Brownian motion calculus:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Chichester
Wiley
2008
|
| Series: | Wiley finance series
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XV, 313 S. graph. Darst. |
| ISBN: | 9780470021705 |
Staff View
MARC
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| 020 | |a 9780470021705 |c pbk. : alk. paper |9 978-0-470-02170-5 | ||
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| 245 | 1 | 0 | |a Brownian motion calculus |c Ubbo F. Wiersema |
| 264 | 1 | |a Chichester |b Wiley |c 2008 | |
| 300 | |a XV, 313 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 0 | |a Wiley finance series | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Mathematisches Modell | |
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Record in the Search Index
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| adam_text | Contents
Preface
xiii
1 Brownian Motion 1
1.1
Origins I
1.2 Brownian Motion
Specification
2
1.3
Use of
Brownian Motion in Stock
Price
Dynamics 4
1.4
Construction of
Brownian Motion
from
a
Symmetrie
Random Walk 6
1.5 Covariance
of
Brownian Motion 12
16
Correlated
Brownian
Motions
14
1.7
Successive
Brownian Motion
Increments
16
1.7.1
Numerical
Illustration 17
1.8 Features
of a
Brownian Motion
Path
19
1.8.1 Simulation
of
Brownian Motion
Paths
19
1.8.2
Slope of Path
20
1.8.3
Non-Differentiability of Brownian Motion
Path
21
1.8.4
Measuring Variability
24
1.9
Exercises
26
1.10
Summary
29
2
Martingales
31
2.1
Simple Example
31
2.2
Filtration 32
2.3
Conditional Expectation
33
2.3.1
General Properties
34
Contents
2.4
Martingale Description
36
2.4.1
Martingale Construction
by Conditioning
36
2.5
Martingale
Analysis Steps
37
2.6
Examples of
Martingale
Analysis
37
2.6.1
Sum of Independent Trials
37
2.6.2
Square of Sum of Independent Trials
38
2.6.3
Product of Independent Identical Trials
39
2.6.4
Random Process B(t)
39
2.6.5
Random Process
exp[ß(i) -
t]
40
2.6.6
Frequently Used Expressions
40
2.7
Process of Independent Increments
41
2.8
Exercises
42
2.9
Summary
42
3
Ito
Stochastic Integral
45
3.1
How a Stochastic Integral Arises
45
3.2
Stochastic Integral for Non-Random Step-Functions
47
3.3
Stochastic Integral for
Non-
Anticipating Random
Step-Functions
49
3.4
Extension to
Non-
Anticipating General Random
Integrands
52
3.5
Properties of an
Ito
Stochastic Integral
57
3.6
Significance of Integrand Position
59
3.7
Ito
integral of Non-Random Integrand
61
3.8
Area under a Brownian Motion Path
62
3.9
Exercises
64
3.10
Summary
67
3.11
A Tribute to Kiyosi
Ito
68
Acknowledgment
72
4
Ito
Calculus
73
4.1
Stochastic Differential Notation
73
4.2
Taylor Expansion in Ordinary Calculus
74
4.3
Ito s Formula as a Set of Rules
75
4.4
Illustrations of Ito s Formula
78
4.4.1
Frequent Expressions for Functions of Two
Processes
78
4.4.2
Function of Brownian Motion
ƒ
[B(t)]
80
4.4.3
Function of Time and Brownian Motion
fit, B(t)]
82
Contents ix
4.4.4
Finding
an Expression
for
f^0 B(t) dB(t) 83
4.4.5
Change of Numeraire
84
4.4.6
Deriving an Expectation via an ODE
85
4.5
Levy Characterization of Brownian Motion
87
4.6
Combinations of Brownian Motions
89
4.7
Multiple Correlated Brownian Motions
92
4.8
Area under a Brownian Motion Path
-
Revisited
95
4.9
Justification of Ito s Formula
96
4.10
Exercises
100
4.11
Summary
101
5
Stochastic Differential Equations
103
5.1
Structure of a Stochastic Differential Equation
103
5.2
Arithmetic Brownian Motion SDE
104
5.3
Geometric Brownian Motion SDE
105
5.4
Ornstein-Uhlenbeck SDE
108
5.5
Mean-Reversion SDE
110
5.6
Mean-Reversion with Square-Root Diffusion SDE
112
5.7
Expected Value of Square-Root Diffusion Process
112
5.8
Coupled SDEs
114
5.9
Checking the Solution of a SDE
115
5.10
General Solution Methods for Linear SDEs
115
5.11
Martingale Representation
120
5.12
Exercises
123
5.13
Summary
124
6
Option Valuation
127
6.1
Partial Differential Equation Method
128
6.2
