Foundations of Quantitative Finance, Book VI: Densities, Transformed Distributions, and Limit Theorems
Densities, Transformed Distributions, and Limit Theorems
1st Edition
Published on 12. November 2024
Book
Paperback/Softback
386 pages
978-1-032-22949-2 (ISBN)
Description
Every finance professional wants and needs a competitive edge. A firm foundation in advanced mathematics can translate into dramatic advantages to professionals willing to obtain it. Many are not-and that is the competitive edge these books offer the astute reader.
Published under the collective title of Foundations of Quantitative Finance, this set of ten books develops the advanced topics in mathematics that finance professionals need to advance their careers. These books expand the theory most do not learn in graduate finance programs, or in most financial mathematics undergraduate and graduate courses.
As an investment executive and authoritative instructor, Robert R. Reitano presents the mathematical theories he encountered and used in nearly three decades in the financial services industry and two decades in academia where he taught in highly respected graduate programs.
Readers should be quantitatively literate and familiar with the developments in the earlier books in the set. While the set offers a continuous progression through these topics, each title can be studied independently.
Features
Extensively referenced to materials from earlier books
Presents the theory needed to support advanced applications
Supplements previous training in mathematics, with more detailed developments
Built from the author's five decades of experience in industry, research, and teaching
Published and forthcoming titles in the Robert R. Reitano Quantitative Finance Series:
Book I: Measure Spaces and Measurable Functions
Book II: Probability Spaces and Random Variables
Book III: The Integrals of Riemann, Lebesgue and (Riemann-)Stieltjes
Book IV: Distribution Functions and Expectations
Book V: General Measure and Integration Theory
Book VI: Densities, Transformed Distributions, and Limit Theorems
Book VII: Brownian Motion and Other Stochastic Processes
Book VIII: Ito Integration and Stochastic Calculus 1
Book IX: Stochastic Calculus 2 and Stochastic Differential Equations
Book X: Classical Models and Applications in Finance
Published under the collective title of Foundations of Quantitative Finance, this set of ten books develops the advanced topics in mathematics that finance professionals need to advance their careers. These books expand the theory most do not learn in graduate finance programs, or in most financial mathematics undergraduate and graduate courses.
As an investment executive and authoritative instructor, Robert R. Reitano presents the mathematical theories he encountered and used in nearly three decades in the financial services industry and two decades in academia where he taught in highly respected graduate programs.
Readers should be quantitatively literate and familiar with the developments in the earlier books in the set. While the set offers a continuous progression through these topics, each title can be studied independently.
Features
Extensively referenced to materials from earlier books
Presents the theory needed to support advanced applications
Supplements previous training in mathematics, with more detailed developments
Built from the author's five decades of experience in industry, research, and teaching
Published and forthcoming titles in the Robert R. Reitano Quantitative Finance Series:
Book I: Measure Spaces and Measurable Functions
Book II: Probability Spaces and Random Variables
Book III: The Integrals of Riemann, Lebesgue and (Riemann-)Stieltjes
Book IV: Distribution Functions and Expectations
Book V: General Measure and Integration Theory
Book VI: Densities, Transformed Distributions, and Limit Theorems
Book VII: Brownian Motion and Other Stochastic Processes
Book VIII: Ito Integration and Stochastic Calculus 1
Book IX: Stochastic Calculus 2 and Stochastic Differential Equations
Book X: Classical Models and Applications in Finance
More details
Series
Language
English
Place of publication
Boca Raton
United Kingdom
Publishing group
Taylor & Francis Ltd
Target group
College/higher education
Professional and scholarly
Postgraduate and Professional
Dimensions
Height: 254 mm
Width: 178 mm
Thickness: 22 mm
Weight
761 gr
ISBN-13
978-1-032-22949-2 (9781032229492)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Other editions
Additional editions
Robert R. Reitano
Foundations of Quantitative Finance, Book VI: Densities, Transformed Distributions, and Limit Theorems
E-Book
11/2024
1st Edition
Chapman and Hall
€104.99
Available for download
Robert R. Reitano
Foundations of Quantitative Finance, Book VI: Densities, Transformed Distributions, and Limit Theorems
E-Book
11/2024
1st Edition
Chapman and Hall
€104.99
Available for download
Robert R. Reitano
Foundations of Quantitative Finance, Book VI: Densities, Transformed Distributions, and Limit Theorems
Densities, Transformed Distributions, and Limit Theorems
Book
11/2024
1st Edition
Chapman & Hall/CRC
€270.00
Shipment within 10-20 days
Person
Robert R. Reitano is Professor of the Practice of Finance at the Brandeis International Business School where he specializes in risk management and quantitative finance, and where he previously served as MSF Program Director, and Senior Academic Director. He has a Ph.D. in Mathematics from MIT, is a Fellow of the Society of Actuaries, and a Chartered Enterprise Risk Analyst. He has taught as Visiting Professor at Wuhan University of Technology School of Economics, Reykjavik University School of Business, and as Adjunct Professor in Boston University's Masters Degree program in Mathematical Finance. Dr. Reitano consults in investment strategy and asset/liability risk management and previously had a 29-year career at John Hancock/Manulife in investment strategy and asset/liability management, advancing to Executive Vice President & Chief Investment Strategist. His research papers have appeared in a number of journals and have won an Annual Prize of the Society of Actuaries and two F.M.
