Mathematical Methods for Finance by Focardi, Sergio M. -- Read -- Imperial Library of Trantor

Index

Cover Series Page Title Page Copyright Page Dedication Preface About the Authors Chapter 1: Basic Concepts

INTRODUCTION SETS AND SET OPERATIONS DISTANCES AND QUANTITIES FUNCTIONS VARIABLES KEY POINTS

Chapter 2: Differential Calculus

INTRODUCTION LIMITS CONTINUITY TOTAL VARIATION THE NOTION OF DIFFERENTIATION COMMONLY USED RULES FOR COMPUTING DERIVATIVES HIGHER-ORDER DERIVATIVES TAYLOR SERIES EXPANSION CALCULUS IN MORE THAN ONE VARIABLE KEY POINTS

Chapter 3: Integral Calculus

INTRODUCTION RIEMANN INTEGRALS LEBESGUE-STIELTJES INTEGRALS INDEFINITE AND IMPROPER INTEGRALS THE FUNDAMENTAL THEOREM OF CALCULUS INTEGRAL TRANSFORMS CALCULUS IN MORE THAN ONE VARIABLE KEY POINTS

Chapter 4: Matrix Algebra

INTRODUCTION VECTORS AND MATRICES DEFINED SQUARE MATRICES DETERMINANTS SYSTEMS OF LINEAR EQUATIONS LINEAR INDEPENDENCE AND RANK HANKEL MATRIX VECTOR AND MATRIX OPERATIONS FINANCE APPLICATION EIGENVALUES AND EIGENVECTORS DIAGONALIZATION AND SIMILARITY SINGULAR VALUE DECOMPOSITION KEY POINTS

Chapter 5: Probability

INTRODUCTION REPRESENTING UNCERTAINTY WITH MATHEMATICS PROBABILITY IN A NUTSHELL OUTCOMES AND EVENTS PROBABILITY MEASURE RANDOM VARIABLES INTEGRALS DISTRIBUTIONS AND DISTRIBUTION FUNCTIONS RANDOM VECTORS STOCHASTIC PROCESSES PROBABILISTIC REPRESENTATION OF FINANCIAL MARKETS INFORMATION STRUCTURES FILTRATION KEY POINTS

Chapter 6: Probability

INTRODUCTION CONDITIONAL PROBABILITY AND CONDITIONAL EXPECTATION MOMENTS AND CORRELATION COPULA FUNCTIONS SEQUENCES OF RANDOM VARIABLES INDEPENDENT AND IDENTICALLY DISTRIBUTED SEQUENCES SUM OF VARIABLES GAUSSIAN VARIABLES APPPROXIMATING THE TAILS OF A PROBABILITY DISTRIBUTION: CORNISH-FISHER EXPANSION AND HERMITE POLYNOMIALS THE REGRESSION FUNCTION FAT TAILS AND STABLE LAWS KEY POINTS

Chapter 7: Optimization

INTRODUCTION MAXIMA AND MINIMA LAGRANGE MULTIPLIERS NUMERICAL ALGORITHMS CALCULUS OF VARIATIONS AND OPTIMAL CONTROL THEORY STOCHASTIC PROGRAMMING APPLICATION TO BOND PORTFOLIO: LIABILITY-FUNDING STRATEGIES KEY POINTS

Chapter 8: Difference Equations

INTRODUCTION THE LAG OPERATOR L HOMOGENEOUS DIFFERENCE EQUATIONS RECURSIVE CALCULATION OF VALUES OF DIFFERENCE EQUATIONS NONHOMOGENEOUS DIFFERENCE EQUATIONS SYSTEMS OF LINEAR DIFFERENCE EQUATIONS SYSTEMS OF HOMOGENEOUS LINEAR DIFFERENCE EQUATIONS KEY POINTS

Chapter 9: Differential Equations

INTRODUCTION DIFFERENTIAL EQUATIONS DEFINED ORDINARY DIFFERENTIAL EQUATIONS SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS CLOSED-FORM SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS NONLINEAR DYNAMICS AND CHAOS PARTIAL DIFFERENTIAL EQUATIONS KEY POINTS

Chapter 10: Stochastic Integrals

INTRODUCTION THE INTUITION BEHIND STOCHASTIC INTEGRALS BROWNIAN MOTION DEFINED PROPERTIES OF BROWNIAN MOTION STOCHASTIC INTEGRALS DEFINED SOME PROPERTIES OF ITÔ STOCHASTIC INTEGRALS MARTINGALE MEASURES AND THE GIRSANOV THEOREM KEY POINTS

Chapter 11: Stochastic Differential Equations

INTRODUCTION THE INTUITION BEHIND STOCHASTIC DIFFERENTIAL EQUATIONS ITÔ PROCESSES STOCHASTIC DIFFERENTIAL EQUATIONS GENERALIZATION TO SEVERAL DIMENSIONS SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATIONS DERIVATION OF ITÔ’S LEMMA DERIVATION OF THE BLACK-SCHOLES OPTION PRICING FORMULA KEY POINTS

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