Undergraduate Introduction To Financial Mathematics, An (Fourth Edition)

by J Robert Buchanan

Overview

Anyone with an interest in learning about the mathematical modeling of prices of financial derivatives such as bonds, futures, and options can start with this book, whereby the only mathematical prerequisite is multivariable calculus. The necessary theory of interest, statistical, stochastic, and differential equations are developed in their respective chapters, with the goal of making this introductory text as self-contained as possible.In this edition, the chapters on hedging portfolios and extensions of the Black-Scholes model have been expanded. The chapter on optimizing portfolios has been completely re-written to focus on the development of the Capital Asset Pricing Model. The binomial model due to Cox-Ross-Rubinstein has been enlarged into a standalone chapter illustrating the wide-ranging utility of the binomial model for numerically estimating option prices. There is a completely new chapter on the pricing of exotic options. The appendix now features linear algebra with sufficient background material to support a more rigorous development of the Arbitrage Theorem.The new edition has more than doubled the number of exercises compared to the previous edition and now contains over 700 exercises. Thus, students completing the book will gain a deeper understanding of the development of modern financial mathematics.

Product Details

ISBN-13:	9789811260308
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	12/22/2022
Pages:	468
Product dimensions:	6.00(w) x 9.00(h) x 1.00(d)

Table of Contents

Preface vii

About the Author xi

1 The Theory of Interest 1

1.1 Simple Interest 1

1.2 Compound Interest 3

1.3 Continuously Compounded Interest 5

1.4 Present Value 7

1.5 Rate of Return 15

1.6 Time-Varying Interest Rates 17

1.7 Continuous Income Streams 20

2 Discrete Probability 23

2.1 Events and Probabilities 24

2.2 Random Variables 28

2.3 Addition Rule 30

2.4 Conditional Probability and Multiplication Rule 32

2.5 Cumulative Distribution Functions 41

2.6 Binomial Random Variables 44

2.7 Expected Value 47

2.8 Variance and Standard Deviation 55

2.9 Covariance and Correlation 58

2.10 Odds and Wagering 65

3 The Arbitrage Theorem 69

3.1 An Introduction to Linear Programming 70

3.2 Primal and Dual Problems 79

3.3 The Fundamental Theorem of Finance 91

3.4 Remarks 96

4 Optimal Portfolio Choice 99

4.1 Return and Risk 100

4.2 The Efficient Frontier 106

4.3 The Capital Asset Pricing Model 115

4.4 Uncorrelated Returns 120

4.5 Utility Functions 125

5 Forwards and Futures 135

5.1 Definition of a Forward Contract 135

5.2 Pricing a Forward Contract 139

5.3 Dividends and Pricing 145

5.4 Incorporating Transaction and Storage Costs 150

5.5 Synthetic Forwards 155

5.6 Futures 157

6 Options 163

6.1 Properties of Options 164

6.2 Including the Effects of Dividends 169

6.3 Parity and American Options 172

6.4 Option Strategies 179

7 Approximating Option Prices Using Binomial Trees 191

7.1 One Period Binomial Model 191

7.2 Multi-Period Binomial Models 196

7.3 Estimating Increase/Decrease Factors 200

7.4 American Options 203

8 Normal Random Variables and Probability 211

8.1 Continuous Random Variables 211

8.2 Expected Value of Continuous Random Variables 218

8.3 Variance and Standard Deviation 223

8.4 Normal Random Variables 226

8.5 Lognormal Random Variables 235

8.6 Application: Portfolio Selection with Utility 238

8.7 Partial Expectation 242

8.8 The Binormal Distribution 246

9 Random Walks and Brownian Motion 249

9.1 Empirical Justification for the Mathematical Model 249

9.2 Advection/Diffusion Equation 256

9.3 Continuous Random Walk 260

9.4 The Stochastic Integral 268

9.5 Ito's Lemma 273

9.6 Maximum and Minimum of a Random Walk 281

10 Black-Scholes Equation and Option Formulas 287

10.1 Black-Scholes Partial Differential Equation 288

10.2 Boundary and Initial Conditions 290

10.3 Changing Variables in the Black-Scholes PDE 292

10.4 Solving the Black-Scholes Equation 296

10.5 Derivation Using the Binomial Model 303

11 Extensions of the Black-Scholes Model 307

11.1 Options on Exchange Rates 307

11.2 Options on Futures 312

11.3 Options on Stocks Paying Discrete Dividends 314

12 Derivatives of Black-Scholes Option Prices 323

12.1 Delta and Gamma 324

12.2 Theta 329

12.3 Vega, Rho, and Psi 334

12.4 Applications of the Greeks 339

13 Hedging 347

13.1 General Principles 348

13.2 Delta Hedging 351

13.3 Self-Financing Portfolios 358

13.4 Gamma-Neutral Portfolios 360

14 Exotic Options 365

14.1 Gap Options 365

14.2 Power Options 370

14.3 Exchange Options 375

14.4 Chooser Options 380

14.5 Forward Start Options 385

14.6 Compound Options 387

14.7 Asian Options 394

14.8 Barrier Options 403

Appendix A Linear Algebra Primer 417

A.1 Matrices and their Properties 417

A.2 Vector and Matrix Norms 423

A.3 Separating Hyperplanes and Convex Cones 426

A.4 Farkas' Lemma 431

Bibliography 435

Index 441