Martingale Method in One-Period Binomial
Framework
130
6.3
Martingale Method in Continuous-Time Framework
135
6.4
Overview of Risk-Neutral Method
138
6.5
Martingale Method Valuation of Some European
Options
139
6.5.1
Digital Call
139
6.5.2
Asset-or-Nothing Call
141
6.5.3
Standard European Call
142
6.6
Links between Methods
144
6.6.1
Feynman-Kač
Link between PDE Method
and Martingale Method
144
6.6.2
Multi-Period Binomial Link to Continuous
146
Contents
6.7
Exercise
147
6.8
Summary
148
7
Change
of Probability
151
7.1
Change of Discrete Probability Mass
151
7.2
Change of Normal Density
153
7.3
Change of Brownian Motion
154
7.4
Girsanov Transformation
155
7.5
Use in Stock Price Dynamics
-
Revisited
160
7.6
General Drift Change
162
7.7
Use in Importance Sampling
163
7.8
Use in Deriving Conditional Expectations
167
7.9
Concept of Change of Probability
172
7.9.1
Relationship between Expected Values under
Equivalent Probabilities
174
7.10
Exercises
174
7.11
Summary
176
8
Numeraire
179
8.1
Change of Numeraire
179
8.1.1
In Discrete Time
179
8.1.2
In Continuous Time
182
8.2
Forward Price Dynamics
184
8.2.1
Dynamics of Forward Price of a Bond
184
8.2.2
Dynamics of Forward Price of any Traded
Asset
185
8.3
Option Valuation under most Suitable Numeraire
187
8.3.1
Exchange Option
187
8.3.2
Option on Bond
188
8.3.3
European Call under Stochastic Interest Rate
188
8.4
Relating Change of Numeraire to Change of
Probability
190
8.5
Change of Numeraire for Geometric Brownian
Motion
192
8.6
Change of Numeraire in
LIBOR
Market
Model
194
8.7
Application in Credit Risk Modelling
198
8.8
Exercises
200
8.9
Summary
201
Contents xi
ANNEXES
A
Annex
A: Computations with Brownian Motion
205
A.I Moment Generating Function and Moments
of Brownian Motion
205
A.2 Probability of Brownian Motion Position
208
A.3 Brownian Motion Reflected at the Origin
208
A.4 First Passage of a Barrier
214
A.5 Alternative Brownian Motion Specification
216
В
Annex B: Ordinary Integration
221
B.I Riemann Integral
221
B.2 Riemann-Stieltjes Integral
226
B.3 Other Useful Properties
231
B.4 References
234
С
Annex C: Brownian Motion Variability
235
C.I Quadratic Variation
235
C.2 First Variation
238
D
Annex
D:
Norms
239
D.
1
Distance between Points
239
242
244
244
246
247
247
248
249
250
250
253
299
303
D.2
Norm of a Function
D.3
Norm of a Random Variable
D.4
Norm of a Random Process
D.5
Reference
E
Annex E: Convergence Concepts
E.I
Central Limit Theorem
E.2
Mean-Square Convergence
E.3
Almost Sure Convergence
E.4
Convergence in Probability
E.5
Summary
Answers
•
to Exercises
References
Index
|
| adam_txt |
Contents
Preface
xiii
1 Brownian Motion 1
1.1
Origins I
1.2 Brownian Motion
Specification
2
1.3
Use of
Brownian Motion in Stock
Price
Dynamics 4
1.4
Construction of
Brownian Motion
from
a
Symmetrie
Random Walk 6
1.5 Covariance
of
Brownian Motion 12
16
Correlated
Brownian
Motions
14
1.7
Successive
Brownian Motion
Increments
16
1.7.1
Numerical
Illustration 17
1.8 Features
of a
Brownian Motion
Path
19
1.8.1 Simulation
of
Brownian Motion
Paths
19
1.8.2
Slope of Path
20
1.8.3
Non-Differentiability of Brownian Motion
Path
21
1.8.4
Measuring Variability
24
1.9
Exercises
26
1.10
Summary
29
2
Martingales
31
2.1
Simple Example
31
2.2
Filtration 32
2.3
Conditional Expectation
33
2.3.1
General Properties
34
Contents
2.4
Martingale Description
36
2.4.1
Martingale Construction
by Conditioning
36
2.5
Martingale
Analysis Steps
37
2.6
Examples of
Martingale
Analysis
37
2.6.1
Sum of Independent Trials
37
2.6.2
Square of Sum of Independent Trials
38
2.6.3
Product of Independent Identical Trials
39
2.6.4
Random Process B(t)
39
2.6.5
Random Process
exp[ß(i) -
\t]
40
2.6.6
Frequently Used Expressions
40
2.7
Process of Independent Increments
41
2.8
Exercises
42
2.9
Summary
42
3
Ito
Stochastic Integral
45
3.1
How a Stochastic Integral Arises
45
3.2
Stochastic Integral for Non-Random Step-Functions
47
3.3
Stochastic Integral for
