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Arbitrage Theory in Continuous Time 3rd Revised edition [Hardback]

4.21/5 (78 ratings by Goodreads)
(Professor of Mathematical Finance, Stockholm School of Economics)
  • Formāts: Hardback, 560 pages, height x width x depth: 241x161x32 mm, weight: 938 g, 23 Figures
  • Sērija : Oxford Finance Series
  • Izdošanas datums: 06-Aug-2009
  • Izdevniecība: Oxford University Press
  • ISBN-10: 019957474X
  • ISBN-13: 9780199574742
Citas grāmatas par šo tēmu:
Arbitrage Theory in Continuous Time 3rd Revised editionPalielināt
  • Formāts: Hardback, 560 pages, height x width x depth: 241x161x32 mm, weight: 938 g, 23 Figures
  • Sērija : Oxford Finance Series
  • Izdošanas datums: 06-Aug-2009
  • Izdevniecība: Oxford University Press
  • ISBN-10: 019957474X
  • ISBN-13: 9780199574742
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9780199574742.html
Keywords:
The third edition of this popular introduction to the classical underpinnings of the mathematics behind finance continues to combine sound mathematical principles with economic applications.

Concentrating on the probabilistic theory of continuous arbitrage pricing of financial derivatives, including stochastic optimal control theory and Merton's fund separation theory, the book is designed for graduate students and combines necessary mathematical background with a solid economic focus. It includes a solved example for every new technique presented, contains numerous exercises, and suggests further reading in each chapter.

In this substantially extended new edition Bjork has added separate and complete chapters on the martingale approach to optimal investment problems, optimal stopping theory with applications to American options, and positive interest models and their connection to potential theory and stochastic discount factors.

More advanced areas of study are clearly marked to help students and teachers use the book as it suits their needs.

Recenzijas

Review from previous edition This book is one of the best of a large number of new books on mathematical and probabilistic models in finance, positioned between the books by Hull and Duffie on a mathematical scale...This is a highly reasonable book and strikes a balance between mathematical development and intuitive explanation * Short Book Reviews *

