Foundations of the Pricing of Financial Derivatives - Robert E. Brooks, Don M. Chance

Foundations of the Pricing of Financial Derivatives

Theory and Analysis
Buch | Hardcover
624 Seiten
2024
John Wiley & Sons Inc (Verlag)
978-1-394-17965-7 (ISBN)
93,50 inkl. MwSt
An accessible and mathematically rigorous resource for masters and PhD students

In Foundations of the Pricing of Financial Derivatives: Theory and Analysis two expert finance academics with professional experience deliver a practical new text for doctoral and masters’ students and also new practitioners. The book draws on the authors extensive combined experience teaching, researching, and consulting on this topic and strikes an effective balance between fine-grained quantitative detail and high-level theoretical explanations.

The authors fill the gap left by books directed at masters’-level students that often lack mathematical rigor. Further, books aimed at mathematically trained graduate students often lack quantitative explanations and critical foundational materials. Thus, this book provides the technical background required to understand the more advanced mathematics used in this discipline, in class, in research, and in practice.

Readers will also find:



Tables, figures, line drawings, practice problems (with a solutions manual), references, and a glossary of commonly used specialist terms
Review of material in calculus, probability theory, and asset pricing
Coverage of both arithmetic and geometric Brownian motion
Extensive treatment of the mathematical and economic foundations of the binomial and Black-Scholes-Merton models that explains their use and derivation, deepening readers’ understanding of these essential models
Deep discussion of essential concepts, like arbitrage, that broaden students’ understanding of the basis for derivative pricing
Coverage of pricing of forwards, futures, and swaps, including arbitrage-free term structures and interest rate derivatives

An effective and hands-on text for masters’-level and PhD students and beginning practitioners with an interest in financial derivatives pricing, Foundations of the Pricing of Financial Derivatives is an intuitive and accessible resource that properly balances math, theory, and practical applications to help students develop a healthy command of a difficult subject.

ROBERT E. BROOKS, PHD, CFA, is Professor Emeritus of Finance at the University of Alabama. He is the President of Financial Risk Management, LLC, a quantitative finance consulting firm. He is the author of several books and maintains a YouTube channel, @FRMHelpForYou. DON M. CHANCE, PHD, CFA, holds the James C. Flores Endowed Chair of MBA Studies and is Professor of Finance at the E.J. Ourso College of Business at Louisiana State University. He is the author of four books on derivatives and risk management. His consulting firm is Omega Risk Advisors, LLC, and his website is donchance.com.

