Introduction to Financial Mathematics

Kevin J. Hastings is emeritus professor of mathematics; Rothwell C. Stephens Distinguished Service Chair at Knox College. He holds a PhD from Northwestern University. His interests include applications to real-world problems affected by random inputs or disturbances. He is the author of Chapman & Hall/CRC books: Financial Mathematics: From Discrete to Continuous Time, 2022; Introduction to Probability with Mathematica®, 2nd ed., 2019; and Introduction to the Mathematics of Operations Research with Mathematica®, 2nd ed., 2019.

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 Additional Topics 335

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