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.
Table of Contents
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
