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Foundations of the Pricing of Financial Derivatives

Theory and Analysis
Frank J. Fabozzi Series 1. Aufl.

von: Robert E. Brooks, Don M. Chance
60,99 €
Verlag:	Wiley
Format:	EPUB
Veröffentl.:	25.01.2024
ISBN/EAN:	9781394179664
Sprache:	englisch
Anzahl Seiten:	624

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Titelbeschreibung

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: <ul> <li>Tables, figures, line drawings, practice problems (with a solutions manual), references, and a glossary of commonly used specialist terms</li> <li>Review of material in calculus, probability theory, and asset pricing</li> <li>Coverage of both arithmetic and geometric Brownian motion </li> <li>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</li> <li>Deep discussion of essential concepts, like arbitrage, that broaden students’ understanding of the basis for derivative pricing</li> <li>Coverage of pricing of forwards, futures, and swaps, including arbitrage-free term structures and interest rate derivatives</li></ul>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.

Inhaltsverzeichnis

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

Autorenportrait

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.

Back cover copy

Praise for FOUNDATIONS of the PRICING of FINANCIAL DERIVATIVES “This book stands out for me in at least two important ways. First, quite incredibly, the authors have succeeded in presenting financial derivatives in a remarkably accessible user-friendly manner that integrates technical derivatives’ mathematics with insightful conceptual understanding, enabling students to easily navigate the complex minefield of ideas and applications involved. Second, it combines a strong academic focus with an equivalent emphasis on addressing real-world problems across different echelons of difficulty levels.” — PRADEEP YADAV, W. Ross Johnston Chair and Professor of Finance, University of Oklahoma “This is a comprehensive and cleverly developed book on derivatives. It is an excellent text for advanced Master’s and Ph.D. students (and for reference by professionals).” — JIMMY HILLIARD, Harbert Eminent Scholar and Professor of Finance, Auburn University “The authors are great storytellers; they make derivatives come alive. The subject is obviously highly technical and intimidating at times, but they have made it so accessible, relevant and, most importantly, fun. The topics covered are comprehensive and yet very selective with all the right choices and emphasis. I wholeheartedly recommend this excellent textbook for both novice and advanced students of derivatives.” —YISONG S. TIAN, Professor of Finance, York University