF ITÔ’S LEMMA

DERIVATION OF THE BLACK-SCHOLES OPTION PRICING FORMULA

KEY POINTS

Index

The Frank J. Fabozzi Series

Fixed Income Securities, Second Edition by Frank J. Fabozzi

Focus on Value: A Corporate and Investor Guide to Wealth Creation by James L. Grant and James A. Abate

Handbook of Global Fixed Income Calculations by Dragomir Krgin

Managing a Corporate Bond Portfolio by Leland E. Crabbe and Frank J. Fabozzi

Real Options and Option-Embedded Securities by William T. Moore

Capital Budgeting: Theory and Practice by Pamela P. Peterson and Frank J. Fabozzi

The Exchange-Traded Funds Manual by Gary L. Gastineau

Professional Perspectives on Fixed Income Portfolio Management, Volume 3 edited by Frank J. Fabozzi

Investing in Emerging Fixed Income Markets edited by Frank J. Fabozzi and Efstathia Pilarinu

Handbook of Alternative Assets by Mark J. P. Anson

The Global Money Markets by Frank J. Fabozzi, Steven V. Mann, and Moorad Choudhry

The Handbook of Financial Instruments edited by Frank J. Fabozzi

Interest Rate, Term Structure, and Valuation Modeling edited by Frank J. Fabozzi

Investment Performance Measurement by Bruce J. Feibel

The Handbook of Equity Style Management edited by T. Daniel Coggin and Frank J. Fabozzi

The Theory and Practice of Investment Management edited by Frank J. Fabozzi and Harry M. Markowitz

Foundations of Economic Value Added, Second Edition by James L. Grant

Financial Management and Analysis, Second Edition by Frank J. Fabozzi and Pamela P. Peterson

Measuring and Controlling Interest Rate and Credit Risk, Second Edition by Frank J. Fabozzi, Steven V. Mann, and Moorad Choudhry

Professional Perspectives on Fixed Income Portfolio Management, Volume 4 edited by Frank J. Fabozzi

The Handbook of European Fixed Income Securities edited by Frank J. Fabozzi and Moorad Choudhry

The Handbook of European Structured Financial Products edited by Frank J. Fabozzi and Moorad Choudhry

The Mathematics of Financial Modeling and Investment Management by Sergio M. Focardi and Frank J. Fabozzi

Short Selling: Strategies, Risks, and Rewards edited by Frank J. Fabozzi

The Real Estate Investment Handbook by G. Timothy Haight and Daniel Singer

Market Neutral Strategies edited by Bruce I. Jacobs and Kenneth N. Levy

Securities Finance: Securities Lending and Repurchase Agreements edited by Frank J. Fabozzi and Steven V. Mann

Fat-Tailed and Skewed Asset Return Distributions by Svetlozar T. Rachev, Christian Menn, and Frank J. Fabozzi

Financial Modeling of the Equity Market: From CAPM to Cointegration by Frank J. Fabozzi, Sergio M. Focardi, and Petter N. Kolm

Advanced Bond Portfolio Management: Best Practices in Modeling and Strategies edited by Frank J. Fabozzi, Lionel Martellini, and Philippe Priaulet

Analysis of Financial Statements, Second Edition by Pamela P. Peterson and Frank J. Fabozzi

Collateralized Debt Obligations: Structures and Analysis, Second Edition by Douglas J. Lucas, Laurie S. Goodman, and Frank J. Fabozzi

Handbook of Alternative Assets, Second Edition by Mark J. P. Anson

Introduction to Structured Finance by Frank J. Fabozzi, Henry A. Davis, and Moorad Choudhry

Financial Econometrics by Svetlozar T. Rachev, Stefan Mittnik, Frank J. Fabozzi, Sergio M. Focardi, and Teo Jasic

Developments in Collateralized Debt Obligations: New Products and Insights by Douglas J. Lucas, Laurie S. Goodman, Frank J. Fabozzi, and Rebecca J. Manning

Robust Portfolio Optimization and Management by Frank J. Fabozzi, Peter N. Kolm, Dessislava A. Pachamanova, and Sergio M. Focardi

