An introduction to the mathematics of financial derivatives

목차 (Table of Contents)

• CONTENTS
• PREFACE TO THE SECOND EDITION = xxi
• INTRODUCTION = xxiii
• 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 $$a_t$$ and $$\sigma _t$$ = 253
• 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 $${\tilde P}$$ = 350
• 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 $$r_t$$ in the HJM Approach = 441
• 3.5 Another Advantage of the HJM Approach = 443
• 3.6 Market Practice = 444
• 4 How to Fit $$r_t$$ to Initial Term Structure = 444
• 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 $$r_t$$ = 460
• 5.2 Case 2 : A Mean-Reverting $$r_t$$ = 461
• 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