Mathematical methods for finance : tools for asset and risk management

322 Pages·2013·9.582 MB·English

by  Sergio Focardi, Turan G Bali, Frank J Fabozzi

Mathematical
Methods
for Finance
TheFrankJ.FabozziSeries
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Mathematical
Methods
for Finance
Tools for Asset
and Risk Management
SERGIO M. FOCARDI
FRANK J. FABOZZI
TURAN G. BALI
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To the memory of my parents
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and my children, Patricia, Karly, and Francesco
FJF
To my wife, Mehtap,
and my son, Kaan
TGB

Contents
Preface xi
AbouttheAuthors xvii
CHAPTER1
BasicConcepts:Sets,Functions,andVariables 1
Introduction 2
SetsandSetOperations 2
DistancesandQuantities 6
Functions 10
Variables 10
KeyPoints 11
CHAPTER2
DifferentialCalculus 13
Introduction 14
Limits 15
Continuity 17
TotalVariation 19
TheNotionofDifferentiation 19
CommonlyUsedRulesforComputingDerivatives 21
Higher-OrderDerivatives 26
TaylorSeriesExpansion 34
CalculusinMoreThanOneVariable 40
KeyPoints 41
CHAPTER3
IntegralCalculus 43
Introduction 44
RiemannIntegrals 44
Lebesgue-StieltjesIntegrals 47
IndefiniteandImproperIntegrals 48
TheFundamentalTheoremofCalculus 51
vii
viii CONTENTS
IntegralTransforms 52
CalculusinMoreThanOneVariable 57
KeyPoints 57
CHAPTER4
MatrixAlgebra 59
Introduction 60
VectorsandMatricesDefined 61
SquareMatrices 63
Determinants 66
SystemsofLinearEquations 68
LinearIndependenceandRank 69
HankelMatrix 70
VectorandMatrixOperations 72
FinanceApplication 78
EigenvaluesandEigenvectors 81
DiagonalizationandSimilarity 82
SingularValueDecomposition 83
KeyPoints 83
CHAPTER5
Probability:BasicConcepts 85
Introduction 86
RepresentingUncertaintywithMathematics 87
ProbabilityinaNutshell 89
OutcomesandEvents 91
Probability 92
Measure 93
RandomVariables 93
Integrals 94
DistributionsandDistributionFunctions 96
RandomVectors 97
StochasticProcesses 100
ProbabilisticRepresentationofFinancialMarkets 102
InformationStructures 103
Filtration 104
KeyPoints 106
CHAPTER6
Probability:RandomVariablesandExpectations 107
Introduction 109
ConditionalProbabilityandConditionalExpectation 110