Foundations of Quantitative Finance Book II: Probability Spaces and Random Variables

276 Pages·2022·13.032 MB·English

by  Robert R. Reitnao

Foundations of Quantitative
Finance
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Commodities: Fundamental Theory of Futures, Forwards, and
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Foundations of Quantitative Finance
Book I: Measure Spaces and Measurable Functions
Robert R. Reitano
Introducing Financial Mathematics: Theory, Binomial Models, and
Applications
Mladen Victor Wickerhauser
Foundations of Quantitative Finance
Book II: Probability Spaces and Random Variables
Robert R. Reitano
Financial Mathematics: From Discrete to Continuous Time
Kevin J. Hastings
Financial Mathematics : A Comprehensive Treatment in Discrete Time
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Foundations of Quantitative
Finance
Book II: Probability Spaces and
Random Variables
Robert R. Reitano
Brandeis International Business School
Waltham, MA
First edition published 2023
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Library of Congress Cataloging-in-Publication Data
Names: Reitano, Robert R., 1950- author.
Title: Foundations of quantitative finance. Book II, Probability spaces and
random variables / Robert R. Reitano.
Other titles: Probability spaces and random variables
Description: First edition. | Boca Raton : CRC Press, 2023. | Includes
bibliographical references and index.
Identifiers: LCCN 2022025709 | ISBN 9781032197180 (hardback) | ISBN
9781032197173 (paperback) | ISBN 9781003260547 (ebook)
Subjects: LCSH: Finance--Mathematical models. | Probabilities. | Random
variables.
Classification: LCC HG106 .R448 2023 | DDC 332.01/5195--dc23/eng/20220601
LC record available at https://lccn.loc.gov/2022025709
ISBN: 978-1-032-19718-0 (hbk)
ISBN: 978-1-032-19717-3 (pbk)
ISBN: 978-1-003-26054-7 (ebk)
DOI: 10.1201/9781003260547
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to Dorothy and Domenic
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Contents
Preface xi
Author xiii
Introduction xv
1 Probability Spaces 1
1.1 Probability Theory: A Very Brief History . . . . . . . . . . . . . . . . . . . 1
1.2 A Finite Measure Space with a “Story” . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Bond Loss Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Some Probability Measures on R . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Measures from Discrete Probability Theory . . . . . . . . . . . . . . 11
1.3.2 Measures from Continuous Probability Theory . . . . . . . . . . . . 16
1.3.3 More General Probability Measures on R . . . . . . . . . . . . . . . 20
1.4 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.1 Independent Classes and Associated Sigma Algebras . . . . . . . . . 24
1.5 Conditional Probability Measures . . . . . . . . . . . . . . . . . . . . . . . 27
1.5.1 Law of Total Probability . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.5.2 Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2 Limit Theorems on Measurable Sets 35
2.1 Introduction to Limit Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 The Borel-Cantelli Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3 Kolmogorov’s Zero-One Law . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3 Random Variables and Distribution Functions 47
3.1 Introduction and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1.1 Bond Loss Example (Continued) . . . . . . . . . . . . . . . . . . . . 50
3.2 “Inverse” of a Distribution Function . . . . . . . . . . . . . . . . . . . . . . 52
3.2.1 Properties of F∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.2 The Function F∗∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Random Vectors and Joint Distribution Functions . . . . . . . . . . . . . . 62
3.3.1 Marginal Distribution Functions . . . . . . . . . . . . . . . . . . . . 65
3.3.2 Conditional Distribution Functions . . . . . . . . . . . . . . . . . . . 67
3.4 Independent Random Variables . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4.1 Sigma Algebras Generated by R.V.s . . . . . . . . . . . . . . . . . . 70
3.4.2 Independent Random Variables and Vectors . . . . . . . . . . . . . . 71
3.4.3 Distribution Functions of Independent R.V.s . . . . . . . . . . . . . 74
3.4.4 Independence and Transformations . . . . . . . . . . . . . . . . . . . 75
vii
viii Contents
4 Probability Spaces and i.i.d. RVs 77
4.1 Probability Space (S(cid:48),E(cid:48),µ(cid:48)) and i.i.d. {X }N . . . . . . . . . . . . . . . . 78
j j=1
4.1.1 First Construction: (S(cid:48) ,E(cid:48) ,µ(cid:48) ) . . . . . . . . . . . . . . . . . . . . 79
F F F
4.2 Simulation of Random Variables - Theory . . . . . . . . . . . . . . . . . . . 81
