1st Edition

# Foundations of Quantitative Finance, Book VI: Densities, Transformed Distributions, and Limit Theorems

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 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: Itô Integration and Stochastic Calculus 1**

**Book IX: Stochastic Calculus 2 and Stochastic Differential Equations**

**Book X: Classical Models and Applications in Finance**

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 Cramér-Wold Theorem - Part 1

3.3.4 Slutsky.s Theorem

3.3.5 The Delta Method

3.3.6 Sche¤é.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 Cramér-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

### Biography

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.