Content
1 Density Functions
1.1 Density Functions of Measures
1.2 Density Functions of Distributions
1.2.1 Distribution Functions and Random Vectors
1.2.2 Distribution Functions and Probability Measures
1.2.3 Existence of Density Functions
1.3 Marginal Density Functions
1.4 Densities and Independent RVs
1.5 Conditional Density Functions
2 Transformations of Random Vectors
2.1 Cavalieri.s Principle
2.2 Sums of Independent Random Vectors
2.2.1 Distribution Functions
2.2.2 Density Functions
2.3 A Result on Convolutions
2.4 Ratios of Independent Random Variables
2.5 Densities of Transformed Random Vectors
3 Weak Convergence of Probability Measures
3.1 Portmanteau Theorem on R
3.2 Portmanteau Theorem on Rm
3.3 Applications
3.3.1 The Mapping Theorem
3.3.2 Mann-Wald Theorem
3.3.3 Cramer-Wold Theorem - Part 1
3.3.4 Slutsky.s Theorem
3.3.5 The Delta Method
3.3.6 Sche?e.s Theorem
3.3.7 Prokhorov.s theorem
4 Expectations of Random Variables 2
4.1 Expectations and Moments
4.1.1 Expectations of Independent RV Products
4.1.2 Moments and the MGF
4.1.3 Properties of Moments
4.2 Weak Convergence and Moment Limits
4.3 Conditional Expectations
4.3.1 Conditional Probability Measures
4.3.2 Conditional Expectation -An Introduction
4.3.3 Conditional Expectation as a Function
4.3.4 Existence of Conditional Expectation
4.4 Properties of Conditional Expectations
4.4.1 Fundamental Properties
4.4.2 Conditional Jensen.s Inequality
4.4.3 Lp(S)-Space Properties
4.5 Conditional Expectations in the Limit
4.5.1 Conditional Monotone Convergence
4.5.2 Conditional Fatou.s Lemma
4.5.3 Conditional Dominated Convergence
5 The Characteristic Function
5.1 The Moment Generating Function
5.2 Integration of Complex-Valued Functions
5.3 The Characteristic Function
5.4 Examples of Characteristic Functions
5.4.1 Discrete Distributions
5.4.2 Continuous Distributions
5.5 Properties of Characteristic Functions on R
5.6 Properties of Characteristic Functions on Rn
5.6.1 The Cramer-Wold Theorem
5.7 Bochner.s Theorem
5.7.1 Positive Semide.nite Functions
5.7.2 Bochner.s Theorem
5.8 A Uniqueness of Moments Result
6 Multivariate Normal Distribution
6.1 Derivation and De.nition
6.1.1 Density Function Approach
6.1.2 Characteristic Function Approach
6.1.3 Multivariate Normal De.nition
6.2 Existence of Densities
6.3 The Cholesky Decomposition
6.4 Properties of Multivariate Normal
6.4.1 Higher Moments
6.4.2 Independent vs. Uncorrelated Normals
6.4.3 Sample Mean and Variance
7 Applications of Characteristic Functions
7.1 Central Limit Theorems
7.1.1 The Classical Central Limit Theorem
7.1.2 Lindeberg.s Central Limit Theorem
7.1.3 Lyapunov.s Central Limit Theorem
7.1.4 A Central Limit Theorem on Rn
7.2 Distribution Families Related Under Addition
7.2.1 Discrete Distributions
7.2.2 Continuous Distributions
7.3 In.nitely Divisible Distributions
7.3.1 De Finetti.s Theorem
7.4 Distribution Families Related Under Multiplication
8 Discrete Time Asset Models in Finance
8.1 Models of Asset Prices
8.1.1 Additive Temporal Models