4.2.1 Distributional Results . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2.2 Independence Results . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2.3 Second Construction: (S(cid:48) ,E(cid:48) ,µ(cid:48) ) . . . . . . . . . . . . . . . . . . . 88
U U U
4.3 An Alternate Construction for Discrete Random Variables . . . . . . . . . 91
4.3.1 Third Construction: (S(cid:48),E(cid:48),µ(cid:48)) . . . . . . . . . . . . . . . . . . . . . 93
p p p
5 Limit Theorems for RV Sequences 99
5.1 Two Limit Theorems for Binomial Sequences . . . . . . . . . . . . . . . . . 99
5.1.1 The Weak Law of Large Numbers . . . . . . . . . . . . . . . . . . . 100
5.1.2 The Strong Law of Large Numbers . . . . . . . . . . . . . . . . . . . 103
5.1.3 Strong Laws versus Weak Laws . . . . . . . . . . . . . . . . . . . . . 108
5.2 Convergence of Random Variables 1 . . . . . . . . . . . . . . . . . . . . . . 108
5.2.1 Notions of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2.2 Convergence Relationships . . . . . . . . . . . . . . . . . . . . . . . 111
5.2.3 Slutsky’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2.4 Kolmogorov’s Zero-One Law . . . . . . . . . . . . . . . . . . . . . . 118
6 Distribution Functions and Borel Measures 123
6.1 Distribution Functions on R . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.1.1 Probability Measures from Distribution Functions . . . . . . . . . . 126
6.1.2 Random Variables from Distribution Functions . . . . . . . . . . . . 129
6.2 Distribution Functions on Rn . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.2.1 Probability Measures from Distribution Functions . . . . . . . . . . 131
6.2.2 Random Vectors from Distribution Functions . . . . . . . . . . . . . 135
6.2.3 Marginal and Conditional Distribution Functions . . . . . . . . . . . 136
7 Copulas and Sklar’s Theorem 137
7.1 Fr´echet Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.2 Copulas and Sklar’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.2.1 Identifying Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.3 Partial Results on Sklar’s Theorem . . . . . . . . . . . . . . . . . . . . . . 145
7.4 Examples of Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.4.1 Archimedean Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.4.2 Extreme Value Copulas . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.5 General Result on Sklar’s Theorem . . . . . . . . . . . . . . . . . . . . . . 158
7.5.1 The Distributional Transform . . . . . . . . . . . . . . . . . . . . . . 160
7.5.2 Sklar’s Theorem - The General Case . . . . . . . . . . . . . . . . . . 164
7.6 Tail Dependence and Copulas . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.6.1 Bivariate Tail Dependence . . . . . . . . . . . . . . . . . . . . . . . . 165
7.6.2 Multivariate Tail Dependence and Copulas . . . . . . . . . . . . . . 170
7.6.3 Survival Functions and Copulas . . . . . . . . . . . . . . . . . . . . . 173
8 Weak Convergence 179
8.1 Definitions of Weak Convergence . . . . . . . . . . . . . . . . . . . . . . . . 180
8.2 Properties of Weak Convergence . . . . . . . . . . . . . . . . . . . . . . . . 184
8.3 Weak Convergence and Left Continuous Inverses . . . . . . . . . . . . . . . 189
8.4 Skorokhod’s Representation Theorem . . . . . . . . . . . . . . . . . . . . . 191
Contents ix
8.4.1 Mapping Theorem on R . . . . . . . . . . . . . . . . . . . . . . . . . 192
8.5 Convergence of Random Variables 2 . . . . . . . . . . . . . . . . . . . . . . 194
8.5.1 Mann-Wald Theorem on R . . . . . . . . . . . . . . . . . . . . . . . 194
8.5.2 The Delta-Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
9 Estimating Tail Events 1 201
9.1 Large Deviation Theory 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
9.2 Extreme Value Theory 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
9.2.1 Introduction and Examples . . . . . . . . . . . . . . . . . . . . . . . 206
9.2.2 Extreme Value Distributions . . . . . . . . . . . . . . . . . . . . . . 210
9.2.3 The Fisher-Tippett-Gnedenko Theorem . . . . . . . . . . . . . . . . 212
9.3 The Pickands-Balkema-de Haan Theorem . . . . . . . . . . . . . . . . . . . 223
9.3.1 Quantile Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
9.3.2 Tail Probability Estimation . . . . . . . . . . . . . . . . . . . . . . . 224
9.4 γ in Theory: von Mises’ Condition . . . . . . . . . . . . . . . . . . . . . . . 229
9.5 Independence vs. Tail Independence . . . . . . . . . . . . . . . . . . . . . . 234
9.6 Multivariate Extreme Value Theory . . . . . . . . . . . . . . . . . . . . . . 235
9.6.1 Multivariate Fisher-Tippett-Gnedenko Theorem . . . . . . . . . . . 236
9.6.2 The Extreme Value Distribution G . . . . . . . . . . . . . . . . . . . 238
9.6.3 The Extreme Value Copula C . . . . . . . . . . . . . . . . . . . . . 241
G
References 249
Index 253