Advanced Stochastic Models, Risk Assessment, and Portfolio Optimizations by Svetlozar T. Rachev, Stogan V. Stoyanov, and Frank J. Fabozzi

How to Select Investment Managers and Evaluate Performance by G. Timothy Haight, Stephen O. Morrell, and Glenn E. Ross

Bayesian Methods in Finance by Svetlozar T. Rachev, John S. J. Hsu, Biliana S. Bagasheva, and Frank J. Fabozzi

The Handbook of Municipal Bonds edited by Sylvan G. Feldstein and Frank J. Fabozzi

Subprime Mortgage Credit Derivatives by Laurie S. Goodman, Shumin Li, Douglas J. Lucas, Thomas A Zimmerman, and Frank J. Fabozzi

Introduction to Securitization by Frank J. Fabozzi and Vinod Kothari

Structured Products and Related Credit Derivatives edited by Brian P. Lancaster, Glenn M. Schultz, and Frank J. Fabozzi

Handbook of Finance: Volume I: Financial Markets and Instruments edited by Frank J. Fabozzi

Handbook of Finance: Volume II: Financial Management and Asset Management edited by Frank J. Fabozzi

Handbook of Finance: Volume III: Valuation, Financial Modeling, and Quantitative Tools edited by Frank J. Fabozzi

Finance: Capital Markets, Financial Management, and Investment Management by Frank J. Fabozzi and Pamela Peterson-Drake

Active Private Equity Real Estate Strategy edited by David J. Lynn

Foundations and Applications of the Time Value of Money by Pamela Peterson-Drake and Frank J. Fabozzi

Leveraged Finance: Concepts, Methods, and Trading of High-Yield Bonds, Loans, and Derivatives by Stephen Antczak, Douglas Lucas, and Frank J. Fabozzi

Modern Financial Systems: Theory and Applications by Edwin Neave

Institutional Investment Management: Equity and Bond Portfolio Strategies and Applications by Frank J. Fabozzi

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ISBN 978-1-118-31263-6 (Hardcover) ISBN 978-1-118-42008-9 (ePDF) ISBN 978-1-118-42149-9 (ePub)

To the memory of my parents SMF

To my wife, Donna, and my children, Patricia, Karly, and Francesco FJF

To my wife, Mehtap, and my son, Kaan TGB

Preface

Since the pioneering work of Harry Markowitz in the 1950s, mathematical tools drawing from the fields of standard and stochastic calculus, set theory, probability theory, stochastic processes, matrix algebra, optimization theory, and differential equations have increasingly made their way into finance. Some of these tools have been used in the development of financial theory, such as asset pricing theory and option pricing theory, as well as like theories in the practice of asset management, risk management, and financial modeling.

Different areas of finance call for different mathematics. For example, asset management, also referred to as investment management and money management, is primarily concerned with understanding hard facts about financial processes. Ultimately, the performance of an asset manager is linked to an understanding of risk and return. This implies the ability to extract information from time series data that are highly noisy and appear nearly random. Mathematical models must be simple, but with a deep economic meaning. In other areas, the complexity of instruments is the key driver behind the growing use of sophisticated mathematics in finance. There is the need to understand how relatively simple assumptions on the probabilistic behavior of basic quantities translate into the potentially very complex probabilistic behavior of financial products. Examples of such products include option-type financial derivatives (such as options, swaptions, caps, and floors), credit derivatives, bonds with embedded option-like payoffs (such as callable bonds and convertible bonds), structured notes, and mortgage-backed securities.

One might question whether all this mathematics is justified in finance. The field of finance is generally considered much less accurate and viable than the physical sciences. Sophisticated mathematical models of financial markets and market agents have been developed but their accuracy is questionable to the point that the recent global financial crisis is often blamed on unwarranted faith on faulty mathematical models. However, we believe that the mathematical handling of finance is reasonably successful and models are not to be blamed for this crisis. Finance does not study laws of nature but complex human artifacts—the financial markets—that are designed to be largely uncertain. We could make financial markets less uncertain and, thereby, mathematical models more faithful by introducing more rules and collecting more data. Collectively, we have decided not to do so and, therefore, models can only be moderately accurate. Still, they offer a valuable design tool to engineer our financial systems. Nevertheless, the mathematics of finance cannot be that of physics. It is the mathematics of learning and complexity, similar to the mathematics used in studying biological and ecological systems.