Preface xv

Chapter 1 Introduction and Overview 1

1.1 Motivation for This Book 2

1.2 What Is a Derivative? 6

1.3 Options Versus Forwards, Futures, and Swaps 8

1.4 Size and Scope of the Financial Derivatives Markets 9

1.5 Outline and Features of the Book 12

1.6 Final Thoughts and Preview 14

Questions and Problems 15

Notes 15

Part I Basic Foundations for Derivative Pricing

Chapter 2 Boundaries, Limits, and Conditions on Option Prices 19

2.1 Setup, Definitions, and Arbitrage 20

2.2 Absolute Minimum and Maximum Values 21

2.3 The Value of an American Option Relative to the Value of a European Option 22

2.4 The Value of an Option at Expiration 22

2.5 The Lower Bounds of European and American Options and the Optimality of Early Exercise 23

2.6 Differences in Option Values by Exercise Price 31

2.7 The Effect of Differences in Time to Expiration 37

2.8 The Convexity Rule 38

2.9 Put-Call Parity 40

2.10 The Effect of Interest Rates on Option Prices 47

2.11 The Effect of Volatility on Option Prices 47

2.12 The Building Blocks of European Options 48

2.13 Recap and Preview 49

Questions and Problems 50

Notes 51

Chapter 3 Elementary Review of Mathematics for Finance 53

3.1 Summation Notation 53

3.2 Product Notation 55

3.3 Logarithms and Exponentials 56

3.4 Series Formulas 58

3.5 Calculus Derivatives 59

3.6 Integration 68

3.7 Differential Equations 70

3.8 Recap and Preview 71

Questions and Problems 71

Notes 73

Chapter 4 Elementary Review of Probability for Finance 75

4.1 Marginal, Conditional, and Joint Probabilities 75

4.2 Expectations, Variances, and Covariances of Discrete Random Variables 80

4.3 Continuous Random Variables 86

4.4 Some General Results in Probability Theory 93

4.5 Technical Introduction to Common Probability Distributions Used in Finance 95

4.6 Recap and Preview 109

Questions and Problems 109

Notes 110

Chapter 5 Financial Applications of Probability Distributions 113

5.1 The Univariate Normal Probability Distribution 113

5.2 Contrasting the Normal with the Lognormal Probability Distribution 119

5.3 Bivariate Normal Probability Distribution 123

5.4 The Bivariate Lognormal Probability Distribution 125

5.5 Recap and Preview 126

Appendix 5A An Excel Routine for the Bivariate Normal Probability 126

Questions and Problems 128

Notes 128

Chapter 6 Basic Concepts in Valuing Risky Assets and Derivatives 129

6.1 Valuing Risky Assets 129

6.2 Risk-Neutral Pricing in Discrete Time 130

6.3 Identical Assets and the Law of One Price 133

6.4 Derivative Contracts 134

6.5 A First Look at Valuing Options 136

6.6 A World of Risk-Averse and Risk-Neutral Investors 137

6.7 Pricing Options Under Risk Aversion 138

6.8 Recap and Preview 138

Questions and Problems 139

Notes 139

Part II Discrete Time Derivatives Pricing Theory

Chapter 7 The Binomial Model 143

7.1 The One-Period Binomial Model for Calls 143

7.2 The One-Period Binomial Model for Puts 146

7.3 Arbitraging Price Discrepancies 149

7.4 The Multiperiod Model 151

7.5 American Options and Early Exercise in the Binomial Framework 154

7.6 Dividends and Recombination 155

7.7 Path Independence and Path Dependence 159

7.8 Recap and Preview 159

Appendix 7A Derivation of Equation (7.9) 159

Appendix 7B Pascal’s Triangle and the Binomial Model 161

Questions and Problems 163

Notes 163

Chapter 8 Calculating the Greeks in the Binomial Model 165

8.1 Standard Approach 165

8.2 An Enhanced Method for Estimating Delta and Gamma 170

8.3 Numerical Examples 172

8.4 Dividends 174

8.5 Recap and Preview 175

Questions and Problems 175

Notes 176

Chapter 9 Convergence of the Binomial Model to the Black-Scholes-Merton Model 177

9.1 Setting Up the Problem 177

9.2 The Hsia Proof 181

9.3 Put Options 187

9.4 Dividends 188

9.5 Recap and Preview 188

Questions and Problems 189

Notes 190

Part III Continuous Time Derivatives Pricing Theory

Chapter 10 The Basics of Brownian Motion and Wiener Processes 193

10.1 Brownian Motion 193

10.2 The Wiener Process 195

10.3 Properties of a Model of Asset Price Fluctuations 196

10.4 Building a Model of Asset Price Fluctuations 199

10.5 Simulating Brownian Motion and Wiener Processes 202

10.6 Formal Statement of Wiener Process Properties 205

10.7 Recap and Preview 207

Appendix 10A Simulation of the Wiener Process and the Square of the Wiener Process for Successively Smaller Time Intervals 207

Questions and Problems 208

Notes 209

Chapter 11 Stochastic Calculus and Itô’s Lemma 211

11.1 A Result from Basic Calculus 211

11.2 Introducing Stochastic Calculus and Itô’s Lemma 212

11.3 Itô’s Integral 215

11.4 The Integral Form of Itô’s Lemma 216

11.5 Some Additional Cases of Itô’s Lemma 217

11.6 Recap and Preview 219

Appendix 11A Technical Stochastic Integral Results 220
11A.1 Selected Stochastic Integral Results 220
11A.2 A General Linear Theorem 224

Questions and Problems 229

Notes 230

Chapter 12 Properties of the Lognormal and Normal Diffusion Processes for Modeling Assets 231

12.1 A Stochastic Process for the Asset Relative Return 232

12.2 A Stochastic Process for the Asset Price Change 235

12.3 Solving the Stochastic Differential Equation 236

12.4 Solutions to Stochastic Differential Equations Are Not Always the Same as Solutions to Corresponding Ordinary Differential Equations 237

