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CONTENTS
PREFACE TO THE SECOND EDITION ... xxi
INTRODUCTION ... xxii
CHAPTER 1 Financial Derivatives : A Brief Introduction
1 Introduction ... 1
2 Definitions ... 2
3 Types of Derivatives ... 2
3.1 Cash-and-Carry Markets ... 3
3.2 Price-Discovery Markets ... 4
3.3 Expiration Date ... 4
4 Forwards and Futures ... 5
4.1 Futures ... 6
5 Options ... 7
5.1 Some Notation ... 7
6 Swaps ... 9
6.1 A Simple Interest Rate Swap ... 10
7 Conclusions ... 11
8 References ... 11
9 Exercises ... 11
CHAPTER 2 A Primer on the Arbitrage Theorem
1 Introduction ... 13
2 Notation ... 14
2.1 Asset Prices ... 15
2.2 States of the World ... 15
2.3 Returns and Payoffs ... 16
2.4 Portfolio ... 17
3 A Basic Example of Asset Pricing ... 17
3.1 A First Glance at the Arbitrage Theorem ... 19
3.2 Relevance of the Arbitrage Theorem ... 20
3.3 The Use of Synthetic Probabilities ... 21
3.4 Martingales and Submartingales ... 24
3.5 Normalization ... 24
3.6 Equalization of Rates of Return ... 25
3.7 The No-Arbitrage Condition ... 26
4 A Numerical Example ... 27
4.1 Case 1 : Arbitrage Possibilities ... 27
4.2 Case 2 : Arbitrage-Free Prices ... 28
4.3 An Indeterminacy ... 29
5 An Application : Lattice Models ... 29
6 Payouts and Foreign Currencies ... 32
6.1 The Case with Dividends ... 32
6.2 The Case with Foreign Currencies ... 34
7 Some Generalizations ... 36
7.1 Time Index ... 36
7.2 States of the World ... 36
7.3 Discounting ... 37
8 Conclusions : A Methodology for Pricing Assets ... 37
9 References ... 38
10 Appendix : Generalization of the Arbitrage Theorem ... 38
11 Exercises ... 40
CHAPTER 3 Calculus in Deterministic and Stochastic Environments
1 Introduction ... 45
1.1 Information Flows ... 46
1.2 Modeling Random Behavior ... 46
2 Some Tools of Standard Calculus ... 47
3 Functions ... 47
3.1 Random Functions ... 48
3.2 Examples of Functions ... 49
4 Convergence and Limit ... 52
4.1 The Derivative ... 53
4.2 The Chain Rule ... 57
4.3 The Integral ... 59
4.4 Integration by Parts ... 65
5 Partial Derivatives ... 66
5.1 Example ... 67
5.2 Total Differentials ... 67
5.3 Taylor Series Expansion ... 68
5.4 Ordinary Differential Equations ... 72
6 Conclusions ... 73
7 References ... 74
8 Exercises ... 74
CHAPTER 4 Pricing Derivatives : Models and Notation
1 Introduction ... 77
2 Pricing Functions ... 78
2.1 Forwards ... 78
2.2 Options ... 80
3 Application : Another Pricing Method ... 84
3.1 Example ... 85
4 The Problem ... 86
4.1 A First Look at Ito's Lemma ... 86
4.2 Conclusions ... 88
5 References ... 88
6 Exercises ... 89
CHAPTER 5 Tools in Probability Theory
1 Introduction ... 91
2 Probability ... 91
2.1 Example ... 92
2.2 Random Variable ... 93
3 Moments ... 94
3.1 First Two Moments ... 94
3.2 Higher-Order Moments ... 95
4 Conditional Expectations ... 97
4.1 Conditional Probability ... 97
4.2 Properties of Conditional Expectations ... 99
5 Some Important Models ... 100
5.1 Binomial Distribution in Financial Markets ... 100
5.2 Limiting Properties ... 101
5.3 Moments ... 102
5.4 The Normal Distribution ... 103
5.5 The Poisson Distribution ... 106
6 Markov Processes and Their Relevance ... 108
6.1 The Relevance ... 109
6.2 The Vector Case ... 110
7 Convergence of Random Variables ... 112
7.1 Types of Convergence and Their Uses ... 112
7.2 Weak Convergence ... 113
8 Conclusions ... 116
9 References ... 116
10 Exercises ... 117
CHAPTER 6 Martingales and Martingale Representations
1 Introduction ... 119
2 Definitions ... 120
2.1 Notation ... 120
