Financial Mathematics From Discrete to Continuous Time

Chapter 1 - Review of Preliminaries

1.1. Risky Assets

1.1.1. Single and Multiple Discrete Time Periods

1.1.2. Continuous-Time Processes

1.1.3. Martingales

1.2. Risk Aversion and Portfolios of Assets

1.2.1. Risk Aversion Constant

1.2.2. The Portfolio Problem

1.3. Expectation, Variance, and Covariance

1.3.1. One Variable Expectation

1.3.2. Expectation for Multiple Random Variables

1.3.3. Variance of a Linear Combination

1.4. Simple Portfolio Optimization

1.5. Derivative Assets and Arbitrage

1.5.1. Futures

1.5.2. Arbitrage and Futures

1.5.3. Options

1.6. Valuation of Derivatives in Single Time Period

1.6.1. Replicating Portfolios

1.6.2. Risk-Neutral Valuation

2.1. Portfolio Optimization with Multiple Assets

2.1.1. Lagrange Multipliers

2.1.1. Qualitative Behavior

2.1.3. Correlated Assets

2.1.4. Portfolio Separation and the Market Portfolio

2.2. Capital Market Theory, Part I

2.2.1. Linear Algebraic Approach

2.2.2. Efficient Mean-Standard Deviation Frontier

2.3. Capital Market Theory, Part II

2.3.1. Capital Market Line

2.3.2. CAPM Formula; Asset β

2.3.3. Systematic and Non-Systematic Risk; Pricing Using CAPM

2.4. Utility Theory

2.4.1. Securities and Axioms for Investor Behavior

2.4.2. Indifference Curves, Certainty Equivalent, Risk Aversion

2.4.3. Examples of Utility Functions

2.4.4. Absolute and Relative Risk Aversion

2.4.5. Utility Maximization

2.5. Multiple Period Portfolio Problems

2.5.1. Problem Description and Dynamic Programming Approach

2.5.2. Examples

2.5.3. Optimal Portfolios and Martingales

3.1. Options Pricing for Multiple Time Periods

3.1.1. Introduction

3.1.2. Valuation by Chaining

3.1.3. Valuation by Martingales

3.2. Key Ideas of Discrete Probability, Part I

3.2.1. Algebras and Measurability

3.2.2. Independence

3.3. Key Ideas of Discrete Probability, Part II

3.3.1. Conditional Expectation

3.3.2. Application to Pricing Models

3.4. Fundamental Theorems of Options Pricing

3.4.1. The Market Model

3.4.2. Gain, Arbitrage, and Attainability

3.4.3. Martingale Measures and the Fundamental Theorems

3.5. Valuation of Non-Vanilla Options

3.5.1. American and Bermudan Options

3.5.2. Barrier Options

3.5.3. Asian Options

3.5.4. Two-Asset Options

3.6. Derivatives Pricing by Simulation

3.6.1. Setup and Algorithm

3.6.2. Examples

3.7. From Discrete to Continuous-Time (A Preview)

4.1. Continuous Distributions and Expectation

4.1.1. Densities and Cumulative Distribution Functions

4.1.2. Expectation

4.1.3. Normal and Lognormal Distributions

4.2. Joint Distributions

4.2.1 Basic Ideas

4.2.2. Marginal and Conditional Distributions

4.2.3. Independence

4.2.4. Covariance and Correlation

4,2,5 Bivariate Normal Distribution

4.3. Measurability and Conditional Expectation

4.3.1. Sigma Algebras

4.3.2. Random Variables and Measurability

4.3.3. Continuous Conditional Expectation

4.4. Brownian Motion and Geometric Brownian Motion

4.4.1. Random Walk Processes

4.4.2. Standard Brownian Motion

4.4.3. Non-standard Brownian motion

4.4.4. Geometric Brownian Motion

4.4.5. Brownian Motion and Binomial Branch Processes

4.5. Introduction to Stochastic Differential Equations

4.5.1. Meaning of the General SDE

4.5.2. Ito's Formula

4.5.3. Geometric Brownian Motion as Solution

5.1. Black-Scholes Via Limits

5.1.1. Black-Scholes Formula

5.1.2. Limiting Approach

5.1.3. Put Options and Put-Call Parity

5.2. Black-Scholes via Martingales

5.2.2. Fundamental Theorems of Asset Pricing

5.2.2. Trading Strategies and Martingale Valuation

5.2.3. Martingale Measures

5.2.4. Asset and Bond Binaries

5.2.5. Other Binary Derivatives

5.3. Black-Scholes via Differential Equations

5.3.1. Deriving the PDE

5.3.2. Boundary Conditions; Solving the PDE

5.4. Checking Black-Scholes Assumptions

5.4.1. Normality of Rates of Return

5.4.2. Stability of Parameters

5.4.3. Independence of Rates of Return

A.1 Review of Matrix Concepts

A,2.Multivariate Normal Distribution

Biography

Kevin J. Hastings is Professor of Mathematics; Rothwell C. Stephens Distinguished Service Chair at Knox College. He holds a Ph.D. from Northwestern University. His interests include applications to real-world problems affected by random inputs or disturbances. He is the author or three other books for CRC Press:

Introduction to Financial Mathematics, CRC Press, 2016. CHOICE Highly Recommended selection and 2017 Top Books for Colleges.

Introduction to Probability with Mathematica, 2nd ed., Chapman & Hall/CRC Press, 2009.

Introduction to the Mathematics of Operations Research with Mathematica, 2nd edition, Taylor & Francis/Marcel Dekker, 2006.

Introduction to Probability with Mathematica. CRC Press/Chapman & Hall, 2000. Also available as an e-book.

I like Kevin Hastings' "Introduction to Financial Mathematics" (Volume 1) very much.

The book is very readable; it builds slowly with many examples and
exercises (and answers to some of the exercises are in the back).
The writing style is good; the exercises are easy to understand.　


The material is comprehensive and covers the topics well.
It is surprising that the book maintains the same clear level of exposition
from the simple early chapters to the more complicated later chapters.

The table of contents covers all the material that should appear in a
financial mathematics course.

Dan Zwillinger

　

In addition to its clear explanations, this volume emphasizes real problem solving with examples and exercises that challenge students to apply knowledge of basic concepts to new situations. Another unique aspect is the application of discrete probability to finance; the author provides an overview and illustrates problems in which the rates of interest are random variables, instead of traditional problems that consider only known constants. Topics covered include the mathematics of interest, valuation of bonds, discrete probability for finance, portfolio selection, and derivatives.

This book is highly recommended for undergraduates and those preparing for actuarial credentialing and exams.

S. J. Chapman Jr.,

Purdue University-NorthWest