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Financial Mathematics

From Discrete to Continuous Time

## Preview

## Book Description

**Financial Mathematics: From Discrete to Continuous Time **is a study of the mathematical ideas and techniques that are important to the two main arms of the area of financial mathematics: portfolio optimization and derivative valuation. The text is authored for courses taken by advanced undergraduates, MBA, or other students in quantitative finance programs.

The approach will be mathematically correct but informal, sometimes omitting proofs of the more difficult results and stressing practical results and interpretation. The text will not be dependent on any particular technology, but it will be laced with examples requiring the numerical and graphical power of the machine.

The text illustrates simulation techniques to stand in for analytical techniques when the latter are impractical. There will be an electronic version of the text that integrates Mathematica functionality into the development, making full use of the computational and simulation tools that this program provides. Prerequisites are good courses in mathematical probability, acquaintance with statistical estimation, and a grounding in matrix algebra.

The highlights of the text are:

- A thorough presentation of the problem of portfolio optimization, leading in a natural way to the Capital Market Theory
- Dynamic programming and the optimal portfolio selection-consumption problem through time
- An intuitive approach to Brownian motion and stochastic integral models for continuous time problems
- The Black-Scholes equation for simple European option values, derived in several different ways
- A chapter on several types of exotic options
- Material on the management of risk in several contexts

## Table of Contents

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

Chapter 2 - More on Portfolio Optimization; Capital Market Theory

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

Chapter 3 - Derivatives Valuation in Multiple Periods

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)

Chapter 4 - Continuous Probability Models

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

Chapter 5 - Derivative Valuation in Continuous Time

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

Appendices

A. Multivariate Normal Distribution

A.1 Review of Matrix Concepts

A,2.Multivariate Normal Distribution

B. Answers to Selected Exercises

## Author(s)

### 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.

## Reviews

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