Mathematical Methods for 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 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.

• Preface xiAbout the Authors xviiCHAPTER 1 Basic Concepts: Sets, Functions, and Variables 1Introduction 2Sets and Set Operations 2Distances and Quantities 6Functions 10Variables 10Key Points 11CHAPTER 2 Differential Calculus 13Introduction 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 41CHAPTER 3 Integral Calculus 43Introduction 44Riemann Integrals 44Lebesgue-Stieltjes Integrals 47Indefinite and Improper Integrals 48The Fundamental Theorem of Calculus 51Integral Transforms 52Calculus in More Than One Variable 57Key Points 57CHAPTER 4 Matrix Algebra 59Introduction 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 83CHAPTER 5 Probability: Basic Concepts 85Introduction 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 106CHAPTER 6 Probability: Random Variables and Expectations 107Introduction 109Conditional Probability and Conditional Expectation 110Moments 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 144CHAPTER 7 Optimization 147Introduction 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 178CHAPTER 8 Difference Equations 181Introduction 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 209CHAPTER 9 Differential Equations 211Introduction 212Differential Equations Defined 213Ordinary Differential Equations 213Systems of Ordinary Differential Equations 216Closed-Form Solutions of Ordinary Differential Equations 218Numerical Solutions of Ordinary Differential Equations 222Nonlinear Dynamics and Chaos 228Partial Differential Equations 231Key Points 237CHAPTER 10 Stochastic Integrals 239Introduction 240The Intuition behind Stochastic Integrals 243Brownian Motion Defined 248Properties of Brownian Motion 254Stochastic Integrals Defined 255Some Properties of Itoˆ Stochastic Integrals 259Martingale Measures and the Girsanov Theorem 260Key Points 266CHAPTER 11 Stochastic Differential Equations 267Introduction 268The Intuition behind Stochastic Differential Equations 269Itoˆ Processes 272Stochastic Differential Equations 273Generalization to Several Dimensions 276Solution of Stochastic Differential Equations 278Derivation of Itoˆ ’s Lemma 282Derivation of the Black-Scholes Option Pricing Formula 284Key Points 291Index 293