8.1.2 Multiplicative Temporal Models
8.1.3 Simulating Asset Price Paths
8.2 Scalable Asset Models
8.2.1 Properties of Scalable Models
8.2.2 Scalable Additive Models
8.2.3 Scalable Multiplicative Models
8.3 Limiting Distributions of Scalable Models
8.3.1 Scalable Additive Models
8.3.2 Scalable Multiplicative Models
9 Pricing of Financial Derivatives
9.1 Binomial Lattice Pricing
9.1.1 European Derivatives
9.1.2 American Options
9.2 Limiting Risk Neutral Asset Distribution
9.2.1 Analysis of the Probability q(_t)
9.2.2 Limiting Asset Distribution Under q (_t)
9.3 A Real World Model Under p (_t)
9.4 Limiting Price of European Derivatives
9.4.1 Black-Scholes-Merton Option Pricing
9.5 Properties of Black-Scholes-Merton Prices
9.5.1 Price Convergence to Payo?
9.5.2 Put-Call Parity
9.5.3 Black-Scholes-Merton PDE
9.5.4 Lattice Approximations for "Greeks"
9.6 Limiting Price of American Derivatives
9.7 Path Dependent Derivatives
9.7.1 Path-Based Pricing of European Derivatives
9.7.2 Lattice Pricing of European PD Derivatives
9.7.3 Lattice Pricing of American PD Derivatives
9.7.4 Monte Carlo Pricing of European PD Derivatives
9.8 Lognormal Pricing Model
9.8.1 European Financial Derivatives
9.8.2 European PD Financial Derivatives
10 Limits of Binomial Motion
10.1 Binomial Paths
10.2 Uniform Limits of Bt(_t)
10.3 Distributional Limits of Bt(_t)
10.4 Nonstandard Binomial Motion
10.4.1 Nonstandard Binomial Motion with p 6= 1=2 Fixed
10.4.2 Nonstandard Binomial Motion with p = q (_t)
10.5 Limits of Binomial Asset Models
References
1.1 Density Functions of Measures
1.2 Density Functions of Distributions
1.2.1 Distribution Functions and Random Vectors
1.2.2 Distribution Functions and Probability Measures
1.2.3 Existence of Density Functions
1.3 Marginal Density Functions
1.4 Densities and Independent RVs
1.5 Conditional Density Functions
2 Transformations of Random Vectors
2.1 Cavalieri.s Principle
2.2 Sums of Independent Random Vectors
2.2.1 Distribution Functions
2.2.2 Density Functions
2.3 A Result on Convolutions
2.4 Ratios of Independent Random Variables
2.5 Densities of Transformed Random Vectors
3 Weak Convergence of Probability Measures
3.1 Portmanteau Theorem on R
3.2 Portmanteau Theorem on Rm
3.3 Applications
3.3.1 The Mapping Theorem
3.3.2 Mann-Wald Theorem
3.3.3 Cramer-Wold Theorem - Part 1
3.3.4 Slutsky.s Theorem
3.3.5 The Delta Method
3.3.6 Sche?e.s Theorem
3.3.7 Prokhorov.s theorem
4 Expectations of Random Variables 2
4.1 Expectations and Moments
4.1.1 Expectations of Independent RV Products
4.1.2 Moments and the MGF
4.1.3 Properties of Moments
4.2 Weak Convergence and Moment Limits
4.3 Conditional Expectations
4.3.1 Conditional Probability Measures
4.3.2 Conditional Expectation -An Introduction
4.3.3 Conditional Expectation as a Function
4.3.4 Existence of Conditional Expectation
4.4 Properties of Conditional Expectations
4.4.1 Fundamental Properties
4.4.2 Conditional Jensen.s Inequality
4.4.3 Lp(S)-Space Properties
4.5 Conditional Expectations in the Limit
4.5.1 Conditional Monotone Convergence
