Details

Mathematical Methods for Finance

Tools for Asset and Risk Management
Frank J. Fabozzi Series 1. Aufl.

von: Sergio M. Focardi, Frank J. Fabozzi, Turan G. Bali
103,99 €
Verlag:	Wiley
Format:	EPUB
Veröffentl.:	04.09.2013
ISBN/EAN:	9781118421499
Sprache:	englisch
Anzahl Seiten:	320

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Beschreibungen

Titelbeschreibung

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. Fabozzi Series, has been created with this in mind. Designed to provide the tools needed to apply finance theory to real world financial markets, this book offers a wealth of insights and guidance in practical applications. It contains applications that are broader in scope from what is covered in a typical book on mathematical techniques. Most books focus almost exclusively on derivatives pricing, the applications in this book cover not only derivatives and asset pricing but also risk management—including credit risk management—and portfolio management. <ul> <li>Includes an overview of the essential math and statistical skills required to succeed in quantitative finance</li> <li>Offers the basic mathematical concepts that apply to the field of quantitative finance, from sets and distances to functions and variables</li> <li>The book also includes information on calculus, matrix algebra, differential equations, stochastic integrals, and much more</li> <li>Written by Sergio Focardi, one of the world's leading authors in high-level finance</li> </ul> Drawing on the author's perspectives as a practitioner and academic, each chapter of this book offers a solid foundation in the mathematical tools and techniques need to succeed in today's dynamic world of finance.

Inhaltsverzeichnis

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

Autorenportrait

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.

Back cover copy

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.