Mathematical Methods for Finance: Tools for Asset and Risk Management

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.

Preface xi About the Authors xvii CHAPTER 1 Basic Concepts: Sets, Functions, and Variables 1 Introduction 2 Sets and Set Operations 2 Distances and Quantities 6 Functions 10 Variables 10 Key Points 11 CHAPTER 2 Differential Calculus 13 Introduction 14 Limits 15 Continuity 17 Total Variation 19 The Notion of Differentiation 19 Commonly Used Rules for Computing Derivatives 21 Higher–Order Derivatives 26 Taylor Series Expansion 34 Calculus in More Than One Variable 40 Key Points 41 CHAPTER 3 Integral Calculus 43 Introduction 44 Riemann Integrals 44 Lebesgue–Stieltjes Integrals 47 Indefinite and Improper Integrals 48 The Fundamental Theorem of Calculus 51 Integral Transforms 52 Calculus in More Than One Variable 57 Key Points 57 CHAPTER 4 Matrix Algebra 59 Introduction 60 Vectors and Matrices Defined 61 Square Matrices 63 Determinants 66 Systems of Linear Equations 68 Linear Independence and Rank 69 Hankel Matrix 70 Vector and Matrix Operations 72 Finance Application 78 Eigenvalues and Eigenvectors 81 Diagonalization and Similarity 82 Singular Value Decomposition 83 Key Points 83 CHAPTER 5 Probability: Basic Concepts 85 Introduction 86 Representing Uncertainty with Mathematics 87 Probability in a Nutshell 89 Outcomes and Events 91 Probability 92 Measure 93 Random Variables 93 Integrals 94 Distributions and Distribution Functions 96 Random Vectors 97 Stochastic Processes 100 Probabilistic Representation of Financial Markets 102 Information Structures 103 Filtration 104 Key Points 106 CHAPTER 6 Probability: Random Variables and Expectations 107 Introduction 109 Conditional Probability and Conditional Expectation 110 Moments and Correlation 112 Copula Functions 114 Sequences of Random Variables 116 Independent and Identically Distributed Sequences 117 Sum of Variables 118 Gaussian Variables 120 Appproximating the Tails of a Probability Distribution: Cornish–Fisher Expansion and Hermite Polynomials 123 The Regression Function 129 Fat Tails and Stable Laws 131 Key Points 144 CHAPTER 7 Optimization 147 Introduction 148 Maxima and Minima 149 Lagrange Multipliers 151 Numerical Algorithms 156 Calculus of Variations and Optimal Control Theory 161 Stochastic Programming 163 Application to Bond Portfolio: Liability–Funding Strategies 164 Key Points 178 CHAPTER 8 Difference Equations 181 Introduction 182 The Lag Operator L 183 Homogeneous Difference Equations 183 Recursive Calculation of Values of Difference Equations 192 Nonhomogeneous Difference Equations 195 Systems of Linear Difference Equations 201 Systems of Homogeneous Linear Difference Equations 202 Key Points 209 CHAPTER 9 Differential Equations 211 Introduction 212 Differential Equations Defined 213 Ordinary Differential Equations 213 Systems of Ordinary Differential Equations 216 Closed–Form Solutions of Ordinary Differential Equations 218 Numerical Solutions of Ordinary Differential Equations 222 Nonlinear Dynamics and Chaos 228 Partial Differential Equations 231 Key Points 237 CHAPTER 10 Stochastic Integrals 239 Introduction 240 The Intuition behind Stochastic Integrals 243 Brownian Motion Defined 248 Properties of Brownian Motion 254 Stochastic Integrals Defined 255 Some Properties of Itoˆ Stochastic Integrals 259 Martingale Measures and the Girsanov Theorem 260 Key Points 266 CHAPTER 11 Stochastic Differential Equations 267 Introduction 268 The Intuition behind Stochastic Differential Equations 269 Itoˆ Processes 272 Stochastic Differential Equations 273 Generalization to Several Dimensions 276 Solution of Stochastic Differential Equations 278 Derivation of Itoˆ ’s Lemma 282 Derivation of the Black–Scholes Option Pricing Formula 284 Key Points 291 Index 293

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 Management 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 .