Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach

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An introduction to the mathematical theory and financial models developed and used on Wall Street

Providing both a theoretical and practical approach to the underlying mathematical theory behind financial models, Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach presents important concepts and results in measure theory, probability theory, stochastic processes, and stochastic calculus. Measure theory is indispensable to the rigorous development of probability theory and is also necessary to properly address martingale measures, the change of numeraire theory, and LIBOR market models. In addition, probability theory is presented to facilitate the development of stochastic processes, including martingales and Brownian motions, while stochastic processes and stochastic calculus are discussed to model asset prices and develop derivative pricing models.

The authors promote a problem-solving approach when applying mathematics in real-world situations, and readers are encouraged to address theorems and problems with mathematical rigor. In addition, Measure, Probability, and Mathematical Finance features:

  • A comprehensive list of concepts and theorems from measure theory, probability theory, stochastic processes, and stochastic calculus
  • Over 500 problems with hints and select solutions to reinforce basic concepts and important theorems
  • Classic derivative pricing models in mathematical finance that have been developed and published since the seminal work of Black and Scholes 
Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach is an ideal textbook for introductory quantitative courses in business, economics, and mathematical finance at the upper-undergraduate and graduate levels. The book is also a useful reference for readers who need to build their mathematical skills in order to better understand the mathematical theory of derivative pricing models.

Author(s): Guojun Gan, Chaoqun Ma, Hong Xie
Edition: 1
Publisher: Wiley
Year: 2014

Language: English
Pages: 744
Tags: Finance;Corporate Finance;Crowdfunding;Financial Engineering;Financial Risk Management;Wealth Management;Business & Money;Statistics;Education & Reference;Business & Money;Probability & Statistics;Applied;Mathematics;Science & Math;Statistics;Applied;Mathematics;Science & Math;Economics;Economic Theory;Macroeconomics;Microeconomics;Business & Finance;New, Used & Rental Textbooks;Specialty Boutique;Finance;Business & Finance;New, Used & Rental Textbooks;Specialty Boutique;Statistics;Mathematics;S