2.2 Continuous-Time Martingales ... 121
3 The Use of Martingales in Asset Pricing ... 122
4 Relevance of Martingales in Stochastic Modeling ... 124
4.1 An Example ... 126
5 Properties of Martingale Trajectories ... 127
6 Examples of Martingales ... 130
6.1 Example 1 : Brownian Motion ... 130
6.2 Example 2 : A Squared Process ... 132
6.3 Example 3 : An Exponential Process ... 133
6.4 Example 4 : Right Continuous Martingales ... 134
7 The Simplest Martingale ... 134
7.1 An Application ... 135
7.2 An Example ... 136
8 Martingale Representations ... 137
8.1 An Example ... 137
8.2 Doob-Meyer Decomposition ... 140
9 The First Stochastic Integral ... 143
9.1 Application to Finance : Trading Gains ... 144
10 Martingale Methods and Pricing ... 145
11 A Pricing Methodology ... 146
11.1 A Hedge ... 147
11.2 Time Dynamics ... 147
11.3 Normalization and Risk-Neutral Probability ... 150
11.4 A Summary ... 152
12 Conclusions ... 152
13 References ... 153
14 Exercises ... 154
CHAPTER 7 Differentiation in Stochastic Environments
1 Introduction ... 156
2 Motivation ... 157
3 A Framework for Discussing Differentiation ... 161
4 The "Size" of Incremental Errors ... 164
5 One Implication ... 167
6 Putting the Results Together ... 169
6.1 Stochastic Differentials ... 170
7 Conclusions ... 171
8 References ... 171
9 Exercises ... 171
CHAPTER 8 The Wiener Process and Rare Events in Financial Markets
1 Introduction ... 173
1.1 Relevance of the Discussion ... 174
2 Two Generic Models ... 175
2.1 The Wiener Process ... 176
2.2 The Poisson Process ... 178
2.3 Examples ... 180
2.4 Back to Rare Events ... 182
3 SDE in Discrete Intervals, Again ... 183
4 Characterizing Rare and Normal Events ... 184
4.1 Normal Events ... 187
4.2 Rare Events ... 189
5 A Model for Rare Events ... 190
6 Moments That Matter ... 193
7 Conclusions ... 195
8 Rare and Normal Events in Practice ... 196
8.1 The Binomial Model ... 196
8.2 Normal Events ... 197
8.3 Rare Events ... 198
8.4 The Behavior of Accumulated Changes ... 199
9 References ... 202
10 Exercises ... 203
CHAPTER 9 Integration in Stochastic Environments : The Ito Integral
1 Introduction ... 204
1.1 The Ito Integral and SDEs ... 206
1.2 The Practical Relevance of the Ito Integral ... 207
2 The Ito Integral ... 208
2.1 The Riemann-Stieltjes Integral ... 209
2.2 Stochastic Integration and Riemann Sums ... 211
2.3 Definition : The Ito Integral ... 213
2.4 An Expository Example ... 214
3 Properties of the Ito Integral ... 220
3.1 The Ito Integral Is a Martingale ... 220
3.2 Pathwise Integrals ... 224
4 Other Properties of the Ito Integral ... 226
4.1 Existence ... 226
4.2 Correlation Properties ... 226
4.3 Addition ... 227
5 Integrals with Respect to Jump Processes ... 227
6 Conclusions ... 228
7 References ... 228
8 Exercises ... 228
CHAPTER 10 Ito's Lemma
1 Introduction ... 230
2 Types of Derivatives ... 231
2.1 Example ... 232
3 Ito's Lemma ... 232
3.1 The Notion of "Size" in Stochastic Calculus ... 235
3.2 First-Order Terms ... 237
3.3 Second-Order Terms ... 238
3.4 Terms Involving Cross Products ... 239
3.5 Terms in the Remainder ... 240
4 The Ito Formula ... 240
5 Uses of Ito's Lemma ... 241
5.1 Ito's Formula as a Chain Rule ... 241
5.2 Ito's Formula as an Integration Tool ... 242
6 Integral Form of Ito's Lemma ... 244
7 Ito's Formula in More Complex Settings ... 245
7.1 Multivariate Case ... 245
7.2 Ito's Formula and Jumps ... 248
8 Conclusions ... 250
9 References ... 251
10 Exercises ... 251
CHAPTER 11 The Dynamics of Derivative Prices : Stochastic Differential Equations
1 Introduction ... 252
