Mathematical Methods for Finance Tools for Asset and Risk Management

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