12.5 Finding the Expected Future Asset Price 238

12.6 Geometric Brownian Motion or Arithmetic Brownian Motion? 240

12.7 Recap and Preview 241

Questions and Problems 242

Notes 242

Chapter 13 Deriving the Black-Scholes-Merton Model 245

13.1 Derivation of the European Call Option Pricing Formula 245

13.2 The European Put Option Pricing Formula 249

13.3 Deriving the Black-Scholes-Merton Model as an Expected Value 250

13.4 Deriving the Black-Scholes-Merton Model as the Solution of a Partial Differential Equation 254

13.5 Decomposing the Black-Scholes-Merton Model into Binary Options 258

13.6 Black-Scholes-Merton Option Pricing When There Are Dividends 259

13.7 Selected Black-Scholes-Merton Model Limiting Results 259

13.8 Computing the Black-Scholes-Merton Option Pricing Model Values 262

13.9 Recap and Preview 265

Appendix 13.A Deriving the Arithmetic Brownian Motion Option Pricing Model 265

Questions and Problems 269

Notes 270

Chapter 14 The Greeks in the Black-Scholes-Merton Model 271

14.1 Delta: The First Derivative with Respect to the Underlying Price 274

14.2 Gamma: The Second Derivative with Respect to the Underlying Price 274

14.3 Theta: The First Derivative with Respect to Time 275

14.4 Verifying the Solution of the Partial Differential Equation 275

14.5 Selected Other Partial Derivatives of the Black-Scholes-Merton Model 277

14.6 Partial Derivatives of the Black-Scholes-Merton European Put Option Pricing Model 278

14.7 Incorporating Dividends 279

14.8 Greek Sensitivities 280

14.9 Elasticities 283

14.10 Extended Greeks of the Black-Scholes-Merton Option Pricing Model 284

14.11 Recap and Preview 284

Questions and Problems 285

Notes 286

Chapter 15 Girsanov’s Theorem in Option Pricing 287

15.1 The Martingale Representation Theorem 287

15.2 Introducing the Radon-Nikodym Derivative by Changing the Drift for a Single Random Variable 289

15.3 A Complete Probability Space 291

15.4 Formal Statement of Girsanov’s Theorem 292

15.5 Changing the Drift in a Continuous Time Stochastic Process 293

15.6 Changing the Drift of an Asset Price Process 297

15.7 Recap and Preview 300

Questions and Problems 301

Notes 302

Chapter 16 Connecting Discrete and Continuous Brownian Motions 303

16.1 Brownian Motion in a Discrete World 303

16.2 Moving from a Discrete to a Continuous World 306

16.3 Changing the Probability Measure with the Radon-Nikodym Derivative in Discrete Time 310

16.4 The Kolmogorov Equations 313

16.5 Recap and Preview 321

Questions and Problems 322

Notes 322

Part IV Extensions and Generalizations of Derivative Pricing

Chapter 17 Applying Linear Homogeneity to Option Pricing 327

17.1 Introduction to Exchange Options 327

17.2 Homogeneous Functions 328

17.3 Euler’s Rule 330

17.4 Using Linear Homogeneity and Euler’s Rule to Derive the Black-Scholes-Merton Model 330

17.5 Exchange Option Pricing 333

17.6 Spread Options 337

17.7 Forward Start Options 339

17.8 Recap and Preview 341

Appendix 17A Linear Homogeneity and the Arithmetic Brownian Motion Model 342

Appendix 17B Multivariate Itô’s Lemma 344

Appendix 17C Greeks of the Exchange Option Model 345

Questions and Problems 347

Notes 347

Chapter 18 Compound Option Pricing 349

18.1 Equity as an Option 350

18.2 Valuing an Option on the Equity as a Compound Option 351

18.3 Compound Option Boundary Conditions and Parities 353

18.4 Geske’s Approach to Valuing a Call on a Call 356

18.5 Characteristics of Geske’s Call on Call Option 358

18.6 Geske’s Call on Call Option Model and Linear Homogeneity 359

18.7 Generalized Compound Option Pricing Model 360

18.8 Installment Options 361

18.9 Recap and Preview 362

Appendix 18A Selected Greeks of the Compound Option 362

Questions and Problems 363

Notes 363

Chapter 19 American Call Option Pricing 365

19.1 Closed-Form American Call Pricing: Roll-Geske-Whaley 366

19.2 The Two-Payment Case 370

19.3 Recap and Preview 372

Appendix 19A Numerical Example of the One-Dividend Model 373

Questions and Problems 374

Notes 374

Chapter 20 American Put Option Pricing 377

20.1 The Nature of the Problem of Pricing an American Put 377

20.2 The American Put as a Series of Compound Options 378

20.3 Recap and Preview 380

Questions and Problems 380

Notes 381

Chapter 21 Min-Max Option Pricing 383

21.1 Characteristics of Stulz’s Min-Max Option 383

21.2 Pricing the Call on the Min 388

21.3 Other Related Options 393