In 1960, the physicist Eugene Wigner, recipient of the 1962 Nobel Prize in Physics, wrote his now famous paper “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Wigner argued that the success of mathematics in describing natural phenomena is so extraordinary that it is in itself a phenomenon that needs explanation.1 Mathematics in finance is reasonably effective and the reasons why it is reasonably effective deserve an explanation. Recently, the world went through the worst financial and economic crisis since the Great Depression. Many have pointed their fingers at the growing use of mathematics in finance and the resulting mathematical models. We would argue that mathematics does not have much to do with that crisis. In a nutshell, we believe that mathematics is reasonably effective in finance because we apply it to study large engineered artifacts—financial markets—that have been designed to have a lot of freedom. Modern financial systems are designed to be relatively unpredictable and uncontrolled in order to leave possibilities of changes and innovations. The level of unpredictability and control is different in different systems. Some systems are prone to crises. Mathematics does a reasonably good job to describe these systems. But the mathematics involved is not the same as that of physics. It is the mathematics of learning and complexity. Mathematics can be perceived as ineffective in finance only if we insist on comparing it with physics.

There are differences between finance and the physical sciences. In the three centuries following the publication of Newton’s Principia in 1687, physics has developed into an axiomatic theory. Physical theories are axiomatic in the sense that that the entire theory can be derived through mathematical deduction from a small number of fundamental laws. Physics is not yet completely unified but the different disciplines that make the body of physics are axiomatic. Even more striking is the fact that physical phenomena can be approximately represented by computational structures, so that physical reality can be mimicked by a computer program.

Though it is clear that finance has made progress and will make additional progress only by adopting the scientific method of empirical science, it should be clear that there are significant differences between finance and physics. We can identify, albeit with some level of arbitrariness, four major differences between finance and the physical sciences:

1. Finance must study a global financial system without the possibility of studying simplified subsystems.

2. Finance is an empirical science, but the ability to conduct experiments in finance is limited when compared with the experimental facilities built in the physical sciences.

3. Finance does not study laws of nature, but it studies a human artifact that is subject to changes due to human decisions.

4. Finance systems are self-reflecting in the sense that the knowledge accumulated on the system changes the system itself.

None of the above four points is in itself an objection to the scientific study of finance as a mathematical science. However, it should be clear that the methods of scientific investigations and the findings of finance might be conceptually different from those of the physical sciences. It would probably be a mistake to expect in finance the same type of generalized axiomatic laws that we find in physics.

One of the major sources of the progress made by physics is due to the ability to isolate elementary subsystems, to come out with laws that apply to these subsystems, and then to recover macroscopic laws by a mathematical process. For example, the study of mechanics was greatly simplified by the study of the material point, a subsystem without structure identified by a small number of continuous variables. After identifying the laws that govern the motion of a material point, the motion of any physical body can be recovered by a process of mathematical integration. Simplifications of this type allow one to both simplify the mathematics and to perform empirical tests in a simplified environment.

In financial economics, however, we cannot study idealized subsystems because we cannot identify subsystems with a simplified behavior. This is not to say that attempts have not been made. Drawing on the principles of microeconomics, financial economics attempts to study the behavior of individuals as the elementary units of the financial system. The real problem, however, is that the study of individuals as economic “atoms” cannot produce simple laws because it is the study of a human financial decision-making process, which is very complex in itself. In addition, we cannot perform experiments. Instead, we have to rely on how the only financial system we know develops in itself.

Note that the possibility to study elementary subsystems does not coincide with the existence of fundamental laws. For example, consider the Schrödinger equation of quantum mechanics formulated in 1926 by the physicist Erwin Schrödinger. The equation is a partial differential equation describing how in some physical system a quantum state evolves over time. Although the Schrödinger equation is indeed a fundamental law, it applies to any system and not only to microscopic entities. Fundamental laws are not necessarily microscopic laws. We might be able to find fundamental laws of finance even if we are unable to isolate elementary subsystems.

