Stochastic calculus for finance II: Continuous-time models

Series Title ......Page 1
Series Contents ......Page 2
Title ......Page 3
Date-line ......Page 4
Preface ......Page 7
Contents ......Page 9
Introduction......Page 15
Summary of Volume I ......Page 17
Summary of Volume II......Page 18
1.1 Infinite Probability Spaces ......Page 21
1.2 Random Variables and Distributions ......Page 27
1.3 Expectations ......Page 33
1.4 Convergence of Integrals ......Page 43
1.5 Computation of Expectations ......Page 47
1.6 Change of Measure ......Page 52
1.7 Summary ......Page 59
1.9 Exercises ......Page 61
2.1 Information and ?-algebras ......Page 69
2.2 Independence ......Page 73
2.3 General Conditional Expectations ......Page 86
2.4 Summary ......Page 95
2.6 Exercises ......Page 97
3.2.1 Symmetric Random Walk ......Page 103
3.2.2 Increments of the Symmetric Random Walk ......Page 104
3.2.4 Quadratic Variation of the Symmetric Random Walk ......Page 105
3.2.5 Scaled Symmetric Random Walk ......Page 106
3.2.6 Limiting Distribution of the Scaled Random Walk ......Page 108
3.2.7 Log-Normal Distribution as the Limit of the Binomial Model ......Page 111
3.3.1 Definition of Brownian Motion ......Page 113
3.3.2 Distribution of Brownian Motion ......Page 115
3.3.3 Filtration for Brownian Motion ......Page 117
3.4 Quadratic Variation ......Page 118
3.4.1 First-Order Variation ......Page 119
3.4.2 Quadratic Variation ......Page 121
3.4.3 Volatility of Geometric Brownian Motion ......Page 126
3.5 Markov Property ......Page 127
3.6 First Passage Time Distribution ......Page 128
3.7.1 Reflection Equality ......Page 131
3.7.2 First Passage Time Distribution ......Page 132
3.7.3 Distribution of Brownian Motion and Its Maximum ......Page 133
3.8 Summary ......Page 135
3.9 Notes ......Page 136
3.10 Exercises ......Page 137
4.2 Ito's Integral for Simple Integrands ......Page 145
4.2.1 Construction of the Integral ......Page 146
4.2.2 Properties of the Integral ......Page 148
4.3 Ito's Integral for General Integrands ......Page 152
4.4.1 Formula for Brownian Motion ......Page 157
4.4.2 Formula for Ito Processes ......Page 163
4.4.3 Examples ......Page 167
4.5 Black-Scholes-Merton Equation ......Page 173
4.5.1 Evolution of Portfolio Value ......Page 174
4.5.2 Evolution of Option Value ......Page 175
4.5.3 Equating the Evolutions ......Page 176
4.5.4 Solution to the Black-Scholes-Merton Equation ......Page 178
4.5.5 The Greeks ......Page 179
4.5.6 Put-Call Parity ......Page 182
4.6.1 Multiple Brownian Motions ......Page 184
4.6.2 Ito-Doeblin Formula for Multiple Processes ......Page 185
4.6.3 Recognizing a Brownian Motion ......Page 188
4.7.1 Gaussian Processes ......Page 192
4.7.2 Brownian Bridge as a Gaussian Process ......Page 195
4.7.3 Brownian Bridge as a Scaled Stochastic Integral ......Page 196
4.7.4 Multidimensional Distribution of the Brownian Bridge ......Page 198
4.7.5 Brownian Bridge as a Conditioned Brownian Motion ......Page 202
4.8 Summary ......Page 203
4.9 Notes ......Page 207
4.10 Exercises ......Page 209
5.1 Introduction ......Page 229
5.2.1 Girsanov's Theorem for a Single Brownian Motion ......Page 230
5.2.2 Stock Under the Risk-Neutral Measure ......Page 234
5.2.3 Value of Portfolio Process Under the Risk-Neutral Measure ......Page 237
5.2.5 Deriving the Black-Scholes-Merton Formula ......Page 238
5.3.1 Martingale Representation with One Brownian Motion ......Page 241
5.3.2 Hedging with One Stock ......Page 242
5.4.1 Girsanov and Martingale Representation Theorems ......Page 244
5.4.2 Multidimensional Market Model ......Page 246
5.4.3 Existence of the Risk-Neutral Measure ......Page 248
5.4.4 Uniqueness of the Risk-Neutral Measure ......Page 251
5.5 Dividend-Paying Stocks ......Page 254
5.5.1 Continuously Paying Dividend ......Page 255
5.5.2 Continuously Paying Dividend with Constant Coefficients ......Page 257
5.5.3 Lump Payments of Dividends ......Page 258
5.5.4 Lump Payments of Dividends with Constant Coefficients ......Page 259
5.6.1 Forward Contracts ......Page 260
5.6.2 Futures Contracts ......Page 261
5.6.3 Forward-Futures Spread ......Page 267
5.7 Summary ......Page 268
5.8 Notes ......Page 270
5.9 Exercises ......Page 271
6.2 Stochastic Differential Equations ......Page 283
6.3 The Markov Property ......Page 286
6.4 Partial Differential Equations ......Page 288
6.5 Interest Rate Models ......Page 292