1.1 Conditions on <?import namespace ... m ur
2 A Geometric Description of Paths Implied by SDEs ... 254
3 Solution of SDEs ... 255
3.1 What Does a Solution Mean? ... 255
3.2 Types of Solutions ... 256
3.3 Which Solution Is to Be Preferred? ... 258
3.4 A Discussion of Strong Solutions ... 258
3.5 Verification of Solutions to SDEs ... 261
3.6 An Important Example ... 262
4 Major Models of SDEs ... 265
4.1 Linear Constant Coefficient SDEs ... 266
4.2 Geometric SDEs ... 267
4.3 Square Root Process ... 269
4.4 Mean Reverting Process ... 270
4.5 Ornstein-Uhlenbeck Process ... 271
5 Stochastic Volatility ... 271
6 Conclusions ... 272
7 References ... 272
8 Exercises ... 273
CHAPTER 12 Pricing Derivative Products : Partial Differential Equations
1 Introduction ... 275
2 Forming Risk-Free Portfolios ... 276
3 Accuracy of the Method ... 280
3.1 An Interpretation ... 282
4 Partial Differential Equations ... 282
4.1 Why Is the PDE an "Equation"? ... 283
4.2 What Is the Boundary Condition? ... 283
5 Classification of PDEs ... 284
5.1 Example 1 : Linear, First-Order PDE ... 284
5.2 Example 2 : Linear, Second-Order PDE ... 286
6 A Reminder : Bivariate, Second-Degree Equations ... 289
6.1 Circle ... 290
6.2 Ellipse ... 290
6.3 Parabola ... 292
6.4 Hyperbola ... 292
7 Types of PDEs ... 292
7.1 Example : Parabolic PDE ... 293
8 Conclusions ... 293
9 References ... 294
10 Exercises ... 294
CHAPTER 13 The Black-Scholes PDE : An Application
1 Introduction ... 296
2 The Black-Scholes PDE ... 296
2.1 A Geometric Look at the Black-Scholes Formula ... 298
3 PDEs in Asset Pricing ... 299
3.1 Constant Dividends ... 300
4 Exotic Options ... 301
4.1 Lookback Options ... 301
4.2 Ladder Options ... 301
4.3 Trigger or Knock-in Options ... 302
4.4 Knock-out Options ... 302
4.5 Other Exotics ... 302
4.6 The Relevant PDEs ... 303
5 Solving PDEs in Practice ... 304
5.1 Closed-Form Solutions ... 304
5.2 Numerical Solutions ... 306
6 Conclusions ... 309
7 References ... 310
8 Exercises ... 310
CHAPTER 14 Pricing Derivative Products : Equivalent Martingale Measures
1 Translations of Probabilities ... 312
1.1 Probability a "Measure" ... 312
2 Changing Means ... 316
2.1 Method 1 : Operating on Possible Values ... 317
2.2 Method 2 : Operating on Probabilities ... 321
3 The Girsanov Theorem ... 322
3.1 A Normally Distributed Random Variable ... 323
3.2 A Normally Distributed Vector ... 325
3.3 The Radon-Nikodym Derivative ... 327
3.4 Equivalent Measures ... 328
4 Statement of the Girsanov Theorem ... 329
5 A Discussion of the Girsanov Theorem ... 331
5.1 Application to SDEs ... 332
6 Which Probabilities? ... 334
7 A Method for Generating Equivalent Probabilities ... 337
7.1 An Example ... 340
8 Conclusions ... 342
9 References ... 342
10 Exercises ... 343
CHAPTER 15 Equivalent Martingale Measures : Applications
1 Introduction ... 345
2 A Martingale Measure ... 346
2.1 The Moment-Generating Function ... 346
2.2 Conditional Expectation of Geometric Processes ... 348
3 Converting Asset Prices into Martingales ... 349
3.1 Determining <m:math xmlns ... '"htt
3.2 The Implied SDEs ... 352
4 Application : The Black-Scholes Formula ... 353
4.1 Calculation ... 356
5 Comparing Martingale and PDE Approaches ... 358
5.1 Equivalence of the Two Approaches ... 359
5.2 Critical Steps of the Derivation ... 363
5.3 Integral Form of the Ito Formula ... 364
6 Conclusions ... 365
7 References ... 366
8 Exercises ... 366
CHAPTER 16 New Results and Tools for Interest-Sensitive Securities
1 Introduction ... 368
2 A Summary ... 369
