2nd Edition

# Introduction to Financial Mathematics

By Kevin J. Hastings Copyright 2025
416 Pages 73 B/W Illustrations
by Chapman & Hall

The second edition of this successful and widely recognized textbook again focuses on discrete topics. The author recognizes two distinct paths of study and careers of actuarial science and financial engineering. This text can be very useful as a common core for both.

Therefore, there is substantial material on the theory of interest (the first half of the book), as well as the probabilistic background necessary for the study of portfolio optimization and derivative valuation (the second half).

The material in the first two chapters should go a long way toward helping students prepare for the Financial Mathematics (FM) actuarial exam. Also, the discrete material will reveal how beneficial it is to know more about loans in student’s personal financial lives.

The notable changes and updates to this edition are itemized in the Preface, however, overall, the presentation has been made more efficient. One example is the chapter on discrete probability, rather unique in its emphasis on giving the deterministic problems studied earlier a probabilistic context.

Probably is now a subsection on Markov chains. Sample spaces and probability measures, random variables and distributions, expectation, conditional probability, independence, and estimation all follow.

Optimal portfolio selection coverage is reorganized and the section on the practicalities of stock transactions has been revised. Market portfolio, and Capital Market Theory coverage is expanded.

This book, like the first edition, was written so that the print edition could stand alone.  At times we simplify complicated algebraic expressions, or solve systems of linear equations, or numerically solve non-linear equations. Also, some attention is given to the use of computer simulation to approximate solutions to problems. A course in multivariable calculus is not required.

The entire text is available digitally from the publisher in the form of a series of Mathematica notebooks, which can be loaded into Mathematica, and which include complete executable commands and programs, and some additional material.

Contents

Preface xi

1 Theory of Interest

1.1 Rate of Return and Present Value

1.2 Compound Interest

1.2.1 Geometric Sequences and Series

1.2.2 Compound Interest

1.2.3 Discounting

1.2.4 Present Value and Net Present Value

1.3 Annuities

1.3.1 Ordinary Annuities

1.3.2 Annuities Due

1.3.3 Variations on Annuities

1.4 Loans

1.4.1 Loan Payments

1.4.2 Loan Amortization

1.4.3 Retrospective and Prospective Forms for Outstanding

Balance

1.4.4 Sinking Fund Loan Repayment

1.5 Measuring Rate of Return

1.5.1 Internal Rate of Return on a Transaction

1.5.2 Approximate Dollar-Weighted Rate of Return

1.5.3 Time-Weighted Rate of Return

1.6 Continuous Time Interest Theory

1.6.1 Continuous Compounding: Effective Rate and Present

Value

1.6.2 Force of Interest

1.6.3 Continuous Annuities

1.6.4 Continuous Loans3

2 Bonds

2.1 Bond Valuation

2.1.1 Bond Value at Issue Date

2.1.2 Bond Value at Coupon Date

2.1.3 Recursive Approach: Bond Amortization Table

2.2 More on Bonds

2.2.1 Value of a Bond between Coupons

2.2.2 Callable and Putable Bonds

2.2.3 Bond Duration

2.3 Term Structure of Interest Rates

2.3.1 Spot Rates, STRIPS, and Yield to Maturity

2.3.2 Forward Rates and Spot Rates

3 Discrete Probability for Finance

3.1 Sample Spaces and Probability Measures

3.1.1 Counting Rules

3.1.2 Probability Models

3.1.3 More Properties of Probability

3.2 Random Variables and Distributions

3.2.1 Cumulative Distribution Functions

3.2.2 Random Vectors and Joint Distributions

3.3 Discrete Expectation

3.3.1 Mean

3.3.2 Variance

3.3.3 Chebyshev’s Inequality

3.3.4 Expectation for Multiple Random Variables

3.4 Conditional Probability

3.4.1 Fundamental Ideas

3.4.2 Conditional Distributions of Random Variables

3.4.3 Conditional Expectation

3.5 Independence and Dependence

3.5.1 Independent Events

3.5.2 Independent Random Variables

3.5.3 Dependence: Covariance and Correlation

3.6 Estimation

3.6.1 The Sample Mean

3.6.2 Sample Variance, Covariance, and Correlation

4 Portfolio Theory

4.1 Portfolios of Risky Assets

4.1.1 Some Practical Background

4.1.2 Stock Transactions

4.1.3 Asset Rates of Return: Modeling and Estimation

4.1.4 Portfolio Rate of Return

4.1.5 Risk Aversion

4.2 Optimal Portfolio Selection

4.2.1 Two-Asset Problems

4.3 Multiple-Assets and Portfolio Separation

4.3.1 Market Portfolio

5 Valuation of Derivatives

5.1 Basic Terminology and Ideas

5.1.1 Derivative Assets

5.1.2 Arbitrage

5.1.3 Arbitrage Valuation of Futures

5.2 Single-Period Options

5.2.1 Pricing Strategies

5.2.2 Put-Call Parity

5.2.3 Δ-Hedging

5.3 Multiple-Period Options

5.3.1 Martingale Valuation

5.3.2 Valuation by Chaining

6.1 Valuation of Exotic Options and Simulation

6.1.1 American Options

6.1.2 Barrier Options

6.1.3 Asian Options

6.1.4 Approximate Valuation by Simulation

6.2 Swaps

6.2.1 Interest Rate Swaps

6.2.2 Commodity Swaps

6.2.3 Currency Swaps

6.3 Value-at-Risk

6.3.1 Computing VaR for Individual Assets and Portfolios

6.3.2 Conditional Value-at-Risk

6.3.3 Simulation Approximations

Appendix A Short Answers to Selected Exercises

Bibliography

Index

### Biography

Kevin 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 of three other books for CRC Press: Introduction to Probability with Mathematica, 2nd ed., Chapman and Hall/CRC Press, 2009; Introduction to the Mathematics of Operations Research with Mathematica, 2nd edition, Taylor & Francis 2006; and Introduction to Probability with Mathematica, CRC Press/Chapman and Hall, 2000.