6.6 Multidimensional Feynman-Kac Theorems ......Page 297
6.7 Summary ......Page 300
6.8 Notes ......Page 301
6.9 Exercises ......Page 302
7.2 Maximum of Brownian Motion with Drift ......Page 315
7.3 Knock-out Barrier Options ......Page 319
7.3.2 Black-Scholes-Merton Equation ......Page 320
7.3.3 Computation of the Price of the Up-and-Out Call ......Page 324
7.4.1 Floating Strike Lookback Option ......Page 328
7.4.2 Black-Scholes-Merton Equation ......Page 329
7.4.3 Reduction of Dimension ......Page 332
7.4.4 Computation of the Price of the Lookback Option ......Page 334
7.5.1 Fixed-Strike Asian Call ......Page 340
7.5.2 Augmentation of the State ......Page 341
7.5.3 Change of Numeraire ......Page 343
7.7 Notes ......Page 351
7.8 Exercises ......Page 352
8.1 Introduction ......Page 359
8.2 Stopping Times ......Page 360
8.3 Perpetual American Put ......Page 365
8.3.1 Price Under Arbitrary Exercise ......Page 366
8.3.2 Price Under Optimal Exercise ......Page 369
8.3.3 Analytical Characterization of the Put Price ......Page 371
8.3.4 Probabilistic Characterization of the Put Price ......Page 373
8.4 Finite-Expiration American Put ......Page 376
8.4.1 Analytical Characterization of the Put Price ......Page 377
8.4.2 Probabilistic Characterization of the Put Price ......Page 379
8.5.1 Underlying Asset Pays No Dividends ......Page 381
8.5.2 Underlying Asset Pays Dividends ......Page 383
8.6 Summary ......Page 388
8.7 Notes ......Page 389
8.8 Exercises ......Page 390
9.1 Introduction ......Page 395
9.2 Numeraire ......Page 396
9.3.1 The Basic Processes ......Page 401
9.3.2 Domestic Risk-Neutral Measure ......Page 403
9.3.3 Foreign Risk-Neutral Measure ......Page 405
9.3.4 Siegel's Exchange Rate Paradox ......Page 407
9.3.5 Forward Exchange Rates ......Page 408
9.3.7 Exchange Rate Put-Call Duality ......Page 410
9.4.2 Zero-Coupon Bond as Numeraire ......Page 412
9.4.3 Option Pricing with a Random Interest Rate ......Page 414
9.5 Summary ......Page 417
9.7 Exercises ......Page 418
10.1 Introduction ......Page 423
10.2 Affine-Yield Models ......Page 425
10.2.1 Two-Factor Vasicek Model ......Page 426
10.2.2 Two-Factor CIR Model ......Page 440
10.2.3 Mixed Model ......Page 442
10.3.1 Forward Rates ......Page 443
10.3.2 Dynamics of Forward Rates and Bond Prices ......Page 445
10.3.3 No-Arbitrage Condition ......Page 446
10.3.4 HJM Under Risk-Neutral Measure ......Page 449
10.3.5 Relation to Affine-Yield Models ......Page 450
10.3.6 Implementation of HJM ......Page 452
10.4.1 The Problem with Forward Rates ......Page 455
10.4.2 LIBOR and Forward LIBOR ......Page 456
10.4.3 Pricing a Backset LIBOR Contract ......Page 457
10.4.4 Black Caplet Formula ......Page 458
10.4.5 Forward LIBOR and Zero-Coupon Bond Volatilities ......Page 460
10.4.6 A Forward LIBOR Term-Structure Model ......Page 462
10.5 Summary ......Page 467
10.6 Notes ......Page 470
10.7 Exercises ......Page 471
11.1 Introduction ......Page 481
11.2.1 Exponential Random Variables ......Page 482
11.2.3 Distribution of Poisson Process Increments ......Page 483
11.2.4 Mean and Variance of Poisson Increments ......Page 486
11.2.5 Martingale Property ......Page 487
11.3.1 Construction of a Compound Poisson Process ......Page 488
11.3.2 Moment-Generating Function ......Page 490
11.4 Jump Processes and Their Integrals ......Page 493
11.4.1 Jump Processes ......Page 494
11.4.2 Quadratic Variation ......Page 499
11.5.1 Ito-Doeblin Formula for One Jump Process ......Page 503
11.5.2 Ito-Doeblin Formula for Multiple Jump Processes ......Page 509
11.6 Change of Measure ......Page 512
11.6.1 Change of Measure for a Poisson Process ......Page 513
11.6.2 Change of Measure for a Compound Poisson Process ......Page 515
11.6.3 Change of Measure for a Compound Poisson Process and a Brownian Motion ......Page 522
11.7.1 Asset Driven by a Poisson Process ......Page 525
11.7.2 Asset Driven by a Brownian Motion and a Compound Poisson Process ......Page 532
11.8 Summary ......Page 543
11.10 Exercises ......Page 545
A.1 Countable Additivity ......Page 547
A.2 Generating $\sigma$-algebras ......Page 550
A.3 Random Variable with Neither Density nor Probability Mass Function ......Page 551
B Existence of Conditional Expectations ......Page 553
C Completion of the Proof of the Second Fundamental Theorem of Asset Pricing ......Page 555
References ......Page 557
Index ......Page 565