Mathematical methods for finance : tools for asset and risk management

 Content: 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 Ito 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  Ito Processes 272  Stochastic Differential Equations 273  Generalization to Several Dimensions 276  Solution of Stochastic Differential Equations 278  Derivation of Ito 's Lemma 282  Derivation of the Black-Scholes Option Pricing Formula 284  Key Points 291  Index 293
 Abstract:             
              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.                 Read more...