Contents
General Probability Theory 1
1.1 Infinite Probability Spaces 1
1.2 Random Variables and Distributions 7
1.3 Expectations 13
1.4 Convergence of Integrals 23
1.5 Computation of Expectations 27
1.6 Change of Measure 32
1.7 Summary 39
1.8 Notes 41
1.9 Exercises 41
Information and Conditioning 49
2.1 Information and ?-algebras 49
2.2 Independence 53
2.3 General Conditional Expectations 66
2.4 Summary 75
2.5 Notes 77
2.6 Exercises 77
Brownian Motion 83
3.1 Introduction · 83
3.2 Scaled Random Walks 83
3.2.1 Symmetric Random Walk 83
3.2.2 Increments of the Symmetric Random Walk 84
3.2.3 Martingale Property for the Symmetric
Random Walk 85
3.2.4 Quadratic Variation of the Symmetric
Random Walk 85
3.2.5 Scaled Symmetric Random Walk 86
3.2.6 Limiting Distribution of the Scaled Random Walk 88
X Contents
3.2.7 Log-Normal Distribution as the Limit of the
Binomial Model 91
3.3 Brownian Motion 93
3.3.1 Definition of Brownian Motion 93
3.3.2 Distribution of Brownian Motion 95
3.3.3 Filtration for Brownian Motion 97
3.3.4 Martingale Property for Brownian Motion 98
3.4 Quadratic Variation 98
3.4.1 First-Order Variation 99
3.4.2 Quadratic Variation 101
3.4.3 Volatility of Geometric Brownian Motion 106
3.5 Markov Property 107
3.6 First Passage Time Distribution 108
3.7 Reflection Principle Ill
3.7.1 Reflection Equality Ill
3.7.2 First Passage Time Distribution 112
3.7.3 Distribution of Brownian Motion and Its Maximum.... 113
3.8 Summary 115
3.9 Notes 116
3.10 Exercises 117
4 Stochastic Calculus 125
4.1 Introduction 125
4.2 Ito's Integral for Simple Integrands 125
4.2.1 Construction of the Integral 126
4.2.2 Properties of the Integral 128
4.3 Ito's Integral for General Integrands 132
4.4 Ito-Doeblin Formula 137
4.4.1 Formula for Brownian Motion 137
4.4.2 Formula for Ito Processes 143
4.4.3 Examples 147
4.5 Black-Scholes-Merton Equation 153
4.5.1 Evolution of Portfolio Value 154
4.5.2 Evolution of Option Value 155
4.5.3 Equating the Evolutions 156
4.5.4 Solution to the Black-Scholes-Merton Equation 158
4.5.5 The Greeks 159
4.5.6 Put-Call Parity 162
4.6 Multivariable Stochastic Calculus 164
4.6.1 Multiple Brownian Motions 164
4.6.2 Ito-Doeblin Formula for Multiple Processes 165
4.6.3 Recognizing a Brownian Motion 168
4.7 Brownian Bridge 172
4.7.1 Gaussian Processes 172
4.7.2 Brownian Bridge as a Gaussian Process 175
Contents XI
4.7.3 Brownian Bridge as a Scaled Stochastic Integral 176
4.7.4 Multidimensional Distribution of the
Brownian Bridge 178
4.7.5 Brownian Bridge as a Conditioned Brownian Motion... 182
4.8 Summary 183
4.9 Notes 187
4.10 Exercises 189
Risk-Neutral Pricing 209
5.1 Introduction 209
5.2 Risk-Neutral Measure 210
5.2.1 Girsanov's Theorem for a Single Brownian Motion 210
5.2.2 Stock Under the Risk-Neutral Measure 214
5.2.3 Value of Portfolio Process Under the
Risk-Neutral Measure 217
5.2.4 Pricing Under the Risk-Neutral Measure 218
5.2.5 Deriving the Black-Scholes-Merton Formula 218
5.3 Martingale Representation Theorem 221
5.3.1 Martingale Representation with One
Brownian Motion 221
5.3.2 Hedging with One Stock 222
5.4 Fundamental Theorems of Asset Pricing 224
5.4.1 Girsanov and Martingale Representation Theorems .... 224
5.4.2 Multidimensional Market Model 226
5.4.3 Existence of the Risk-Neutral Measure 228
5.4.4 Uniqueness of the Risk-Neutral Measure 231
5.5 Dividend-Paying Stocks 234
5.5.1 Continuously Paying Dividend 235
5.5.2 Continuously Paying Dividend with
Constant Coefficients 237
5.5.3 Lump Payments of Dividends 238
5.5.4 Lump Payments of Dividends with
Constant Coefficients 239
5.6 Forwards and Futures 240
5.6.1 Forward Contracts 240
5.6.2 Futures Contracts 241
5.6.3 Forward-Futures Spread 247
5.7 Summary 248
5.8 Notes 250
5.9 Exercises 251
Connections with Partial Differential Equations 263
6.1 Introduction 263
6.2 Stochastic Differential Equations 263
6.3 The Markov Property 266
XII Contents
6.4 Partial Differential Equations 268
6.5 Interest Rate Models 272
6.6 Multidimensional Feynman-Kac Theorems 277
6.7 Summary 280
6.8 Notes 281
6.9 Exercises 282
7 Exotic Options 295