MEASURE, PROBABILITY, AND MATHEMATICAL FINANCE: A Problem-Oriented Approach ... 5
Copyright ... 6
CONTENTS ... 9
Preface ... 19
Financial Glossary ... 24
PART I MEASURE THEORY ... 27
1 Sets and Sequences ... 29
1.1 Basic Concepts and Facts ... 29
1.2 Problems ... 32
1.3 Hints ... 34
1.4 Solutions ... 34
1.5 Bibliographic Notes ... 39
2 MEASURES ... 41
2.1 Basic Concepts and Facts ... 41
2.2 Problems ... 44
2.3 Hints ... 46
2.4 Solutions ... 47
2.5 Bibliographic Notes ... 54
3 EXTENSION OF MEASURES ... 55
3.1 Basic Concepts and Facts ... 55
3.2 Problems ... 56
3.3 Hints ... 58
3.4 Solutions ... 58
3.5 Bibliographic Notes ... 62
4 LEBESGUE-STIELT JES MEASURES ... 63
4.1 Basic Concepts and Facts ... 63
4.2 Problems ... 65
4.3 Hints ... 67
4.4 Solutions ... 67
4.5 Bibliographic Notes ... 71
5 MEASURABLE FUNCTIONS ... 73
5.1 Basic Concepts and Facts ... 73
5.2 Problems ... 74
5.3 Hints ... 76
5.4 Solutions ... 77
5.5 Bibliographic Notes ... 82
6 LEBESGUE INTEGRATION ... 83
6.1 Basic Concepts and Facts ... 83
6.2 Problems ... 85
6.3 Hints ... 88
6.4 Solutions ... 90
6.5 Bibliographic Notes ... 102
7 THE RADON-NIKODYM THEOREM ... 103
7.1 Basic Concepts and Facts ... 103
7.2 Problems ... 105
7.3 Hints ... 106
7.4 Solutions ... 106
7.5 Bibliographic Notes ... 109
8 LP SPACES ... 111
8.1 Basic Concepts and Facts ... 111
8.2 Problems ... 114
8.3 Hints ... 115
8.4 Solutions ... 116
8.5 Bibliographic Notes ... 121
9 CONVERGENCE ... 123
9.1 Basic Concepts and Facts ... 123
9.2 Problems ... 124
9.3 Hints ... 126
9.4 Solutions ... 128
9.5 Bibliographic Notes ... 137
10 PRODUCT MEASURES ... 139
10.1 Basic Concepts and Facts ... 139
10.2 Problems ... 141
10.3 Hints ... 144
10.4 Solutions ... 144
10.5 Bibliographic Notes ... 149
PART II PROBABILITY THEORY ... 151
11 EVENTS AND RANDOM VARIABLES ... 153
11.1 Basic Concepts and Facts ... 153
11.2 Problems ... 156
11.3 Hints ... 158
11.4 Solutions ... 159
11.5 Bibliographic Notes ... 165
12 INDEPENDENCE ... 167
12.1 Basic Concepts and Facts ... 167
12.2 Problems ... 168
12.3 Hints ... 171
12.4 Solutions ... 172
12.5 Bibliographic Notes ... 185
13 EXPECTATION ... 187
13.1 Basic Concepts and Facts ... 187
13.2 Problems ... 189
13.3 Hints ... 191
13.4 Solutions ... 192
13.5 Bibliographic Notes ... 198
14 CONDITIONAL EXPECTATION ... 199
14.1 Basic Concepts and Facts ... 199
14.2 Problems ... 201
14.3 Hints ... 204
14.4 Solutions ... 205
14.5 Bibliographic Notes ... 213
15 INEQUALITIES ... 215
15.1 Basic Concepts and Facts ... 215
15.2 Problems ... 216
15.3 Hints ... 217
15.4 Solutions ... 218
15.5 Bibliographic Notes ... 224
16 LAW OF LARGE NUMBERS ... 225
16.1 Basic Concepts and Facts ... 225
16.2 Problems ... 226
16.3 Hints ... 229
16.4 Solutions ... 231
16.5 Bibliographic Notes ... 241
17 CHARACTERISTIC FUNCTIONS ... 243
17.1 Basic Concepts and Facts ... 243
17.2 Problems ... 244
17.3 Hints ... 246
17.4 Solutions ... 247
17.5 Bibliographic Notes ... 252
18 DISCRETE DISTRIBUTIONS ... 253
18.1 Basic Concepts and Facts ... 253
18.2 Problems ... 254
18.3 Hints ... 256
18.4 Solutions ... 257
18.5 Bibliographic Notes ... 263
19 CONTINUOUS DISTRIBUTIONS ... 265
19.1 Basic Concepts and Facts ... 265
19.2 Problems ... 267
19.3 Hints ... 270
19.4 Solutions ... 272
19.5 Bibliographic Notes ... 282
20 CENTRAL LIMIT THEOREMS ... 283
20.1 Basic Concepts and Facts ... 283
20.2 Problems ... 284
20.3 Hints ... 286
20.4 Solutions ... 287
20.5 Bibliographic Notes ... 293
PART III STOCHASTIC PROCESSES ... 295
21 STOCHASTIC PROCESSES ... 297
21.1 Basic Concepts and Facts ... 297
21.2 Problems ... 301
21.3 Hints ... 304
21.4 Solutions ... 306
21.5 Bibliographic Notes ... 315
22 MARTINGALES ... 317
22.1 Basic Concepts and Facts ... 317
22.2 Problems ... 318
22.3 Hints ... 320
22.4 Solutions ... 321
22.5 Bibliographic Notes ... 326
23 STOPPING TIMES ... 327
23.1 Basic Concepts and Facts ... 327
23.2 Problems ... 329
23.3 Hints ... 331
23.4 Solutions ... 333
23.5 Bibliographic Notes ... 345
24 MARTINGALE INEQUALITIES ... 347
24.1 Basic Concepts and Facts ... 347
24.2 Problems ... 348
24.3 Hints ... 349
