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Mathematical Methods for Finance: Tools for Asset and Risk Management

Book Description

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.

  • Includes an overview of the essential math and statistical skills required to succeed in quantitative finance

  • Offers the basic mathematical concepts that apply to the field of quantitative finance, from sets and distances to functions and variables

  • The book also includes information on calculus, matrix algebra, differential equations, stochastic integrals, and much more

  • Written by Sergio Focardi, one of the world's leading authors in high-level finance

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.

Table of Contents

  1. Cover
  2. Series Page
  3. Title Page
  4. Copyright Page
  5. Dedication
  6. Preface
  7. About the Authors
  8. Chapter 1: Basic Concepts
    1. INTRODUCTION
    2. SETS AND SET OPERATIONS
    3. DISTANCES AND QUANTITIES
    4. FUNCTIONS
    5. VARIABLES
    6. KEY POINTS
  9. Chapter 2: Differential Calculus
    1. INTRODUCTION
    2. LIMITS
    3. CONTINUITY
    4. TOTAL VARIATION
    5. THE NOTION OF DIFFERENTIATION
    6. COMMONLY USED RULES FOR COMPUTING DERIVATIVES
    7. HIGHER-ORDER DERIVATIVES
    8. TAYLOR SERIES EXPANSION
    9. CALCULUS IN MORE THAN ONE VARIABLE
    10. KEY POINTS
  10. Chapter 3: Integral Calculus
    1. INTRODUCTION
    2. RIEMANN INTEGRALS
    3. LEBESGUE-STIELTJES INTEGRALS
    4. INDEFINITE AND IMPROPER INTEGRALS
    5. THE FUNDAMENTAL THEOREM OF CALCULUS
    6. INTEGRAL TRANSFORMS
    7. CALCULUS IN MORE THAN ONE VARIABLE
    8. KEY POINTS
  11. Chapter 4: Matrix Algebra
    1. INTRODUCTION
    2. VECTORS AND MATRICES DEFINED
    3. SQUARE MATRICES
    4. DETERMINANTS
    5. SYSTEMS OF LINEAR EQUATIONS
    6. LINEAR INDEPENDENCE AND RANK
    7. HANKEL MATRIX
    8. VECTOR AND MATRIX OPERATIONS
    9. FINANCE APPLICATION
    10. EIGENVALUES AND EIGENVECTORS
    11. DIAGONALIZATION AND SIMILARITY
    12. SINGULAR VALUE DECOMPOSITION
    13. KEY POINTS
  12. Chapter 5: Probability
    1. INTRODUCTION
    2. REPRESENTING UNCERTAINTY WITH MATHEMATICS
    3. PROBABILITY IN A NUTSHELL
    4. OUTCOMES AND EVENTS
    5. PROBABILITY
    6. MEASURE
    7. RANDOM VARIABLES
    8. INTEGRALS
    9. DISTRIBUTIONS AND DISTRIBUTION FUNCTIONS
    10. RANDOM VECTORS
    11. STOCHASTIC PROCESSES
    12. PROBABILISTIC REPRESENTATION OF FINANCIAL MARKETS
    13. INFORMATION STRUCTURES
    14. FILTRATION
    15. KEY POINTS
  13. Chapter 6: Probability
    1. INTRODUCTION
    2. CONDITIONAL PROBABILITY AND CONDITIONAL EXPECTATION
    3. MOMENTS AND CORRELATION
    4. COPULA FUNCTIONS
    5. SEQUENCES OF RANDOM VARIABLES
    6. INDEPENDENT AND IDENTICALLY DISTRIBUTED SEQUENCES
    7. SUM OF VARIABLES
    8. GAUSSIAN VARIABLES
    9. APPPROXIMATING THE TAILS OF A PROBABILITY DISTRIBUTION: CORNISH-FISHER EXPANSION AND HERMITE POLYNOMIALS
    10. THE REGRESSION FUNCTION
    11. FAT TAILS AND STABLE LAWS
    12. KEY POINTS
  14. Chapter 7: Optimization
    1. INTRODUCTION
    2. MAXIMA AND MINIMA
    3. LAGRANGE MULTIPLIERS
    4. NUMERICAL ALGORITHMS
    5. CALCULUS OF VARIATIONS AND OPTIMAL CONTROL THEORY
    6. STOCHASTIC PROGRAMMING
    7. APPLICATION TO BOND PORTFOLIO: LIABILITY-FUNDING STRATEGIES
    8. KEY POINTS
  15. Chapter 8: Difference Equations
    1. INTRODUCTION
    2. THE LAG OPERATOR L
    3. HOMOGENEOUS DIFFERENCE EQUATIONS
    4. RECURSIVE CALCULATION OF VALUES OF DIFFERENCE EQUATIONS
    5. NONHOMOGENEOUS DIFFERENCE EQUATIONS
    6. SYSTEMS OF LINEAR DIFFERENCE EQUATIONS
    7. SYSTEMS OF HOMOGENEOUS LINEAR DIFFERENCE EQUATIONS
    8. KEY POINTS
  16. Chapter 9: Differential Equations
    1. INTRODUCTION
    2. DIFFERENTIAL EQUATIONS DEFINED
    3. ORDINARY DIFFERENTIAL EQUATIONS
    4. SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS
    5. CLOSED-FORM SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
    6. NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
    7. NONLINEAR DYNAMICS AND CHAOS
    8. PARTIAL DIFFERENTIAL EQUATIONS
    9. KEY POINTS
  17. Chapter 10: Stochastic Integrals
    1. INTRODUCTION
    2. THE INTUITION BEHIND STOCHASTIC INTEGRALS
    3. BROWNIAN MOTION DEFINED
    4. PROPERTIES OF BROWNIAN MOTION
    5. STOCHASTIC INTEGRALS DEFINED
    6. SOME PROPERTIES OF ITÔ STOCHASTIC INTEGRALS
    7. MARTINGALE MEASURES AND THE GIRSANOV THEOREM
    8. KEY POINTS
  18. Chapter 11: Stochastic Differential Equations
    1. INTRODUCTION
    2. THE INTUITION BEHIND STOCHASTIC DIFFERENTIAL EQUATIONS
    3. ITÔ PROCESSES
    4. STOCHASTIC DIFFERENTIAL EQUATIONS
    5. GENERALIZATION TO SEVERAL DIMENSIONS
    6. SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATIONS
    7. DERIVATION OF ITÔ’S LEMMA
    8. DERIVATION OF THE BLACK-SCHOLES OPTION PRICING FORMULA
    9. KEY POINTS
  19. Index