Mathematical Methods for Finance -- Contents -- Preface -- About the Authors -- CHAPTER 1 Basic Concepts: Sets, Functions, and Variables -- INTRODUCTION -- SETS AND SET OPERATIONS -- Proper Subsets -- Empty Sets -- Union of Sets -- Intersection of Sets -- Elementary Properties of Sets -- DISTANCES AND QUANTITIES -- n-tuples -- Distance -- Density of Points -- 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 -- Application to Bond Analysis -- Application of the Chain Rule -- TAYLOR SERIES EXPANSION -- Application to Bond Analysis -- CALCULUS IN MORE THAN ONE VARIABLE -- KEY POINTS -- CHAPTER 3 Integral Calculus -- INTRODUCTION -- RIEMANN INTEGRALS -- Properties of Riemann Integrals -- LEBESGUE-STIELTJES INTEGRALS -- INDEFINITE AND IMPROPER INTEGRALS -- THE FUNDAMENTAL THEOREM OF CALCULUS -- INTEGRAL TRANSFORMS -- Laplace Transforms -- Fourier Transforms -- CALCULUS IN MORE THAN ONE VARIABLE -- KEY POINTS -- CHAPTER 4 Matrix Algebra -- INTRODUCTION -- VECTORS AND MATRICES DEFINED -- Vectors -- Matrices -- SQUARE MATRICES -- Diagonals and Antidiagonals -- Identity Matrix -- Diagonal Matrix -- Upper and Lower Triangular Matrix -- DETERMINANTS -- SYSTEMS OF LINEAR EQUATIONS -- LINEAR INDEPENDENCE AND RANK -- HANKEL MATRIX -- VECTOR AND MATRIX OPERATIONS -- Vector Operations -- Matrix Operations -- FINANCE APPLICATION -- EIGENVALUES AND EIGENVECTORS -- DIAGONALIZATION AND SIMILARITY -- SINGULAR VALUE DECOMPOSITION -- KEY POINTS -- CHAPTER 5 Probability: Basic Concepts -- 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: Random Variables and Expectations -- 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 -- Cornish-Fisher Expansion -- Hermite Polynomials -- Cornish-Fisher Expansion with Hermite Polynomials -- THE REGRESSION FUNCTION -- Linear Regression -- FAT TAILS AND STABLE LAWS -- Fat Tails -- The Class L of Fat-Tailed Distributions -- The Law of Large Numbers and the Central Limit Theorem -- Stable Distributions -- KEY POINTS -- CHAPTER 7 Optimization -- INTRODUCTION -- MAXIMA AND MINIMA -- LAGRANGE MULTIPLIERS -- NUMERICAL ALGORITHMS -- Linear Programming -- Quadratic Programming -- CALCULUS OF VARIATIONS AND OPTIMAL CONTROL THEORY -- STOCHASTIC PROGRAMMING -- APPLICATION TO BOND PORTFOLIO: LIABILITY-FUNDING STRATEGIES -- Cash Flow Matching -- Portfolio Immunization -- Scenario Optimization -- Stochastic Programming -- KEY POINTS -- CHAPTER 8 Difference Equations -- INTRODUCTION -- THE LAG OPERATOR L -- HOMOGENEOUS DIFFERENCE EQUATIONS -- Real Roots -- Complex Roots -- RECURSIVE CALCULATION OF VALUES OF DIFFERENCE EQUATIONS -- Solving Homogeneous Higher-Order Difference Equations -- NONHOMOGENEOUS DIFFERENCE EQUATIONS -- Real Roots -- Complex Roots -- SYSTEMS OF LINEAR DIFFERENCE EQUATIONS -- SYSTEMS OF HOMOGENEOUS LINEAR DIFFERENCE EQUATIONS -- KEY POINTS -- CHAPTER 9 Differential Equations -- INTRODUCTION -- DIFFERENTIAL EQUATIONS DEFINED.

ORDINARY DIFFERENTIAL EQUATIONS -- Order and Degree of an ODE -- Solution to an ODE -- SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS -- CLOSED-FORM SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS -- Linear Differential Equation -- NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS -- The Finite Difference Method -- NONLINEAR DYNAMICS AND CHAOS -- Fractals -- PARTIAL DIFFERENTIAL EQUATIONS -- Diffusion Equation -- Solution of the Diffusion Equation -- Numerical Solution of PDEs -- 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 -- Risk-Neutral Measure -- Continuous-Time Martingales -- 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 -- The Greeks -- KEY POINTS -- Index.