Non-
Anticipating Random
Step-Functions
49
3.4
Extension to
Non-
Anticipating General Random
Integrands
52
3.5
Properties of an
Ito
Stochastic Integral
57
3.6
Significance of Integrand Position
59
3.7
Ito
integral of Non-Random Integrand
61
3.8
Area under a Brownian Motion Path
62
3.9
Exercises
64
3.10
Summary
67
3.11
A Tribute to Kiyosi
Ito
68
Acknowledgment
72
4
Ito
Calculus
73
4.1
Stochastic Differential Notation
73
4.2
Taylor Expansion in Ordinary Calculus
74
4.3
Ito's Formula as a Set of Rules
75
4.4
Illustrations of Ito's Formula
78
4.4.1
Frequent Expressions for Functions of Two
Processes
78
4.4.2
Function of Brownian Motion
ƒ
[B(t)]
80
4.4.3
Function of Time and Brownian Motion
fit, B(t)]
82
Contents ix
4.4.4
Finding
an Expression
for
f^0 B(t) dB(t) 83
4.4.5
Change of Numeraire
84
4.4.6
Deriving an Expectation via an ODE
85
4.5
Levy Characterization of Brownian Motion
87
4.6
Combinations of Brownian Motions
89
4.7
Multiple Correlated Brownian Motions
92
4.8
Area under a Brownian Motion Path
-
Revisited
95
4.9
Justification of Ito's Formula
96
4.10
Exercises
100
4.11
Summary
101
5
Stochastic Differential Equations
103
5.1
Structure of a Stochastic Differential Equation
103
5.2
Arithmetic Brownian Motion SDE
104
5.3
Geometric Brownian Motion SDE
105
5.4
Ornstein-Uhlenbeck SDE
108
5.5
Mean-Reversion SDE
110
5.6
Mean-Reversion with Square-Root Diffusion SDE
112
5.7
Expected Value of Square-Root Diffusion Process
112
5.8
Coupled SDEs
114
5.9
Checking the Solution of a SDE
115
5.10
General Solution Methods for Linear SDEs
115
5.11
Martingale Representation
120
5.12
Exercises
123
5.13
Summary
124
6
Option Valuation
127
6.1
Partial Differential Equation Method
128
6.2
Martingale Method in One-Period Binomial
Framework
130
6.3
Martingale Method in Continuous-Time Framework
135
6.4
Overview of Risk-Neutral Method
138
6.5
Martingale Method Valuation of Some European
Options
139
6.5.1
Digital Call
139
6.5.2
Asset-or-Nothing Call
141
6.5.3
Standard European Call
142
6.6
Links between Methods
144
6.6.1
Feynman-Kač
Link between PDE Method
and Martingale Method
144
6.6.2
Multi-Period Binomial Link to Continuous
146
Contents
6.7
Exercise
147
6.8
Summary
148
7
Change
of Probability
151
7.1
Change of Discrete Probability Mass
151
7.2
Change of Normal Density
153
7.3
Change of Brownian Motion
154
7.4
Girsanov Transformation
155
7.5
Use in Stock Price Dynamics
-
Revisited
160
7.6
General Drift Change
162
7.7
Use in Importance Sampling
163
7.8
Use in Deriving Conditional Expectations
167
7.9
Concept of Change of Probability
172
7.9.1
Relationship between Expected Values under
Equivalent Probabilities
174
7.10
Exercises
174
7.11
Summary
176
8
Numeraire
179
8.1
Change of Numeraire
179
8.1.1
In Discrete Time
179
8.1.2
In Continuous Time
182
8.2
Forward Price Dynamics
184
8.2.1
Dynamics of Forward Price of a Bond
184
8.2.2
Dynamics of Forward Price of any Traded
Asset
185
8.3
Option Valuation under most Suitable Numeraire
187
8.3.1
Exchange Option
187
8.3.2
Option on Bond
188
8.3.3
European Call under Stochastic Interest Rate
188
8.4
Relating Change of Numeraire to Change of
Probability
190
8.5
Change of Numeraire for Geometric Brownian
Motion
192
8.6
Change of Numeraire in
LIBOR
Market
Model
194
8.7
Application in Credit Risk Modelling
198
8.8
Exercises
200
8.9
Summary
201
Contents xi
ANNEXES
A
Annex
A: Computations with Brownian Motion
205
A.I Moment Generating Function and Moments
of Brownian Motion
205
A.2 Probability of Brownian Motion Position
208
A.3 Brownian Motion Reflected at the Origin
208
A.4 First Passage of a Barrier
214
A.5 Alternative Brownian Motion Specification
216
В
Annex B: Ordinary Integration
221
B.I Riemann Integral
221
B.2 Riemann-Stieltjes Integral
226
B.3 Other Useful Properties
231
B.4 References
234
С
Annex C: Brownian Motion Variability
235
C.I Quadratic Variation
235
C.2 First Variation
238
D
Annex
D:
Norms
239
D.
