measure probability and mathematical finance a problem oriented approach pdf

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MEASURE, PROBABILITY,
AND MATHEMATICAL
FINANCE

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MEASURE, PROBABILITY,
AND MATHEMATICAL
FINANCE
A Problem-Oriented Approach

Guojun Gan
Manulife Financial
Toronto, ON, Canada

Chaoqun Ma
School of Business Administration
Hunan University
Changsha, Hunan, P.R. China

HongXie
Manulife Financial
Toronto, ON, Canada

WILEY
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Copyright© 2014 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or
by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as
permitted under Section I 07 or I 08 of the 1976 United States Copyright Act, without either the prior
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the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax
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be addressed to the Permissions Department, John Wiley & Sons, Inc., Ill River Street, Hoboken, NJ
07030, (201) 748-60 II, fax (201) 748-6008, or online at />Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in
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completeness of the contents of this book and specifically disclaim any implied warranties of
merchantability or fitness for a particular purpose. No warranty may be created or extended by sales
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Library of Congress Cataloging-in-Publication Data is available.
Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach
Guojun Gan, Chaoqun Ma, and Hong Xie
ISBN 978-1-118-83196-0
Printed in the United States of America.
10 9 8 7 6 5 4 3 2 I

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To my parents
-Guojun Gan
To my wife and my
daughter
-ChaoqunMa
To my family and
friends
-Hong Xie