1 Introduction 1
1.1 Problem Formulation
1
2 The Binomial Model 5
2.1 The One Period Model
5
2.1.1 Model Description
5
2.1.2 Portfolios and Arbitrage
6
2.1.3 Contingent Claims
9
2.1.4 Risk Neutral Valuation
11
2.2 The Multiperiod Model
15
2.2.1 Portfolios and Arbitrage
15
2.2.2 Contingent Claims
17
2.3 Exercises
25
2.4 Notes
25
3 A More General One Period Model 26
3.1 The Model
26
3.2 Absence of Arbitrage
27
3.3 Martingale Measures
32
3.4 Martingale Pricing
34
3.5 Completeness
35
3.6 Stochastic Discount Factors
38
3.7 Exercises
39
4 Stochastic Integrals 40
4.1 Introduction
40
4.2 Information
42
4.3 Stochastic Integrals
44
4.4 Martingales
46
4.5 Stochastic Calculus and the Ito Formula
49
4.6 Examples
54
4.7 The Multidimensional Ito Formula
57
4.8 Correlated Wiener Processes
59
4.9 Exercises
63
4.10 Notes
65
5 Differential Equations 66
5.1 Stochastic Differential Equations
66
5.2 Geometric Brownian Motion
67
5.3 The Linear SDE
70
5.4 The Infinitesimal Operator
71
5.5 Partial Differential Equations
72
5.6 The Kolmogorov Equations
76
5.7 Exercises
79
5.8 Notes
83
6 Portfolio Dynamics 84
6.1 Introduction
84
6.2 Self-financing Portfolios
87
6.3 Dividends
89
6.4 Exercises
91
7 Arbitrage Pricing 92
7.1 Introduction
92
7.2 Contingent Claims and Arbitrage
93
7.3 The Black–Scholes Equation
98
7.4 Risk Neutral Valuation
102
7.5 The Black–Scholes Formula
104
7.6 Options on Futures
106
7.6.1 Forward Contracts
106
7.6.2 Futures Contracts and the Black Formula
107
7.7 Volatility
108
7.7.1 Historic Volatility
109
7.7.2 Implied Volatility
110
7.8 American Options
110
7.9 Exercises
112
7.10 Notes
114
8 Completeness and Hedging 115
8.1 Introduction
115
8.2 Completeness in the Black–Scholes Model
116
8.3 Completeness Absence of Arbitrage
121
8.4 Exercises
122
8.5 Notes
124
9 Parity Relations and Delta Hedging 125
9.1 Parity Relations
125
9.2 The Greeks
127
9.3 Delta and Gamma Hedging
130
9.4 Exercises
134
10 The Martingale Approach to Arbitrage Theory* 137
10.1 The Case with Zero Interest Rate
137
10.2 Absence of Arbitrage
140
10.2.1 A Rough Sketch of the Proof
141
10.2.2 Precise Results
144
10.3 The General Case
146
10.4 Completeness
149
10.5 Martingale Pricing
151
10.6 Stochastic Discount Factors
153
10.7 Summary for the Working Economist
154
10.8 Notes
156
11 The Mathematics of the Martingale Approach* 158
11.1 Stochastic Integral Representations
158
11.2 The Girsanov Theorem: Heuristics
162
11.3 The Girsanov Theorem
164
11.4 The Converse of the Girsanov Theorem
168
11.5 Girsanov Transformations and Stochastic Differentials
168
11.6 Maximum Likelihood Estimation
169
11.7 Exercises
171
11.8 Notes
172
12 Black-Scholes from a Martingale Point of View* 173
12.1 Absence of Arbitrage
173
12.2 Pricing
175
12.3 Completeness
176
13 Multidimensional Models: Classical Approach 179
13.1 Introduction
179
13.2 Pricing
181
13.3 Risk Neutral Valuation
187
13.4 Reducing the State Space
188
13.5 Hedging
192
13.6 Exercises
195
14 Multidimensional Models: Martingale Approach* 196
14.1 Absence of Arbitrage
197
14.2 Completeness
199
14.3 Hedging
200
14.4 Pricing
202
14.5 Markovian Models and PDEs
203
14.6 Market Prices of Risk
204
14.7 Stochastic Discount Factors
205
14.8 The Hansen-Jagannathan Bounds
205
14.9 Exercises
208
14.10 Notes
208
15 Incomplete Markets 209
15.1 Introduction
209
15.2 A Scalar Nonpriced Underlying Asset
209
15.3 The Multidimensional Case
218
15.4 A Stochastic Short Rate
222
15.5 The Martingale Approach*
223
15.6 Summing Up
224
15.7 Exercises
227
15.8 Notes
228
16 Dividends 229
16.1 Discrete Dividends
229
16.1.1 Price Dynamics and Dividend Structure
229
16.1.2 Pricing Contingent Claims
230
16.2 Continuous Dividends
235
16.2.1 Continuous Dividend Yield
236
16.2.2 The General Case
239
16.3 The Martingale Approach*
241
16.3.1 The Bank Account as Numeraire
242
16.3.2 An Arbitrary Numeraire
243
16.4 Exercises
246
17 Currency Derivatives 247
17.1 Pure Currency Contracts
247
17.2 Domestic and Foreign Equity Markets
250
17.3 Domestic and Foreign Market Prices of Risk
256
17.4 The Martingale Approach*
260
17.5 Exercises
263
17.6 Notes
264
18 Barrier Options 265
18.1 Mathematical Background
265
18.2 Out Contracts
267
18.2.1 Down-and-out Contracts
267
18.2.2 Up-and-out Contracts
271
18.2.3 Examples
272
18.3 In Contracts
276
18.4 Ladders
278
18.5 Lookbacks
279
18.6 Exercises
281
18.7 Notes
281
19 Stochastic Optimal Control 282
19.1 An Example
282
19.2 The Formal Problem
283
19.3 The Hamilton Jacobi Hellman Equation
286
19.4 Handling the HJB Equation
294
19.5 The Linear Regulator
295
19.6 Optimal Consumption and Investment
297
19.6.1 A Generalization
297
19.6.2 Optimal Consumption
299
19.7 The Mutual Fund Theorems
302
19.7.1 The Case with No Risk Free Asset
302
19.7.2 The Case with a Risk Free Asset
306
19.8 Exercises
308