7.1 Introduction 295
7.2 Maximum of Brownian Motion with Drift 295
7.3 Knock-out Barrier Options 299
7.3.1 Up-and-Out Call 300
7.3.2 Black-Scholes-Merton Equation 300
7.3.3 Computation of the Price of the Up-and-Out Call 304
7.4 Lookback Options 308
7.4.1 Floating Strike Lookback Option 308
7.4.2 Black-Scholes-Merton Equation 309
7.4.3 Reduction of Dimension 312
7.4.4 Computation of the Price of the Lookback Option 314
7.5 Asian Options 320
7.5.1 Fixed-Strike Asian Call 320
7.5.2 Augmentation of the State 321
7.5.3 Change of Numeraire 323
7.6 Summary 331
7.7 Notes 331
7.8 Exercises 332
8 American Derivative Securities 339
8.1 Introduction 339
8.2 Stopping Times 340
8.3 Perpetual American Put 345
8.3.1 Price Under Arbitrary Exercise 346
8.3.2 Price Under Optimal Exercise 349
8.3.3 Analytical Characterization of the Put Price 351
8.3.4 Probabilistic Characterization of the Put Price 353
8.4 Finite-Expiration American Put 356
8.4.1 Analytical Characterization of the Put Price 357
8.4.2 Probabilistic Characterization of the Put Price 359
8.5 American Call 361
8.5.1 Underlying Asset Pays No Dividends 361
8.5.2 Underlying Asset Pays Dividends 363
8.6 Summary 368
8.7 Notes 369
8.8 Exercises 370
Contents XIII
9 Change of Numeraire 375
9.1 Introduction 375
9.2 Numeraire 376
9.3 Foreign and Domestic Risk-Neutral Measures 381
9.3.1 The Basic Processes 381
9.3.2 Domestic Risk-Neutral Measure 383
9.3.3 Foreign Risk-Neutral Measure 385
9.3.4 Siegel's Exchange Rate Paradox 387
9.3.5 Forward Exchange Rates 388
9.3.6 Garman-Kohlhagen Formula 390
9.3.7 Exchange Rate Put-Call Duality 390
9.4 Forward Measures 392
9.4.1 Forward Price 392
9.4.2 Zero-Coupon Bond as Numeraire 392
9.4.3 Option Pricing with a Random Interest Rate 394
9.5 Summary 397
9.6 Notes 398
9.7 Exercises 398
10 Term-Structure Models 403
10.1 Introduction 403
10.2 Affine-Yield Models 405
10.2.1 Two-Factor Vasicek Model 406
10.2.2 Two-Factor CIR Model 420
10.2.3 Mixed Model 422
10.3 Heath-Jarrow-Morton Model 423
10.3.1 Forward Rates 423
10.3.2 Dynamics of Forward Rates and Bond Prices 425
10.3.3 No-Arbitrage Condition 426
10.3.4 HJM Under Risk-Neutral Measure 429
10.3.5 Relation to Affine-Yield Models 430
10.3.6 Implementation of HJM 432
10.4 Forward LIBOR Model 435
10.4.1 The Problem with Forward Rates 435
10.4.2 LIBOR and Forward LIBOR 436
10.4.3 Pricing a Backset LIBOR Contract 437
10.4.4 Black Caplet Formula 438
10.4.5 Forward LIBOR and Zero-Coupon Bond Volatilities ... 440
10.4.6 A Forward LIBOR Term-Structure Model 442
10.5 Summary 447
10.6 Notes 450
10.7 Exercises 451
XIV Contents
11 Introduction to Jump Processes 461
11.1 Introduction 461
11.2 Poisson Process 462
11.2.1 Exponential Random Variables 462
11.2.2 Construction of a Poisson Process 463
11.2.3 Distribution of Poisson Process Increments 463
11.2.4 Mean and Variance of Poisson Increments 466
11.2.5 Martingale Property 467
11.3 Compound Poisson Process 468
11.3.1 Construction of a Compound Poisson Process 468
11.3.2 Moment-Generating Function 470
11.4 Jump Processes and Their Integrals 473
11.4.1 Jump Processes 474
11.4.2 Quadratic Variation 479
11.5 Stochastic Calculus for Jump Processes 483
11.5.1 Ito-Doeblin Formula for One Jump Process 483
11.5.2 Ito-Doeblin Formula for Multiple Jump Processes 489
11.6 Change of Measure 492
11.6.1 Change of Measure for a Poisson Process 493
11.6.2 Change of Measure for a Compound Poisson Process ... 495
11.6.3 Change of Measure for a Compound Poisson Process
and a Brownian Motion 502
11.7 Pricing a European Call in a Jump Model 505
11.7.1 Asset Driven by a Poisson Process 505
11.7.2 Asset Driven by a Brownian Motion and a Compound
Poisson Process 512
11.8 Summary 523
11.9 Notes 525
ll.lOExercises 525
A Advanced Topics in Probability Theory 527
A.I Countable Additivity 527
A.2 Generating ?-algebras 530
A.3 Random Variable with Neither Density nor Probability Mass
Function 531
? Existence of Conditional Expectations 533
C Completion of the Proof of the Second Fundamental
Theorem of Asset Pricing 535
References 537
Index 545