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CONTENTS

Preface

XVll

Financial Glossary

xxii
PART I

1

Sets and Sequences
1.1
1.2
1.3
1.4

1.5

2

MEASURE THEORY

3

Basic Concepts and Facts

3

Problems
Hints

6

Solutions
Bibliographic Notes

8
8
13

Measures

15

2.1

Basic Concepts and Facts
Problems

15

2.2
2.3

Hints

20

2.4

Solutions

21

18

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viii

CONTENTS

2.5

3

4

5

6

7

8

Bibliographic Notes

28

Extension of Measures

29

3.1
3.2
3.3
3.4
3.5

29
30
32
32
36

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

Lebesgue-Stieltjes Measures

37

4.1
4.2
4.3

4.4
4.5

37
39
41
41
45

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

Measurable Functions

47

5.1
5.2
5.3
5.4
5.5

47
48
50
51
56

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

Lebesgue Integration

57

6.1
6.2
6.3
6.4
6.5

57
59
62

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

64

76

The Radon-Nikodym Theorem

77

7.1
7.2
7.3
7.4
7.5

77

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

79
80
80
83

85

LP Spaces

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CONTENTS

8.1
8.2
8.3
8.4
8.5
9

10

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

12

13

85
88
89
90
95

Convergence

97

9.1
9.2
9.3
9.4
9.5

97
98
100
102
111

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

Product Measures

113

10.1
10.2
10.3
10.4
10.5

113
115
118
118
123

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes
PART II

11

ix

PROBABILITY THEORY

Events and Random Variables

127

11.1
11.2
11.3
11.4
11.5

127

130
132
133
139

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

Independence

141

12.1
12.2
12.3
12.4
12.5

141
142
145
146
159

Basic Concepts and Facts
Problems
Hints

Solutions
Bibliographic Notes

Expectation

161

13.1

161

Basic Concepts and Facts

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X

14

15

16

CONTENTS

13.2

Problems

163

13.3
13.4

Hints
Solutions

165
166

13.5

Bibliographic Notes

172

Conditional Expectation

173

14.1

Basic Concepts and Facts

14.2
14.3
14.4
14.5

Problems
Hints
Solutions
Bibliographic Notes

173
175
178
179
187

Inequalities

189

15.1
15.2
15.3
15.4
15.5

189
190

18

191
192
198

Law of Large Numbers

199

16.1
16.2

Basic Concepts and Facts

199

Problems
Hints
Solutions
Bibliographic Notes

200
203
205
215

16.3
16.4
16.5

17

Basic Concepts and Facts
Problems
Hints

Solutions
Bibliographic Notes

Characteristic Functions

217

17.1
17.2
17.3
17.4
17.5

217
218
220
221
226

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

Discrete Distributions

227

18.1

18.2

Basic Concepts and Facts
Problems

18.3
18.4

Hints
Solutions

227
228
230
231

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CONTENTS

18.5

19

Continuous Distributions
19.1
19.2
19.3
19.4

19.5

20

Bibliographic Notes

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

22

23

237

239
239
241
244
246
256

Central Limit Theorems

257

20.1

20.2
20.3
20.4
20.5

257
258
260
261
267

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

PART Ill

21

xi

STOCHASTIC PROCESSES

Stochastic Processes

271

21.1

21.2
21.3
21.4
21.5

271
275
278
280
289

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

Martingales

291

22.1
22.2
22.3
22.4
22.5

291
292
294

295
300

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

Stopping Times

301

23.1
23.2
23.3
23.4
23.5

301
303
305
307
319

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

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xii

24

CONTENTS

Martingale Inequalities

321

24.1
24.2
24.3

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

321
322
323
324
331

Martingale Convergence Theorems

333

25.1
25.2
25.3
25.4
25.5

333
334
336
336
342

24.4
24.5

25

26

27

28

Random Walks

343

26.1
26.2
26.3
26.4
26.5

343
344
346
347
355

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

Poisson Processes

357

27.1
27.2
27.3
27.4
27.5

357

359
361
361

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

371

Brownian Motion

373

28.1
28.2
28.3
28.4

373
375
377
378
387

28.5

29

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

Markov Processes

389

29.1

389
391

29.2

Basic Concepts and Facts
Problems

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CONTENTS

29.3
29.4
29.5
30

Hints
Solutions
Bibliographic Notes

32

401

30.1
30.2
30.3
30.4
30.5

401
404
407
408
417

Basic Concepts and Facts
Problems
Hints

Solutions
Bibliographic Notes

34

STOCHASTIC CALCULUS

The Wiener Integral

421

31.1
31.2
31.3
31.4
31.5

421
423
424
425
429

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

The Ito Integral

431

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

431
433
437
438
452

Extension of the Ito Integral

453

33.1
33.2
33.3
33.4
33.5

453
455
456
457
462

32.1
32.2
32.3
32.4
32.5
33

393
394
399

Levy Processes

PART IV
31

xiii

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

Martingale Stochastic Integrals

463

34.1

34.2
34.3

463
468
469

Basic Concepts and Facts
Problems
Hints

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xiv

CONTENTS

34.4
34.5

35

36

477

35.1
35.2
35.3

35.4
35.5

477
481
483
485
494

39

40

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

Martingale Representation Theorem
Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