19.9 Notes
312
20 The Martingale Approach to Optimal Investment* 313
20.1 Generalities
313
20.2 The Basic Idea
314
20.3 The Optimal Terminal Wealth
315
20.4 The Optimal Portfolio
317
20.5 Power Utility
318
20.5.1 The Optimal Terminal Wealth Profile
318
20.5.2 The Optimal Wealth Process
320
20.5.3 The Optimal Portfolio
321
20.6 The Markovian Case
322
20.7 Log Utility
324
20.8 Exponential Utility
324
20.8.1 The Optimal Terminal Wealth
325
20.8.2 The Optimal Wealth Process
325
20.8.3 The Optimal Portfolio
326
20.9 Exercises
327
20.10 Notes
328
21 Optimal Stopping Theory and American Options* 329
21.1 Introduction
329
21.2 Generalities
329
21.3 Some Simple Results
330
21.4 Discrete Time
331
21.4.1 The General Case
331
21.4.2 Markovian Models
335
21.4.3 Infinite Horizon
337
21.5 Continuous Time
339
21.5.1 General Theory
339
21.5.2 Diffusion Models
341
21.5.3 Connections to the General Theory
345
21.6 American Options
345
21.6.1 The American Call Without Dividends
345
21.6.2 The American Put Option
346
21.6.3 The Perpetual American Put
347
21.7 Exercises
348
21.8 Notes
349
22 Bonds and Interest Rates 350
22.1 Zero Coupon Bonds
350
22.2 Interest Rates
351
22.2.1 Definitions
351
22.2.2 Relations between df(t,T), dp(t,T) and dr(t)
353
22.2.3 An Alternative View of the Money Account
356
22.3 Coupon Bonds, Swaps and Yields
357
22.3.1 Fixed Coupon Bonds
358
22.3.2 Floating Rate Bonds
358
22.3.3 Interest Rate Swaps
360
22.3.4 Yield and Duration
361
22.4 Exercises
362
22.5 Notes
363
23 Short Rate Models 364
23.1 Generalities
364
23.2 The Term Structure Equation
367
23.3 Exercises
372
23.4 Notes
373
24 Martingale Models for the Short Rate 374
24.1 Q-dynamics
374
24.2 Inversion of the Yield Curve
375
24.3 Affine Term Structures
377
24.3.1 Definition and Existence
377
24.3.2 A Probabilistic Discussion
379
24.4 Sonic Standard Models
381
24.4.1 The Vasileek Model
381
24.4.2 The Ho-Lee Model
382
24.4.3 The CIR Model
383
24.4.4 The Hull-White Model
383
24.5 Exercises
386
24.6 Notes
387
25 Forward Rate Models 388
25.1 The Heath-Jarrow-Morton Framework
388
25.2 Martingale Modeling
390
25.3 The Musiela Parameterization
392
25.4 Exercises
393
25.5 Notes
395
26 Change of Numeraire* 396
26.1 Introduction
396
26.2 Generalities
397
26.3 Changing the Numeraire
401
26.4 Forward Measures
403
26.4.1 Using the T-bond as Numeraire
403
26.4.2 An Expectation Hypothesis
405
26.5 A General Option Pricing Formula
406
26.6 The Hull-White Model
409
26.7 The General Gaussian Model
411
26.8 Caps and Floors
413
26.9 The Numeraire Portfolio
414
26.10 Exercises
415
26.11 Notes
415
27 LIBOR and Swap Market Models 417
27.1 Caps: Definition and Market Practice
418
27.2 The LIBOR Market Model
420
27.3 Pricing Caps in the LIBOR Model
421
27.4 Terminal Measure Dynamics and Existence
422
27.5 Calibration and Simulation
425
27.6 The Discrete Savings Account
427
27.7 Swaps
428
27.8 Swaptions: Definition and Market Practice
430
27.9 The Swap Market Models
431
27.10 Pricing Swaptions in the Swap Market Model
432
27.11 Drift Conditions for the Regular Swap Market Model
433
27.12 Concluding Comment
436
27.13 Exercises
436
27.14 Notes
437
28 Potentials and Positive Interest 438
28.1 Generalities
438
28.2 The Flesaker- Hughston Framework
439
28.3 Changing Base Measure
443
28.4 Decomposition of a Potential
444
28.5 The Markov Potential Approach of Rogers
445
28.6 Exercises
449
28.7 Notes
451
29 Forwards and Futures 452
29.1 Forward Contracts
452
29.2 Futures Contracts
454
29.3 Exercises
457
29.4 Notes
457
A Measure and Integration* 458
A.1 Sets and Mappings
458
A.2 Measures and Sigma Algebras
460
A.3 Integration
462
A.4 Sigma-Algebras and Partitions
467
A.5 Sets of Measure Zero
468
A.6 The L1 Spaces
469
A.7 Hilbert Spaces
470
A.8 Sigma-Algebras and Generators
473
A.9 Product Measures
476
A.10 The Lebesgue Integral
477
A.11 The Radon–Nikodym Theorem
478
A.12 Exercises
482
A.13 Notes
483
B Probability Theory* 484
B.1 Random Variables and Processes
484
B.2 Partitions and Information
487
B.3 Sigma-algebras and Information
489
B.4 Independence
492
B.5 Conditional Expectations
493
B.6 Equivalent Probability Measures
500
B.7 Exercises
502
B.8 Notes
503
C Martingales and Stopping Times* 504
C.1 Martingales
504
C.2 Discrete Stochastic Integrals
507
C.3 Likelihood Processes
508
C.4 Stopping Times
509
C.5 Exercises
512
References 514
Index 521
Tomas Björk is Professor of Mathematical Finance at the Stockholm School of Economics. His background is in probability theory and he was formerly at the Mathematics Department of the Royal Institute of Technology in Stockholm. He is co-editor of Mathematical Finance and Associate Editor of Finance and Stochastics. He has published numerous journal articles on mathematical finance in general, and in particular on interest rate theory.