21.4 Recap and Preview 395

Appendix 21A Multivariate Feynman-Kac Theorem 395

Appendix 21B An Alternative Derivation of the Min-Max Option Model 396

Questions and Problems 397

Notes 397

Chapter 22 Pricing Forwards, Futures, and Options on Forwards and Futures 399

22.1 Forward Contracts 399

22.2 Pricing Futures Contracts 404

22.3 Options on Forwards and Futures 409

22.4 Recap and Preview 412

Questions and Problems 413

Notes 414

Part V Numerical Methods

Chapter 23 Monte Carlo Simulation 417

23.1 Standard Monte Carlo Simulation of the Lognormal Diffusion 417

23.2 Reducing the Standard Error 421

23.3 Simulation with More Than One Random Variable 424

23.4 Recap and Preview 424

Questions and Problems 425

Notes 426

Chapter 24 Finite Difference Methods 429

24.1 Setting Up the Finite Difference Problem 429

24.2 The Explicit Finite Difference Method 431

24.3 The Implicit Finite Difference Method 434

24.4 Finite Difference Put Option Pricing 435

24.5 Dividends and Early Exercise 435

24.6 Recap and Preview 436

Questions and Problems 436

Notes 436

Part VI Interest Rate Derivatives

Chapter 25 The Term Structure of Interest Rates 439

25.1 The Unbiased Expectations Hypothesis 440

25.2 The Local Expectations Hypothesis 442

25.3 The Difference Between the Local and Unbiased Expectations Hypotheses 446

25.4 Other Term Structure of Interest Rate Hypotheses 447

25.5 Recap and Preview 450

Questions and Problems 450

Notes 450

Chapter 26 Interest Rate Contracts: Forward Rate Agreements, Swaps, and Options 453

26.1 Interest Rate Forwards 454

26.2 Interest Rate Swaps 459

26.3 Interest Rate Options 469

26.4 Recap and Preview 471

Questions and Problems 471

Notes 472

Chapter 27 Fitting an Arbitrage-Free Term Structure Model 475

27.1 Basic Structure of the HJM Model 476

27.2 Discretizing the HJM Model 479

27.3 Fitting a Binomial Tree to the HJM Model 481

27.4 Filling in the Remainder of the HJM Binomial Tree 485

27.5 Recap and Preview 489

Questions and Problems 490

Notes 491

Chapter 28 Pricing Fixed-Income Securities and Derivatives Using an Arbitrage-Free Binomial Tree 493

28.1 Zero-Coupon Bonds 493

28.2 Coupon Bonds 496

28.3 Options on Zero-Coupon Bonds 497

28.4 Options on Coupon Bonds 498

28.5 Callable Bonds 499

28.6 Forward Rate Agreements (FRAs) 501

28.7 Interest Rate Swaps 503

28.8 Interest Rate Options 505

28.9 Interest Rate Swaptions 506

28.10 Interest Rate Futures 508

28.11 Recap and Preview 510

Questions and Problems 510

Notes 510

Part VII Miscellaneous Topics

Chapter 29 Option Prices and the Prices of State-Contingent Claims 513

29.1 Pure Assets in the Market 514

29.2 Pricing Pure and Complex Assets 514

29.3 Numerical Example 518

29.4 State Pricing and Options in a Binomial Framework 519

29.5 State Pricing and Options in Continuous Time 522

29.6 Recap and Preview 525

Questions and Problems 525

Notes 526

Chapter 30 Option Prices and Expected Returns 527

30.1 The Basic Framework 527

30.2 Expected Returns on Options 529

30.3 Volatilities of Options 531

30.4 Options and the Capital Asset Pricing Model 531

30.5 Options and the Sharpe Ratio 532

30.6 The Stochastic Process Followed by the Option 533

30.7 Recap and Preview 535

Questions and Problems 535

Notes 536

Chapter 31 Implied Volatility and the Volatility Smile 537

31.1 Historical Volatility and the VIX 538

31.2 An Example of Implied Volatility 539

31.3 The Volatility Surface 546

31.4 The Perfect Substitutability of Options 547

31.5 Other Attempts to Explain the Implied Volatility Smile 549

31.6 How Practitioners Use the Implied Volatility Surface 550

31.7 Recap and Preview 551

Questions and Problems 551

Notes 553

Chapter 32 Pricing Foreign Currency Options 555

32.1 Definition of Terms 556

32.2 Option Payoffs 556

32.3 Valuation of the Options 557

32.4 Probability of Exercise 561

32.5 Some Terminology Confusion 563

32.6 Recap 563

Questions and Problems 564

Notes 565

References 567

Symbols Used 573

Symbols 573

Time-Related Notation 573

Instrument-Related Notation 574

About the Website 581

Index 583

Erscheinungsdatum
Reihe/Serie Frank J. Fabozzi Series
Verlagsort New York
Sprache englisch
Maße 185 x 257 mm
Gewicht 1066 g
Themenwelt Wirtschaft Betriebswirtschaft / Management Finanzierung
ISBN-10 1-394-17965-0 / 1394179650
ISBN-13 978-1-394-17965-7 / 9781394179657
Zustand Neuware
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