There is a strong connection between fundamental laws and the ability to make experiments. By their nature, fundamental laws are very general and can be applied, albeit after difficult mathematical manipulations, to any phenomena. Therefore, after discovering a fundamental law it is generally possible to design experiments specific to test that same law. In many instances in the history of physics, crucial experiments have suggested rejection of a theory in favor of a new competing theory. However, in finance the ability to conduct experiments is limited though important research in this field has been carried on. In the 1970s, Daniel Kahneman and Amos Tversky performed groundbreaking research on cognitive biases in decision making. Vernon Smith studied different types of market organization, in particular auctions. This type of research has changed the perspective of finance as an empirical science. Still, we cannot make a close parallel between experimental finance and experimental physics where we can design experiments to decide between theories.

Perhaps the deepest difference between finance and physics is the fact that finance studies a human artifact which is subject to change in function of human decisions. Physics aims at discovering fundamental physical laws while finance determines laws that apply to a specific artifact. The level of generality of finance is intrinsically lower than that of physics. In addition, financial systems tend to change in function of the knowledge accumulated so that the object of inquiry is not stable.

As a result of all the above, it is unlikely that the kind of mathematics used in physics is appropriate to the study of financial theories. For example, we cannot expect to find any simple law that might be expressed with a closed formula. Hence, empirical testing cannot be done by comparing the results of closed-form solutions with experiments but more likely by comparing the results of long calculations. Thus the mathematical description of financial systems was delayed until researchers in finance had high-performance computers to perform the requisite large number of calculations. Nor can we expect a great level of accuracy in our descriptions of financial phenomena. If we want to compare finance to the natural sciences, we have to compare our knowledge of finance with our knowledge of the laws that govern macrosystems. While physicists have been able to determine extremely precise laws that govern subsystems such as atoms, their ability to predict macroscopic phenomena such as earthquakes or weather remains quite limited. Parallels between finance and the natural sciences are to be found more in the theory of complex systems than in fundamental physics.

In this book, special emphasis has been put on describing concepts and mathematical techniques, leaving aside lengthy demonstrations, which, while the substance of mathematics, are of limited interest to the practitioner and student of financial economics. From the practitioner’s point of view, what is important is to have a firm grasp of the concepts and techniques so as to understand the appropriate application. There is no prerequisite mathematical knowledge for reading this book: all mathematical concepts used in the book are explained, starting from ordinary calculus and matrix algebra. It is, however, a demanding book given the breadth and depth of concepts covered. Each chapter begins with a brief description of how the tool it covers is used in finance, which is then followed by the learning objectives for the chapter. Each chapter concludes with its key points.

In writing this book, special attention was given to bridging the gap between the intuition of the practitioner and academic mathematical analysis. Often there are simple compelling reasons for adopting sophisticated concepts and techniques that are obscured by mathematical details. That said, whenever possible, we tried to give the reader an understanding of the reasoning behind these concepts. The book has many examples of how quantitative analysis is used in the practice of asset management.

SERGIO M. FOCARDI FRANK J. FABOZZI TURAN G. BALI

1. E. Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences, ” Communications in Pure and Applied Mathematics 13 (1960): 1–14.

About the Authors

Sergio M. Focardi is a Visiting Professor at Stony Brook University, SUNY, where he holds a joint appointment in the College of Business and the Department of Applied Mathematics and Statistics. Prior to that, he was a Professor of Finance at the EDHEC Business School in Nice. Professor Focardi is a founding partner of the Paris-based consulting firm The Intertek Group. A member of the editorial board of the Journal of Portfolio Management, he has authored numerous articles and books on financial modeling and risk management including the following Wiley books: Probability and Statistics for Finance (2010), Quantitative Equity Investing: Techniques and Strategies (2010), Robust Portfolio Optimization and Management (2007), Financial Econometrics (2007), Financial Modeling of the Equity Market (2006), The Mathematics of Financial Modeling and Investment Management (2004), Risk Management: Framework, Methods and Practice (1998), and Modeling the Markets: New Theories and Techniques (1997). He also coauthored three monographs published by the Research Foundation of the CFA Institute: Challenges in Quantitative Equity Management (2008), The Impact of the Financial Crisis on the Asset Management Industry (2010), Trends in Quantitative Finance (2006). His research interests include the econometrics of large equity portfolios and the modeling of regime changes. Professor Focardi holds a degree in Electronic Engineering from the University of Genoa and a PhD in Mathematical Finance and Financial Econometrics from the University of Karlsruhe.