4.5.2 Conditional Fatou.s Lemma
4.5.3 Conditional Dominated Convergence
5 The Characteristic Function
5.1 The Moment Generating Function
5.2 Integration of Complex-Valued Functions
5.3 The Characteristic Function
5.4 Examples of Characteristic Functions
5.4.1 Discrete Distributions
5.4.2 Continuous Distributions
5.5 Properties of Characteristic Functions on R
5.6 Properties of Characteristic Functions on Rn
5.6.1 The Cramer-Wold Theorem
5.7 Bochner.s Theorem
5.7.1 Positive Semide.nite Functions
5.7.2 Bochner.s Theorem
5.8 A Uniqueness of Moments Result
6 Multivariate Normal Distribution
6.1 Derivation and De.nition
6.1.1 Density Function Approach
6.1.2 Characteristic Function Approach
6.1.3 Multivariate Normal De.nition
6.2 Existence of Densities
6.3 The Cholesky Decomposition
6.4 Properties of Multivariate Normal
6.4.1 Higher Moments
6.4.2 Independent vs. Uncorrelated Normals
6.4.3 Sample Mean and Variance
7 Applications of Characteristic Functions
7.1 Central Limit Theorems
7.1.1 The Classical Central Limit Theorem
7.1.2 Lindeberg.s Central Limit Theorem
7.1.3 Lyapunov.s Central Limit Theorem
7.1.4 A Central Limit Theorem on Rn
7.2 Distribution Families Related Under Addition
7.2.1 Discrete Distributions
7.2.2 Continuous Distributions
7.3 In.nitely Divisible Distributions
7.3.1 De Finetti.s Theorem
7.4 Distribution Families Related Under Multiplication
8 Discrete Time Asset Models in Finance
8.1 Models of Asset Prices
8.1.1 Additive Temporal Models
8.1.2 Multiplicative Temporal Models
8.1.3 Simulating Asset Price Paths
8.2 Scalable Asset Models
8.2.1 Properties of Scalable Models
8.2.2 Scalable Additive Models
8.2.3 Scalable Multiplicative Models
8.3 Limiting Distributions of Scalable Models
8.3.1 Scalable Additive Models
8.3.2 Scalable Multiplicative Models
9 Pricing of Financial Derivatives
9.1 Binomial Lattice Pricing
9.1.1 European Derivatives
9.1.2 American Options
9.2 Limiting Risk Neutral Asset Distribution
9.2.1 Analysis of the Probability q(_t)
9.2.2 Limiting Asset Distribution Under q (_t)
9.3 A Real World Model Under p (_t)
9.4 Limiting Price of European Derivatives
9.4.1 Black-Scholes-Merton Option Pricing
9.5 Properties of Black-Scholes-Merton Prices
9.5.1 Price Convergence to Payo?
9.5.2 Put-Call Parity
9.5.3 Black-Scholes-Merton PDE
9.5.4 Lattice Approximations for "Greeks"
9.6 Limiting Price of American Derivatives
9.7 Path Dependent Derivatives
9.7.1 Path-Based Pricing of European Derivatives
9.7.2 Lattice Pricing of European PD Derivatives
9.7.3 Lattice Pricing of American PD Derivatives
9.7.4 Monte Carlo Pricing of European PD Derivatives
9.8 Lognormal Pricing Model
9.8.1 European Financial Derivatives
9.8.2 European PD Financial Derivatives
10 Limits of Binomial Motion
10.1 Binomial Paths
10.2 Uniform Limits of Bt(_t)
10.3 Distributional Limits of Bt(_t)
10.4 Nonstandard Binomial Motion
10.4.1 Nonstandard Binomial Motion with p 6= 1=2 Fixed
10.4.2 Nonstandard Binomial Motion with p = q (_t)
10.5 Limits of Binomial Asset Models
References