3 Interest Rate Derivatives ... 371
4 Complications ... 375
4.1 Drift Adjustment ... 376
4.2 Term Structure ... 377
5 Conclusions ... 377
6 References ... 378
7 Exercises ... 378
CHAPTER 17 Arbitrage Theorem in a New Setting : Normalization and Random Interest Rates
1 Introduction ... 379
2 A Model for New Instruments ... 381
2.1 The New Environment ... 383
2.2 Normalization ... 389
2.3 Some Undesirable Properties ... 392
2.4 A New Normalization ... 395
2.5 Some Implications ... 399
3 Conclusions ... 404
4 References ... 404
5 Exercises ... 404
CHAPTER 18 Modeling Term Structure and Related Concepts
1 Introduction ... 407
2 Main Concepts ... 408
2.1 Three Curves ... 409
2.2 Movements on the Yield Curve ... 412
3 A Bond Pricing Equation ... 414
3.1 Constant Spot Rate ... 414
3.2 Stochastic Spot Rates ... 416
3.3 Moving to Continuous Time ... 417
3.4 Yields and Spot Rates ... 418
4 Forward Rates and Bond Prices ... 419
4.1 Discrete Time ... 419
4.2 Moving to Continuous Time ... 420
5 Conclusions : Relevance of the Relationships ... 423
6 References ... 424
7 Exercises ... 424
CHAPTER 19 Classical and HJM Approaches to Fixed Income
1 Introduction ... 426
2 The Classical Approach ... 427
2.1 Example 1 ... 428
2.2 Example 2 ... 429
2.3 The General Case ... 429
2.4 Using the Spot Rate Model ... 432
2.5 Comparison with the Black-Scholes World ... 434
3 The HJM Approach to Term Structure ... 435
3.1 Which Forward Rate? ... 436
3.2 Arbitrage-Free Dynamics in HJM ... 437
3.3 Interpretation ... 440
3.4 The <m:math xmlns ... '"htt
3.5 Another Advantage of the HJM Approach ... 443
3.6 Market Practice ... 444
4 How to Fit <m:math xmlns ... '"htt
4.1 Monte Carlo ... 445
4.2 Tree Models ... 446
4.3 Closed-Form Solutions ... 447
5 Conclusions ... 447
6 References ... 447
7 Exercises ... 448
CHAPTER 20 Classical PDE Analysis for Interest Rate Derivatives
1 Introduction ... 451
2 The Framework ... 454
3 Market Price of Interest Rate Risk ... 455
4 Derivation of the PDE ... 457
4.1 A Comparison ... 459
5 Closed-Form Solutions of the PDE ... 460
5.1 Case 1 : A Deterministic <m:math xmlns ... '"htt
5.2 Case 2 : A Mean-Reverting <m:math xmlns ... '"htt
5.3 Case 3 : More Complex Forms ... 464
6 Conclusions ... 465
7 References ... 465
8 Exercises ... 465
CHAPTER 21 Relating Conditional Expectations to PDEs
1 Introduction ... 467
2 From Conditional Expectations to PDEs ... 469
2.1 Case 1 : Constant Discount Factors ... 469
2.2 Case 2 : Bond Pricing ... 472
2.3 Case 3 : A Generalization ... 475
2.4 Some Clarifications ... 475
2.5 Which Drift? ... 476
2.6 Another Bond Price Formula ... 477
2.7 Which Formula? ... 479
3 From PDEs to Conditional Expectations ... 479
4 Generators, Feynman-Kac Formula, and Other Tools ... 482
4.1 Ito Diffusions ... 482
4.2 Markov Property ... 483
4.3 Generator of an Ito Diffusion ... 483
4.4 A Representation for A ... 484
4.5 Kolmogorov's Backward Equation ... 485
5 Feynman-Kac Formula ... 487
6 Conclusions ... 487
7 References ... 487
8 Exercises ... 487
CHAPTER 22 Stopping Times and American-Type Securities
1 Introduction ... 489
2 Why Study Stopping Times? ... 491
2.1 American-Style Securities ... 492
3 Stopping Times ... 492
4 Uses of Stopping Times ... 493
5 A Simplified Setting ... 494
5.1 The Model ... 494
6 A Simple Example ... 499
7 Stopping Times and Martingales ... 504
7.1 Martingales ... 504
7.2 Dynkin's Formula ... 504
8 Conclusions ... 505
9 References ... 505
10 Exercises ... 505
BIBLIOGRAPHY ... 509
INDEX ... 513
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