Change of Measure
37.1
37.2
37.3

37.4
37.5

38

470
475

The Ito Formula

36.1
36.2
36.3
36.4
36.5

37

Solutions
Bibliographic Notes

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

495
495
496

497
498
501

503
503
504
508
508
513

Stochastic Differential Equations

515

38.1
38.2
38.3
38.4
38.5

515
517
521
522
530

Basic Concepts and Facts
Problems
Hints

Solutions
Bibliographic Notes

Diffusion

531

39.1
39.2
39.3
39.4
39.5

531
534
536
537
545

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

The Feynman-Kac Formula

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547

CONTENTS

40.1
40.2
40.3
40.4
40.5

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes
PART V

41

42

43

44

45

XV

547

549
551
552
557

STOCHASTIC FINANCIAL MODELS

Discrete-Time Models

561

41.1
41.2
41.3
41.4
41.5

561
565
568
569
576

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

Black-Scholes Option Pricing Models

579

42.1
42.2
42.3
42.4
42.5

579
583
585
586
591

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

Path-Dependent Options

593

43.1
43.2
43.3
43.4
43.5

593
598
600
601
608

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

American Options

609

44.1
44.2
44.3
44.4
44.5

609
613
616
617
626

Basic Concepts and Facts

Problems
Hints
Solutions
Bibliographic Notes

Short Rate Models

629

45.1

629

Basic Concepts and Facts

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xvi

CONTENTS

45.2
45.3
45.4
45.5

46

47

Problems
Hints
Solutions
Bibliographic Notes

631
635
635
644

Instantaneous Forward Rate Models

647

46.1
46.2
46.3
46.4
46.5

647
650
654
654
665

Basic Concepts and Facts
Problems
Hints

Solutions
Bibliographic Notes

LIBOR Market Models

667

47.1
47.2
47.3
47.4
47.5

667
668
672
673
685

Basic Concepts and Facts
Problems
Hints
Solutions
Bibliographic Notes

References

687

List of Symbols

703

Subject Index

707

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PREFACE

Mathematical finance, a new branch of mathematics concerned with financial markets, is experiencing rapid growth. During the last three decades, many books and
papers in the area of mathematical finance have been published. However, understanding the literature requires that the reader have a good background in measuretheoretic probability, stochastic processes, and stochastic calculus. The purpose of
this book is to provide the reader with an introduction to the mathematical theory
underlying the financial models being used and developed on Wall Street. To this
end, this book covers important concepts and results in measure theory, probability theory, stochastic processes, and stochastic calculus so that the reader will be in
a position to understand these financial models. Problems as well as solutions are
included to help the reader learn the concepts and results quickly.
In this book, we adopted the definitions and theorems from various books and
presented them in a mathematically rigorous way. We tried to cover the most of the
basic concepts and the important theorems. We selected the problems in this book
in such a way that the problems will help readers understand and know how to apply
the concepts and theorems. This book includes 516 problems, most of which are not
difficult and can be solved by applying the definitions, theorems, and the results of
previous problems.
This book is organized into five parts, each of which is further organized into several chapters. Each chapter is divided into five sections. The first section presents
xvii

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XViii

PREFACE

the definitions of important concepts and theorems. The second, third, and fourth
sections present the problems, hints on how to solve the problems, and the full solutions to the problems, respectively. The last section contains bibliographic notes.
Interdependencies between all chapters are shown in Table 0.1.
Table 0.1: Interdependencies between Chapters.
Chapter

Related to Chapter(s)

