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THE FRANK J. FABOZZI SERIES

MATHEMATICALMETHODS FORFINANCE

MATH

EMATICA

L METH

OD

S FOR

FINA

NCE

MATHEMATICAL METHODS FOR FINANCE

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

FOCARDI•

FABOZZI •

BALI

Tools for Asset and Risk Management

Tools for Asset and Risk Managem

ent

Cover Design: Wiley Cover Image: © Brownstock / Alamy

Modern finance draws upon many fields

of mathematics—from probability and

statistics to stochastic calculus—and the

level of mathematical skill needed to master

today’s financial markets is extremely high.

Nobody understands this better than the author

team of Sergio Focardi, Frank Fabozzi, and

Turan Bali. Now, in Mathematical Methods

for Finance, they draw upon their extensive

experience in this important area in order to

help both practitioners and students gain a firm

understanding of the subject.

Covering a wide range of technical topics in

mathematics and finance, this reliable resource

opens with an informative discussion of three

basic concepts—which are used in financial

theory, financial modeling, and financial

econometrics—found throughout the book:

sets, functions, and variables. From there, it

introduces and explains key mathematical

techniques, ranging from differential and

integral calculus, matrix algebra, and probability

theory to difference and differential equations,

optimization, and stochastic integrals. Page by

page, you’ll discover how these techniques are

successfully implemented in asset management

and risk management.

Each chapter begins with a brief description of

how the tools and concepts covered are used

in finance, followed by learning objectives.

And a wealth of real-world examples—of how

quantitative analysis is used in practice—

skillfully highlights the connection between this

analysis and financial decision-making.

Bridging the gap between the intuition of

a practitioner and academic mathematical

analysis, Mathematical Methods for Finance is

an essential guide for anyone who intends on

exceling in today’s demanding world of finance.

SERGIO M. FOCARDI, PhD, is a Visiting

Professor in the College of Business at the State

University of New York at Stony Brook and

founding partner of the Paris-based consulting

firm The Intertek Group. He is a member of

the editorial board of the Journal of Portfolio

Management. Focardi has authored numerous

articles and books on financial modeling and

risk management and three monographs for the

Research Foundation of the CFA Institute.

FRANK J. FABOZZI, PhD, CFA, is Professor

of Finance at EDHEC Business School and a

member of the EDHEC-Risk Institute. Prior

to joining EDHEC in August 2011, he held

various professorial positions in finance at Yale

University’s School of Management from 1994

to 2011 and was a visiting professor of finance

and accounting at MIT’s Sloan School of Man-

agement from 1986 to 1992. He is also Editor of

the Journal of Portfolio Management.

TURAN G. BALI, PhD, is the Robert S. Parker

Chair Professor of Business Administration at the

McDonough School of Business at Georgetown

University. Before joining Georgetown, 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 has published more than fifty

articles in economics and finance journals. He

is currently an associate editor of the Journal

of Banking and Finance, Journal of Futures

Markets, Journal of Portfolio Management, and

Journal of Risk. 

With the rapid growth of quantitative finance, practitioners and students

alike must become more proficient in various areas of mathematics in

order to excel in the demanding world of finance. Mathematical Methods

for Finance, part of the Frank J. Fabozzi Series, has been created with

this in mind. Designed to provide the tools and techniques needed to

apply proven mathematical techniques to real-world financial markets,

this book offers a wealth of insights and guidance.

Drawing on the authors’ perspectives as practitioners and academics, this

practical guide covers a wide range of technical topics in mathematics and

finance. It opens with an informative discussion of three basic concepts—

which are used in financial theory, financial modeling, and financial

econometrics—found throughout the book: sets, functions, and variables.

From there, it introduces and explains key mathematical techniques,

ranging from differential and integral calculus, matrix algebra, and

probability theory to difference and differential equations, optimization,

and stochastic integrals. Along the way, you’ll discover exactly how

these techniques are successfully implemented in asset management and

risk management.

Written with both students and practitioners in mind, Mathematical

Methods for Finance is an essential resource that will show you how a

better understanding of specific mathematical techniques can enhance

your financial decision-making.