Frank J. Fabozzi is Professor of Finance at EDHEC Business School and a member of the EDHEC Risk Institute. He has held various professorial positions at Yale and MIT. In 2013–2014 he will hold the position of James Wei Visiting Professor in Entrepreneurship at Princeton University. Since the 2011–2012 academic year, he has been a Visiting Fellow in the Department of Operations Research and Financial Engineering at Princeton University. A trustee for the BlackRock family of closed-end funds, Professor Fabozzi has authored and edited many books in asset management and quantitative finance. In addition to his position as editor of the Journal of Portfolio Management and editorial board member of Quantitative Finance, he serves on the advisory board of The Wharton School’s Jacobs Levy Equity Management Center for Quantitative Financial Research, the Q Group Selection Committee, and from 2003 to 2011 on the Council for the Department of Operations Research and Financial Engineering at Princeton University. He is a Fellow of of the International Center for Finance at Yale University. He is the CFA Institute’s 2007 recipient of the C. Stewart Sheppard Award and an inductee into the Fixed Income Analysts Society Hall of Fame. Professor Fabozzi earned a PhD in Economics in September 1972 from the City University of New York and holds the professional designations of Chartered Financial Analyst (1977) and Certified Public Accountant (1982). In 1994, he was awarded an Honorary Doctorate of Humane Letter from Nova Southeastern University.

Turan G. Bali is the Robert S. Parker Chair Professor of Finance at the McDonough School of Business at Georgetown University. Before joining Georgetown University, Professor Bali was the David Krell Chair Professor of Finance at Baruch College and the Graduate School and University Center of the City University of New York. He also held visiting faculty positions at New York University and Princeton University. Professor Bali specializes in asset pricing, risk management, fixed income securities, and financial derivatives. A founding member of the Society for Financial Econometrics, he has worked on consulting projects sponsored by major financial institutions and government organizations in the United States and other countries. In addition, he currently serves as an Associate Editor for the following journals: Journal of Banking and Finance, Journal of Futures Markets, Journal of Portfolio Management, Review of Financial Economics, and Journal of Risk. He served on the review committee of several research foundations such as the National Science Foundation, Research Grants Council of Hong Kong, Scientific and Technological Research Council of Turkey, and Social Sciences and Humanities Research Council of Canada. With more than 50 published articles in economics and finance journals, Professor Bali’s work has appeared in the Journal of Finance, Journal of Financial Economics, Review of Financial Studies, Journal of Monetary Economics, Management Science, Review of Economics and Statistics, Journal of Business, and Journal of Financial and Quantitative Analysis.

CHAPTER 1

Basic Concepts

Sets, Functions, and Variables

In mathematics, sets, functions, and variables are three fundamental concepts. First, a set is a well-defined collection of objects. A set is a gathering together into a whole of definite, distinct objects of our perception, which are called elements of the set. Sets are one of the most fundamental concepts in mathematics. Set theory is seen as the foundation from which virtually all of mathematics can be derived. For example, structures in abstract algebra, such as groups, fields, and rings, are sets closed under one or more operations. One of the main applications of set theory is constructing relations. Second, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Functions are the central objects of investigation in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function. A function can be described through its relationship with other functions, for example, as an inverse function or as a solution of a differential equation. Finally, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a is a value that remains unchanged, though often unknown or undetermined. Variables are further distinguished as being either a dependent variable or an independent variable. Independent variables are regarded as inputs to a system and may take on different values freely. Dependent variables are those values that change as a consequence of changes in other values in the system.