24.4 Solutions ... 350
24.5 Bibliographic Notes ... 357
25 MARTINGALE CONVERGENCE THEOREMS ... 359
25.1 Basic Concepts and Facts ... 359
25.2 Problems ... 360
25.3 Hints ... 362
25.4 Solutions ... 362
25.5 Bibliographic Notes ... 368
26 RANDOM WALKS ... 369
26.1 Basic Concepts and Facts ... 369
26.2 Problems ... 370
26.3 Hints ... 372
26.4 Solutions ... 373
26.5 Bibliographic Notes ... 381
27 POISSON PROCESSES ... 383
27.1 Basic Concepts and Facts ... 383
27.2 Problems ... 385
27.3 Hints ... 387
27.4 Solutions ... 387
27.5 Bibliographic Notes ... 397
28 BROWNIAN MOTION ... 399
28.1 Basic Concepts and Facts ... 399
28.2 Problems ... 401
28.3 Hints ... 403
28.4 Solutions ... 404
28.5 Bibliographic Notes ... 413
29 MARKOV PROCESSES ... 415
29.1 Basic Concepts and Facts ... 415
29.2 Problems ... 417
29.3 Hints ... 419
29.4 Solutions ... 420
29.5 Bibliographic Notes ... 425
30 LEVY PROCESSES ... 427
30.1 Basic Concepts and Facts ... 427
30.2 Problems ... 430
30.3 Hints ... 433
30.4 Solutions ... 434
30.5 Bibliographic Notes ... 443
PART IV STOCHASTIC CALCULUS ... 445
31THE WIENER INTEGRAL ... 447
31.1 Basic Concepts and Facts ... 447
31.2 Problems ... 449
31.3 Hints ... 450
31.4 Solutions ... 451
31.5 Bibliographic Notes ... 455
32 THE ITO INTEGRAL ... 457
32.1 Basic Concepts and Facts ... 457
32.2 Problems ... 459
32.3 Hints ... 463
32.4 Solutions ... 464
32.5 Bibliographic Notes ... 478
33 EXTENSION OF THE ITO INTEGRAL ... 479
33.1 Basic Concepts and Facts ... 479
33.2 Problems ... 481
33.3 Hints ... 482
33.4 Solutions ... 483
33.5 Bibliographic Notes ... 488
34 MARTINGALE STOCHASTIC INTEGRALS ... 489
34.1 Basic Concepts and Facts ... 489
34.2 Problems ... 494
34.3 Hints ... 495
34.4 Solutions ... 496
34.5 Bibliographic Notes ... 501
35 THE ITO FORMULA ... 503
35.1 Basic Concepts and Facts ... 503
35.2 Problems ... 507
35.3 Hints ... 509
35.4 Solutions ... 511
35.5 Bibliographic Notes ... 520
36 MARTINGALE REPRESENTATION THEOREM ... 521
36.1 Basic Concepts and Facts ... 521
36.2 Problems ... 522
36.3 Hints ... 523
36.4 Solutions ... 524
36.5 Bibliographic Notes ... 527
37 CHANGE OF MEASURE ... 529
37.1 Basic Concepts and Facts ... 529
37.2 Problems ... 530
37.3 Hints ... 534
37.4 Solutions ... 534
37.5 Bibliographic Notes ... 539
38 STOCHASTIC DIFFERENTIAL EQUATIONS ... 541
38.1 Basic Concepts and Facts ... 541
38.2 Problems ... 543
38.3 Hints ... 547
38.4 Solutions ... 548
38.5 Bibliographic Notes ... 556
39 DIFFUSION ... 557
39.1 Basic Concepts and Facts ... 557
39.2 Problems ... 560
39.3 Hints ... 562
39.4 Solutions ... 563
39.5 Bibliographic Notes ... 571
40 THE FEYNMAN-KAC FORMULA ... 573
40.1 Basic Concepts and Facts ... 573
40.2 Problems ... 575
40.3 Hints ... 577
40.4 Solutions ... 578
40.5 Bibliographic Notes ... 583
PART V STOCHASTIC FINANCIAL MODELS ... 585
41 DISCRETE-TIME MODELS ... 587
41.1 Basic Concepts and Facts ... 587
41.2 Problems ... 591
41.3 Hints ... 594
41.4 Solutions ... 595
41.5 Bibliographic Notes ... 602
42 BLACK-SCHOLES OPTION PRICING MODELS ... 605
42.1 Basic Concepts and Facts ... 605
42.2 Problems ... 609
42.3 Hints ... 611
42.4 Solutions ... 612
42.5 Bibliographic Notes ... 617
43 PATH-DEPENDENT OPTIONS ... 619
43.1 Basic Concepts and Facts ... 619
43.2 Problems ... 624
43.3 Hints ... 626
43.4 Solutions ... 627
43.5 Bibliographic Notes ... 634
44 AMERICAN OPTIONS ... 635
44.1 Basic Concepts and Facts ... 635
44.2 Problems ... 639
44.3 Hints ... 642
44.4 Solutions ... 643
44.5 Bibliographic Notes ... 652
45 SHORT RATE MODELS ... 655
45.1 Basic Concepts and Facts ... 655
45.2 Problems ... 657
45.3 Hints ... 661
45.4 Solutions ... 661
45.5 Bibliographic Notes ... 670
46 INSTANTANEOUS FORWARD RATEMODELS ... 673
46.1 Basic Concepts and Facts ... 673
46.2 Problems ... 676
46.3 Hints ... 680
46.4 Solutions ... 680
46.5 Bibliographic Notes ... 691
47 LIBOR MARKET MODELS ... 693
47.1 Basic Concepts and Facts ... 693
47.2 Problems ... 694
47.3 Hints ... 698
47.4 Solutions ... 699
47.5 Bibliographic Notes ... 711
References ... 713
List of Symbols ... 729
Subject Index ... 733