$125.00 USA/$138.00 CAN

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Contents

Preface xi

About the Authors xvii

CHAPTER 1Basic Concepts: Sets, Functions, and Variables 1

Introduction 2Sets and Set Operations 2Distances and Quantities 6Functions 10Variables 10Key Points 11

CHAPTER 2Differential Calculus 13

Introduction 14Limits 15Continuity 17Total Variation 19The Notion of Differentiation 19Commonly Used Rules for Computing Derivatives 21Higher-Order Derivatives 26Taylor Series Expansion 34Calculus in More Than One Variable 40Key Points 41

CHAPTER 3Integral Calculus 43

Introduction 44Riemann Integrals 44Lebesgue-Stieltjes Integrals 47Indefinite and Improper Integrals 48The Fundamental Theorem of Calculus 51

vii

COPYRIG

HTED M

ATERIAL

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viii CONTENTS

Integral Transforms 52Calculus in More Than One Variable 57Key Points 57

CHAPTER 4Matrix Algebra 59

Introduction 60Vectors and Matrices Defined 61Square Matrices 63Determinants 66Systems of Linear Equations 68Linear Independence and Rank 69Hankel Matrix 70Vector and Matrix Operations 72Finance Application 78Eigenvalues and Eigenvectors 81Diagonalization and Similarity 82Singular Value Decomposition 83Key Points 83

CHAPTER 5Probability: Basic Concepts 85

Introduction 86Representing Uncertainty with Mathematics 87Probability in a Nutshell 89Outcomes and Events 91Probability 92Measure 93Random Variables 93Integrals 94Distributions and Distribution Functions 96Random Vectors 97Stochastic Processes 100Probabilistic Representation of Financial Markets 102Information Structures 103Filtration 104Key Points 106

CHAPTER 6Probability: Random Variables and Expectations 107

Introduction 109Conditional Probability and Conditional Expectation 110

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Contents ix

Moments and Correlation 112Copula Functions 114Sequences of Random Variables 116Independent and Identically Distributed Sequences 117Sum of Variables 118Gaussian Variables 120Appproximating the Tails of a Probability Distribution:

Cornish-Fisher Expansion and Hermite Polynomials 123The Regression Function 129Fat Tails and Stable Laws 131Key Points 144

CHAPTER 7Optimization 147

Introduction 148Maxima and Minima 149Lagrange Multipliers 151Numerical Algorithms 156Calculus of Variations and Optimal Control Theory 161Stochastic Programming 163Application to Bond Portfolio: Liability-Funding Strategies 164Key Points 178

CHAPTER 8Difference Equations 181

Introduction 182The Lag Operator L 183Homogeneous Difference Equations 183Recursive Calculation of Values of Difference Equations 192Nonhomogeneous Difference Equations 195Systems of Linear Difference Equations 201Systems of Homogeneous Linear Difference Equations 202Key Points 209

CHAPTER 9Differential Equations 211

Introduction 212Differential Equations Defined 213Ordinary Differential Equations 213Systems of Ordinary Differential Equations 216Closed-Form Solutions of Ordinary Differential Equations 218Numerical Solutions of Ordinary Differential Equations 222

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x CONTENTS

Nonlinear Dynamics and Chaos 228Partial Differential Equations 231Key Points 237

CHAPTER 10Stochastic Integrals 239

Introduction 240The Intuition behind Stochastic Integrals 243Brownian Motion Defined 248Properties of Brownian Motion 254Stochastic Integrals Defined 255Some Properties of Ito Stochastic Integrals 259Martingale Measures and the Girsanov Theorem 260Key Points 266

CHAPTER 11Stochastic Differential Equations 267

Introduction 268The Intuition behind Stochastic Differential Equations 269Ito Processes 272Stochastic Differential Equations 273Generalization to Several Dimensions 276Solution of Stochastic Differential Equations 278Derivation of Ito’s Lemma 282Derivation of the Black-Scholes Option Pricing Formula 284Key Points 291

Index 293