I. Sets and Sequences
2. Measures
3. Extension of Measures

1;2

4. Lebesgue-Stieltjes Measures

2;3

5. Measurable Functions

2

6. Lebesgue Integration

1;2;5

7. The Radon-Nikodym Theorem

2;6

8. LP Spaces

2;6

9. Convergence

1;2;6;8

10. Product Measures

2;3;5;6

11. Events and Random Variables

1;2;4;5

12. Independence

2;3;5;11

13. Expectation

2;6;8;10;11;12

14. Conditional Expectation

1;2;5;6;7;8;10;11; 12;13

15. Inequalities

8;11;14

16. Law of Large Numbers

2;8;9;1 0; 12; 13; 15

17. Characteristic Functions

5;6;8;11 ;12;13; 15

18. Discrete Distributions

12;14;17

19. Continuous Distributions

6;10;12;13;17

20. Central Limit Theorems

6;9; II

21. Stochastic Processes

2;5;10;11;12;19

22. Martingales

2;5;11;13;14;15

23. Stopping Times

2;5;9; 11; 14;21 ;22

24. Martingale Inequalities

2;6;8; 13;14;15;23

25. Martingale Convergence Theorems

1;6;9; 11; 14; 15;22

26. Random Walks

8;9; 13; 14; 15; 19;20;22;23;24

27. Poisson Processes

11; 12; 14; 17;21 ;22

28. Brownian Motion

8;9; 11; 12; 14; 15; 16; 17; 19

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PREFACE

XiX

29. Markov Processes

2;6;11;14;21

30. Levy Processes

1;5;6;11 ;12;14;17;19;22;27;28;29

31. The Wiener Integral

6;9;15;19;28

32. The Ito Integral

5;6;8;10;14;15;22;24;28

33. Extension of the Ito Integrals

9; 10; 14;22;23;32

34. Martingale Stochastic Integrals

14;15;19;27;32

35. The Ito Formula

6;8;9;22;24;32;34

36. Martingale Representation Theorem

9; 14;25 ;28;32;33 ;35

37. Change of Measure

7; 14;32;34;35

38. Stochastic Differential Equations

8;11;13;32;34;35

39. Diffusion

6;9; 11; 14; 19;21 ;24;32;35 ;38

40. The Feynman-Kac Formula

6; 14;32;35;38;39

41. Discrete-Time Models

7;12;14;22;23

42. Black-Scholes Option Pricing Models

9; 14; 19;24;32;33;35;36;37 ;38;41

43. Path-Dependent Options

10;14;19;28;37;38;42

44. American Options

14; 15 ;21 ;22;23;32;35 ;36;37 ;42;43

45. Short Rate Models

11;14;19;29;32;35;37;38;39;40

46. Instantaneous Forward Rate Models

10; 14; 19;32;34;35;37 ;38;40;45

47. LIBOR Market Models

14;32;37;45;46

In Part I, we present measure theory, which is indispensable to the rigorous development of probability theory. Measure theory is also necessary for us to discuss
recently developed theories and models in finance, such as the martingale measures,
the change of numeraire theory, and the London interbank offered rate (LIBOR)
market models.
In Part II, we present probability theory in a measure-theoretic mathematical
framework, which was introduced by A.N. Kolmogorov in 1937 in order to deal
with David Hilbert's sixth problem. The material presented in this part was selected

to facilitate the development of stochastic processes in Part III.
In Part III, we present stochastic processes, which include martingales and Brownian motion. In Part IV, we discuss stochastic calculus. Both stochastic processes
and stochastic calculus are important to modem mathematical finance as they are
used to model asset prices and develop derivative pricing models.
In Part V, we present some classic models in mathematical finance. Many pricing
models have been developed and published since the seminal work of Black and
Scholes. This part covers only a small portion of many models.
In this book, we tried to use a uniform set of symbols and notation. For example,
we used N, R, and 0 to denote the set of natural numbers (i.e., nonnegative integers),

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XX

PREFACE

the set of real numbers, and the empty set, respectively. A comprehensive list of
symbols is also provided at the end of this book.
We have taken great pains to ensure the accuracy of the formulas and statements
in this book. However, a few errors are inevitable in almost every book of this size.
Please feel free to contact us if you spot errors or have any other constructive suggestions.
How to Use This Book

This book can be used by individuals in various ways:
(a) It can be used as a self-study book on mathematical finance. The prerequisite is
linear algebra and calculus at the undergraduate level. This book will provide
you with a series of concepts, facts, and problems. You should explore each
problem and write out your solution in such a way that it can be shared with
others. By doing this you will be able to actively develop an in-depth and comprehensive understanding of the concepts and principles that cannot be archived

by passively reading or listening to comments of others.
(b) It can be used as a reference book. This book contains the most important
concepts and theorems from mathematical finance. The reader can find the
definition of a concept or the statement of a theorem in the book through the
index at the end of this book.
(c) It can be used as a supplementary book for individuals who take advanced
courses in mathematical finance. This book starts with measure theory and
builds up to stochastic financial models. It provides necessary prerequisites for
students who take advanced courses in mathematical finance without completing background courses.
Acknowledgments

