Every financial professional wants and needs an advantage. A firm foundation in advanced mathematics can translate into dramatic advantages to professionals willing to obtain it. Many are not―and that is the advantage these books offer the astute reader.
Published under the collective title of Foundations of Quantitative Finance, this set of ten books presents the advanced mathematics finance professionals need to advantage their careers, these books present the theory most do not learn in graduate finance programs, or in most financial mathematics undergraduate and graduate courses.
As a high-level industry executive and authoritative instructor, Robert R. Reitano presents the mathematical theories he encountered in nearly three decades working in the financial industry and two decades teaching in highly respected graduate programs.
Readers should be quantitatively literate and familiar with the developments in the first book in the set, Foundations of Quantitative Finance Book I: Measure Spaces and Measurable Functions.
Author(s): Robert R. Reitano
Series: Chapman & Hall/CRC Finance Series
Publisher: CRC Press/Chapman & Hall
Year: 2022
Language: English
Pages: 275
City: Boca Raton
Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
Author
Introduction
1. Probability Spaces
1.1. Probability Theory: A Very Brief History
1.2. A Finite Measure Space with a “Story”
1.2.1. Bond Loss Example
1.3. Some Probability Measures on R
1.3.1. Measures from Discrete Probability Theory
1.3.2. Measures from Continuous Probability Theory
1.3.3. More General Probability Measures on R
1.4. Independent Events
1.4.1. Independent Classes and Associated Sigma Algebras
1.5. Conditional Probability Measures
1.5.1. Law of Total Probability
1.5.2. Bayes' Theorem
2. Limit Theorems on Measurable Sets
2.1. Introduction to Limit Sets
2.2. The Borel-Cantelli Lemma
2.3. Kolmogorov's Zero-One Law
3. Random Variables and Distribution Functions
3.1. Introduction and Definitions
3.1.1. Bond Loss Example (Continued)
3.2. “Inverse” of a Distribution Function
3.2.1. Properties of F*
3.2.2. The Function F**
3.3. Random Vectors and Joint Distribution Functions
3.3.1. Marginal Distribution Functions
3.3.2. Conditional Distribution Functions
3.4. Independent Random Variables
3.4.1. Sigma Algebras Generated by R.V.s
3.4.2. Independent Random Variables and Vectors
3.4.3. Distribution Functions of Independent R.V.s
3.4.4. Independence and Transformations
4. Probability Spaces and i.i.d. RVs
4.1. Probability Space (S',E',µ') and i.i.d. {Xj}Nj=1
4.1.1. First Construction: (S′F, E′F, µ′F)
4.2. Simulation of Random Variables - Theory
4.2.1. Distributional Results
4.2.2. Independence Results
4.2.3. Second Construction: (S′U, E′U, µ′U)
4.3. An Alternate Construction for Discrete Random Variables
4.3.1. Third Construction: (S′p, E′p, µ′p)
5. Limit Theorems for RV Sequences
5.1. Two Limit Theorems for Binomial Sequences
5.1.1. The Weak Law of Large Numbers
5.1.2. The Strong Law of Large Numbers
5.1.3. Strong Laws versus Weak Laws
5.2. Convergence of Random Variables 1
5.2.1. Notions of Convergence
5.2.2. Convergence Relationships
5.2.3. Slutsky's Theorem
5.2.4. Kolmogorov's Zero-One Law
6. Distribution Functions and Borel Measures
6.1. Distribution Functions on R
6.1.1. Probability Measures from Distribution Functions
6.1.2. Random Variables from Distribution Functions
6.2. Distribution Functions on Rn
6.2.1. Probability Measures from Distribution Functions
6.2.2. Random Vectors from Distribution Functions
6.2.3. Marginal and Conditional Distribution Functions
7. Copulas and Sklar's Theorem
7.1. Fréchet Classes
7.2. Copulas and Sklar's Theorem
7.2.1. Identifying Copulas
7.3. Partial Results on Sklar's Theorem
7.4. Examples of Copulas
7.4.1. Archimedean Copulas
7.4.2. Extreme Value Copulas
7.5. General Result on Sklar's Theorem
7.5.1. The Distributional Transform
7.5.2. Sklar's Theorem - The General Case
7.6. Tail Dependence and Copulas
7.6.1. Bivariate Tail Dependence
7.6.2. Multivariate Tail Dependence and Copulas
7.6.3. Survival Functions and Copulas
8. Weak Convergence
8.1. Definitions of Weak Convergence
8.2. Properties of Weak Convergence
8.3. Weak Convergence and Left Continuous Inverses
8.4. Skorokhod's Representation Theorem
8.4.1. Mapping Theorem on R
8.5. Convergence of Random Variables 2
8.5.1. Mann-Wald Theorem on R
8.5.2. The Delta-Method
9. Estimating Tail Events 1
9.1. Large Deviation Theory 1
9.2. Extreme Value Theory 1
9.2.1. Introduction and Examples
9.2.2. Extreme Value Distributions
9.2.3. The Fisher-Tippett-Gnedenko Theorem
9.3. The Pickands-Balkema-de Haan Theorem
9.3.1. Quantile Estimation
9.3.2. Tail Probability Estimation
9.4. γ in Theory: von Mises' Condition
9.5. Independence vs. Tail Independence
9.6. Multivariate Extreme Value Theory
9.6.1. Multivariate Fisher-Tippett-Gnedenko Theorem
9.6.2. The Extreme Value Distribution G
9.6.3. The Extreme Value Copula CG
References
Index