1
Distance between Points
239
242
244
244
246
247
247
248
249
250
250
253
299
303
D.2
Norm of a Function
D.3
Norm of a Random Variable
D.4
Norm of a Random Process
D.5
Reference
E
Annex E: Convergence Concepts
E.I
Central Limit Theorem
E.2
Mean-Square Convergence
E.3
Almost Sure Convergence
E.4
Convergence in Probability
E.5
Summary
Answers
•
to Exercises
References
Index |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Wiersema, Ubbo F. |
| author_facet | Wiersema, Ubbo F. |
| author_role | aut |
| author_sort | Wiersema, Ubbo F. |
| author_variant | u f w uf ufw |
| building | Verbundindex |
| bvnumber | BV035139682 |
| callnumber-first | H - Social Science |
| callnumber-label | HG106 |
| callnumber-raw | HG106 |
| callnumber-search | HG106 |
| callnumber-sort | HG 3106 |
| callnumber-subject | HG - Finance |
| classification_rvk | QH 237 SK 820 |
| classification_tum | WIR 651f MAT 902f MAT 606f |
| ctrlnum | (OCoLC)199462405 (DE-599)BVBBV035139682 |
| dewey-full | 332.64/2701519233 |
| dewey-hundreds | 300 - Social sciences |
| dewey-ones | 332 - Financial economics |
| dewey-raw | 332.64/2701519233 |
| dewey-search | 332.64/2701519233 |
| dewey-sort | 3332.64 102701519233 |
| dewey-tens | 330 - Economics |
| discipline | Mathematik Wirtschaftswissenschaften |
| discipline_str_mv | Mathematik Wirtschaftswissenschaften |
| format | Book |
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| id | DE-604.BV035139682 |
| illustrated | Illustrated |
| index_date | 2024-07-02T22:26:40Z |
| indexdate | 2024-07-09T21:23:11Z |
| institution | BVB |
| isbn | 9780470021705 |
| language | English |
| lccn | 2008007641 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016807090 |
| oclc_num | 199462405 |
| open_access_boolean | |
| owner | DE-703 DE-20 DE-91G DE-BY-TUM DE-739 DE-824 |
| owner_facet | DE-703 DE-20 DE-91G DE-BY-TUM DE-739 DE-824 |
| physical | XV, 313 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Wiley |
| record_format | marc |
| series2 | Wiley finance series |
| spelling | Wiersema, Ubbo F. Verfasser aut Brownian motion calculus Ubbo F. Wiersema Chichester Wiley 2008 XV, 313 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Wiley finance series Includes bibliographical references and index Mathematisches Modell Finance Mathematical models Brownian motion processes Brownsche Bewegung (DE-588)4128328-4 gnd rswk-swf Stochastische Analysis (DE-588)4132272-1 gnd rswk-swf Finanzmathematik (DE-588)4017195-4 gnd rswk-swf Brownsche Bewegung (DE-588)4128328-4 s Stochastische Analysis (DE-588)4132272-1 s Finanzmathematik (DE-588)4017195-4 s DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016807090&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wiersema, Ubbo F. Brownian motion calculus Mathematisches Modell Finance Mathematical models Brownian motion processes Brownsche Bewegung (DE-588)4128328-4 gnd Stochastische Analysis (DE-588)4132272-1 gnd Finanzmathematik (DE-588)4017195-4 gnd |
| subject_GND | (DE-588)4128328-4 (DE-588)4132272-1 (DE-588)4017195-4 |
| title | Brownian motion calculus |
| title_auth | Brownian motion calculus |
| title_exact_search | Brownian motion calculus |
| title_exact_search_txtP | Brownian motion calculus |
| title_full | Brownian motion calculus Ubbo F. Wiersema |
| title_fullStr | Brownian motion calculus Ubbo F. Wiersema |
| title_full_unstemmed | Brownian motion calculus Ubbo F. Wiersema |
| title_short | Brownian motion calculus |
| title_sort | brownian motion calculus |
| topic | Mathematisches Modell Finance Mathematical models Brownian motion processes Brownsche Bewegung (DE-588)4128328-4 gnd Stochastische Analysis (DE-588)4132272-1 gnd Finanzmathematik (DE-588)4017195-4 gnd |
| topic_facet | Mathematisches Modell Finance Mathematical models Brownian motion processes Brownsche Bewegung Stochastische Analysis Finanzmathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016807090&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT wiersemaubbof brownianmotioncalculus |