We would like to thank all the academics and practitioners who have contributed to
the knowledge of mathematical finance. In particular, we would like to thank the
following academics and practitioners whose work constitutes the backbone of this
book: Robert B. Ash, Krishna B. Athreya, Rabi Bhattacharya, Patrick Billingsley,
Tomas Bjork, Fischer Sheffey Black, Kai Lai Chung, Erhan <;:inlar, Catherine A.
Doleans-Dade, Darrell Duffie, Richard Durrett, Robert J. Elliott, Damir Filipovic,
Allan Gut, John Hull, Ioannis Karatzas, Fima C. Klebaner, P. Ekkehard Kopp, HuiHsiung Kuo, Soumendra N. Lahiri, Damien Lamberton, Bernard Lapeyre, Gregory
F. Lawler, Robert C. Merton, Marek Musiela, Bernt Oksendal, Andrea Pascucci,
Jeffrey S. Rosenthal, Sheldon M. Ross, Marek Rutkowski, Myron Scholes, Steven
Shreve, J. Michael Steele, and Edward C. Waymire.
We are grateful to Roman Naryshkin and several anonymous reviewers for their
helpful comments. Guojun Gan and Hong Xie would like to thank their friends

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PREFACE

XXi

and colleagues at the Global Variable Annuity Hedging Department of Manulife
Financial for the pleasant cooperation over the last 4 years.
Guojun Gan gratefully acknowledges support from the CAIS (Canadian Academy
of Independent Scholars) grant and thanks Simon Fraser University for giving him
full access to its libraries. Guojun Gan wants to thank his parents and parents-in-law
for all their love and support. He wants to thank his wife, Xiaoying, for taking care
of their children.
This work was supported in part by the National Science Foundation for Distinguished Young Scholars of China (grant 70825006), Program for Changjiang Scholars and Innovative Research Team in University of Ministry of Education of China
(grant IRT0916), the Foundation for Innovative Research Groups of the National
Natural Science Foundation of China (grant 71221001), and the Furong Scholar Program.
GUOJUN GAN, CHAOQUN MA, AND HONG XIE

Toronto, ON, Canada and Changsha, Hunan, P.R. China, February 28, 2014

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Financial Glossary

American option An option that can be exercised at any time prior to the expiration
date.
Asia option An option whose payoff is dependent on the average price of the underlying asset during a certain period.
barrier option An option whose payoff is dependent on whether the path of the
underlying asset has reached a barrier, which is a certain predetermined level.
call option An option that gives the holder the right to buy an asset.
derivative A financial instrument whose price depends on the price of another asset
(called the underlying asset); also referred to as derivative security or financial
derivative.
down-and-in option A barrier option that comes into existence when the price of

the underlying asset declines to the barrier.
down-and-out option A barrier option that ceases to exist when the price of the
underlying asset declines to the barrier.
xxii

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FINANCIAL GLOSSARY

XXiii

European option An option that can be exercised only on the expiration date. Let
K be the strike price of an option. Let Sr be the price of the underlying asset
at maturity. The terminal payoff of a long position (the holder's position) of
a European call is given by max(Sr - K, 0). The terminal payoff of a long
position (the holder's position) of a European put is given by max(K- Sr, 0).
forward contract A nonstandardized agreement between two parties to buy or sell
an asset at a certain future time for a certain price.
futures contract A standardized agreement between two parties to buy or sell an
asset at a certain future time for a certain price.
LIBOR London interbank offered rate.
lookback option An option whose payoff is dependent on the maximum or minimum price of the underlying asset in a certain period.
option A derivative that gives the holder the right (not the obligation) to buy or
sell an asset by a certain date for a predetermined price. The date is called the
expiration date and the predetermined price is called the strike price or exercise
price. An option is said to be exercised if the holder chooses to buy or sell the
underlying asset.
put option An option that gives the holder the right to sell an asset.
term structure The relationship between interest rates and their maturities.

up-and-in option A barrier option that comes into existence when the price of the
underlying asset increases to the barrier.
up-and-out option A barrier option that ceases to exist when the price of the underlying asset increases to the barrier.
zero-coupon bond A bond that does not pay coupons.

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